Optimal. Leaf size=800 \[ -\frac {d \cos (c+d x)}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {\sqrt [3]{b} 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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {\sqrt [3]{b} 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}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.15, antiderivative size = 800, normalized size of antiderivative = 1.00, number of steps
used = 51, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3424, 3426,
3378, 3384, 3380, 3383, 3414, 3427} \begin {gather*} -\frac {\text {CosIntegral}(d x) \sin (c) d^2}{2 a^2}-\frac {\cos (c) \text {Si}(d x) d^2}{2 a^2}-\frac {\cos (c+d x) d}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} \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}}-\frac {\sqrt [3]{b} \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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} \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}}-\frac {(-1)^{2/3} \sqrt [3]{b} \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}}+\frac {\sqrt [3]{b} \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}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sin (c+d x)}{3 b x^5 \left (b x^3+a\right )}-\frac {5 \sin (c+d x)}{6 a^2 x^2}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rule 3424
Rule 3426
Rule 3427
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx &=-\frac {\sin (c+d x)}{3 b 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}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b 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}{3 b}+\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}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac {5 \int \frac {\sin (c+d x)}{x^3} \, dx}{3 a^2}-\frac {5 \int \frac {\sin (c+d x)}{x^6} \, dx}{3 a b}-\frac {(5 b) \int \frac {\sin (c+d x)}{a+b x^3} \, dx}{3 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^2} \, dx}{3 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{3 a b}+\frac {(b d) \int \frac {x \cos (c+d x)}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac {d \cos (c+d x)}{12 a b x^4}+\frac {d \cos (c+d x)}{3 a^2 x}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {(5 b) \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}{3 a^2}+\frac {(5 d) \int \frac {\cos (c+d x)}{x^2} \, dx}{6 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^5} \, dx}{3 a b}+\frac {(b 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}{3 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x} \, dx}{3 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{12 a b}\\ &=-\frac {d \cos (c+d x)}{2 a^2 x}+\frac {\sin (c+d x)}{3 a b x^5}+\frac {d^2 \sin (c+d x)}{36 a b x^3}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac {(5 b) \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac {(5 b) \int \frac {\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}+\frac {(5 b) \int \frac {\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac {\left (b^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3}}+\frac {\left (\sqrt [3]{-1} b^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac {\left ((-1)^{2/3} b^{2/3} d\right ) \int \frac {\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{7/3}}-\frac {\left (5 d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx}{6 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^4} \, dx}{12 a b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{36 a b}+\frac {\left (d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{3 a^2}+\frac {\left (d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{3 a^2}\\ &=\frac {d^3 \cos (c+d x)}{72 a b x^2}-\frac {d \cos (c+d x)}{2 a^2 x}+\frac {d^2 \text {Ci}(d x) \sin (c)}{3 a^2}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}+\frac {d^2 \cos (c) \text {Si}(d x)}{3 a^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^3} \, dx}{36 a b}+\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{72 a b}-\frac {\left (5 d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{6 a^2}+\frac {\left (5 b \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}{9 a^{8/3}}-\frac {\left (b^{2/3} 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}{9 a^{7/3}}-\frac {\left (5 b \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}{9 a^{8/3}}+\frac {\left (\sqrt [3]{-1} b^{2/3} 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}{9 a^{7/3}}+\frac {\left (5 b \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}{9 a^{8/3}}-\frac {\left ((-1)^{2/3} b^{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}{9 a^{7/3}}-\frac {\left (5 d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{6 a^2}+\frac {\left (5 b \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}{9 a^{8/3}}+\frac {\left (b^{2/3} 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}{9 a^{7/3}}+\frac {\left (5 b \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}{9 a^{8/3}}+\frac {\left (\sqrt [3]{-1} b^{2/3} 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}{9 a^{7/3}}+\frac {\left (5 b \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}{9 a^{8/3}}+\frac {\left ((-1)^{2/3} b^{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}{9 a^{7/3}}\\ &=-\frac {d \cos (c+d x)}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {\sqrt [3]{b} 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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {d^4 \sin (c+d x)}{72 a b x}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {\sqrt [3]{b} 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}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^4 \int \frac {\sin (c+d x)}{x^2} \, dx}{72 a b}+\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{72 a b}\\ &=-\frac {d \cos (c+d x)}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {\sqrt [3]{b} 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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {\sqrt [3]{b} 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}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^5 \int \frac {\cos (c+d x)}{x} \, dx}{72 a b}+\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{72 a b}-\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{72 a b}\\ &=-\frac {d \cos (c+d x)}{2 a^2 x}+\frac {d^5 \cos (c) \text {Ci}(d x)}{72 a b}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {\sqrt [3]{b} 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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {d^5 \sin (c) \text {Si}(d x)}{72 a b}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {\sqrt [3]{b} 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}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {\left (d^5 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{72 a b}+\frac {\left (d^5 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{72 a b}\\ &=-\frac {d \cos (c+d x)}{2 a^2 x}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {\sqrt [3]{b} 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}}+\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}-\frac {d^2 \text {Ci}(d x) \sin (c)}{2 a^2}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}+\frac {\sin (c+d x)}{3 a b x^5}-\frac {5 \sin (c+d x)}{6 a^2 x^2}-\frac {\sin (c+d x)}{3 b x^5 \left (a+b x^3\right )}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a^2}-\frac {5 \sqrt [3]{-1} b^{2/3} \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 )}{9 a^{8/3}}-\frac {(-1)^{2/3} \sqrt [3]{b} 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}}-\frac {5 b^{2/3} \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 )}{9 a^{8/3}}+\frac {\sqrt [3]{b} 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}}-\frac {5 (-1)^{2/3} b^{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 )}{9 a^{8/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} 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}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 0.60, size = 470, normalized size = 0.59 \begin {gather*} \frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-5 i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-5 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-5 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+5 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {5 i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-5 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-5 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-5 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}+i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}+i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\frac {3 \left (3 a d x \cos (c+d x)+3 b d x^4 \cos (c+d x)+3 d^2 x^2 \left (a+b x^3\right ) \text {Ci}(d x) \sin (c)+3 a \sin (c+d x)+5 b x^3 \sin (c+d x)+3 d^2 x^2 \left (a+b x^3\right ) \cos (c) \text {Si}(d x)\right )}{x^2 \left (a+b x^3\right )}}{18 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.33, size = 388, normalized size = 0.48
method | result | size |
risch | \(-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-i c +\textit {\_R1} -5\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2}}-\frac {i d^{2} \expIntegral \left (1, -i d x \right ) {\mathrm e}^{i c}}{4 a^{2}}-\frac {i d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-i c +\textit {\_R1} +5\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{18 a^{2}}+\frac {i d^{2} \expIntegral \left (1, i d x \right ) {\mathrm e}^{-i c}}{4 a^{2}}+\frac {\left (-b \,x^{4} d^{4}-a \,d^{4} x \right ) \cos \left (d x +c \right )}{2 a^{2} x^{2} \left (d^{3} x^{3} b +a \,d^{3}\right )}-\frac {\left (5 d^{3} x^{3} b +3 a \,d^{3}\right ) \sin \left (d x +c \right )}{6 a^{2} x^{2} \left (d^{3} x^{3} b +a \,d^{3}\right )}\) | \(315\) |