Optimal. Leaf size=1161 \[ \frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.08, antiderivative size = 1161, normalized size of antiderivative = 1.00, number of steps
used = 99, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {3412, 3424,
3426, 3378, 3384, 3380, 3383, 3414, 3427, 3425} \begin {gather*} -\frac {\text {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^2 b}-\frac {\text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^2 b}-\frac {\text {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^2 b}+\frac {\cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d^2}{54 a^2 b}-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^2 b}-\frac {\cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^2 b}-\frac {\cos (c+d x) d}{18 a^2 b x}-\frac {\cos (c+d x) d}{18 b^2 x^4 \left (b x^3+a\right )}+\frac {\cos (c+d x) d}{18 a b^2 x^4}+\frac {(-1)^{2/3} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{9 a^{7/3} b^{2/3}}+\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3} b^{2/3}}+\frac {(-1)^{2/3} \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) d}{9 a^{7/3} b^{2/3}}-\frac {\sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3} b^{2/3}}+\frac {\sqrt [3]{-1} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d}{9 a^{7/3} b^{2/3}}+\frac {5 \text {CosIntegral}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \sqrt [3]{-1} \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 (-1)^{2/3} \text {CosIntegral}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (b x^3+a\right )}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (b x^3+a\right )^2}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3412
Rule 3414
Rule 3424
Rule 3425
Rule 3426
Rule 3427
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^2 \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \frac {\sin (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d \int \frac {\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \left (\frac {\sin (c+d x)}{a x^6}-\frac {b \sin (c+d x)}{a^2 x^3}+\frac {b^2 \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^5}-\frac {b \cos (c+d x)}{a^2 x^2}+\frac {b^2 x \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {(2 d) \int \left (\frac {\cos (c+d x)}{a x^5}-\frac {b \cos (c+d x)}{a^2 x^2}+\frac {b^2 x \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^4}-\frac {b \sin (c+d x)}{a^2 x}+\frac {b^2 x^2 \sin (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{9 a^2}+\frac {5 \int \frac {\sin (c+d x)}{x^6} \, dx}{9 a b^2}-\frac {5 \int \frac {\sin (c+d x)}{x^3} \, dx}{9 a^2 b}-\frac {d \int \frac {x \cos (c+d x)}{a+b x^3} \, dx}{9 a^2}-\frac {(2 d) \int \frac {x \cos (c+d x)}{a+b x^3} \, dx}{9 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {(2 d) \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{9 a^2 b}+\frac {(2 d) \int \frac {\cos (c+d x)}{x^2} \, dx}{9 a^2 b}-\frac {d^2 \int \frac {x^2 \sin (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{18 a^2 b}\\ &=\frac {d \cos (c+d x)}{12 a b^2 x^4}-\frac {d \cos (c+d x)}{3 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {d^2 \sin (c+d x)}{54 a b^2 x^3}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \int \left (-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {d \int \left (-\frac {\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}-\frac {(2 d) \int \left (-\frac {\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{9 a b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^2} \, dx}{18 a^2 b}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sin (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\sin (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{36 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{18 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx}{9 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}+\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}+\frac {d^3 \cos (c+d x)}{108 a b^2 x^2}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {d^2 \text {Ci}(d x) \sin (c)}{18 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}-\frac {d^2 \sin (c+d x)}{108 a b^2 x^3}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {d^2 \cos (c) \text {Si}(d x)}{18 a^2 b}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {5 \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {(2 d) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{-1} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left ((-1)^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 (-1)^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{36 a b^2}+\frac {\left (5 d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx}{18 a^2 b}-\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{108 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{54 a b^2}+\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{108 a b^2}-\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}-\frac {\left (2 d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{9 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d^3 \cos (c+d x)}{216 a b^2 x^2}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {5 d^2 \text {Ci}(d x) \sin (c)}{18 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {d^4 \sin (c+d x)}{108 a b^2 x}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}-\frac {5 d^2 \cos (c) \text {Si}(d x)}{18 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{108 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{216 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{108 a b^2}+\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{108 a b^2}+\frac {\left (5 d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {\left (d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (5 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (\sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {\left ((-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}+\frac {\left (2 (-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}+\frac {\left (5 d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (5 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left (\sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 \sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}-\frac {\left (5 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{8/3}}-\frac {\left ((-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (2 (-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{7/3} \sqrt [3]{b}}-\frac {\left (d^2 \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^2 b^{2/3}}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}+\frac {d^4 \sin (c+d x)}{216 a b^2 x}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{216 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{216 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{108 a b^2}+\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}-\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {d^5 \cos (c) \text {Ci}(d x)}{108 a b^2}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}-\frac {d^5 \sin (c) \text {Si}(d x)}{108 a b^2}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{108 a b^2}+\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{216 a b^2}+\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{108 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}-\frac {d^5 \cos (c) \text {Ci}(d x)}{216 a b^2}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {d^5 \sin (c) \text {Si}(d x)}{216 a b^2}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{216 a b^2}-\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{216 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^4}-\frac {d \cos (c+d x)}{18 a^2 b x}-\frac {d \cos (c+d x)}{18 b^2 x^4 \left (a+b x^3\right )}+\frac {(-1)^{2/3} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {5 \sqrt [3]{-1} \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}+\frac {5 (-1)^{2/3} \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^2 b}-\frac {\sin (c+d x)}{9 a b^2 x^5}+\frac {5 \sin (c+d x)}{18 a^2 b x^2}-\frac {\sin (c+d x)}{6 b x^2 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{9 b^2 x^5 \left (a+b x^3\right )}+\frac {5 \sqrt [3]{-1} \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^2 \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^2 b}+\frac {(-1)^{2/3} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}+\frac {5 (-1)^{2/3} \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^2 \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^2 b}+\frac {\sqrt [3]{-1} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{7/3} b^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 0.32, size = 675, normalized size = 0.58 \begin {gather*} \frac {-\frac {i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-10 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+10 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+10 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+10 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-6 i d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-6 d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-6 d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+6 i d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2-i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2-i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]}{b}+\frac {i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-10 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-10 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-10 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+10 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+6 i d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-6 d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-6 d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-6 i d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2+i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2+i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]}{b}-\frac {6 x \cos (d x) \left (d x \left (a+b x^3\right ) \cos (c)-\left (8 a+5 b x^3\right ) \sin (c)\right )}{\left (a+b x^3\right )^2}+\frac {6 x \left (\left (8 a+5 b x^3\right ) \cos (c)+d x \left (a+b x^3\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^3\right )^2}}{108 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 392, normalized size = 0.34
method | result | size |
risch | \(-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}+6 i c -6 \textit {\_R1} +10\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}-6 i c +6 \textit {\_R1} +10\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {d^{2} \left (-b \,x^{5} d^{5}-a \,d^{5} x^{2}\right ) \cos \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}+\frac {d^{2} \left (5 b \,x^{4} d^{4}+8 a \,d^{4} x \right ) \sin \left (d x +c \right )}{18 a^{2} \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\) | \(340\) |
derivativedivides | \(d^{8} \left (-\frac {\sin \left (d x +c \right ) \left (8 a c \,d^{3}-8 a \,d^{3} \left (d x +c \right )-5 b \,c^{4}+20 b \,c^{3} \left (d x +c \right )-30 b \,c^{2} \left (d x +c \right )^{2}+20 b c \left (d x +c \right )^{3}-5 b \left (d x +c \right )^{4}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (c^{2}-2 \left (d x +c \right ) c +\left (d x +c \right )^{2}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-10\right ) \left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{54 a^{2} d^{6} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a^{2} d^{6} b}\right )\) | \(392\) |
default | \(d^{8} \left (-\frac {\sin \left (d x +c \right ) \left (8 a c \,d^{3}-8 a \,d^{3} \left (d x +c \right )-5 b \,c^{4}+20 b \,c^{3} \left (d x +c \right )-30 b \,c^{2} \left (d x +c \right )^{2}+20 b c \left (d x +c \right )^{3}-5 b \left (d x +c \right )^{4}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (c^{2}-2 \left (d x +c \right ) c +\left (d x +c \right )^{2}\right )}{18 a^{2} d^{6} \left (a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-10\right ) \left (-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{54 a^{2} d^{6} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a^{2} d^{6} b}\right )\) | \(392\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.44, size = 1223, normalized size = 1.05 \begin {gather*} \frac {{\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + 3 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b - \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 10 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} + {\left (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} - 6 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 10 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 6 \, {\left (a b^{2} d^{2} x^{5} + a^{2} b d^{2} x^{2}\right )} \cos \left (d x + c\right ) + 6 \, {\left (5 \, a b^{2} d x^{4} + 8 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{108 \, {\left (a^{3} b^{3} d x^{6} + 2 \, a^{4} b^{2} d x^{3} + a^{5} b d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________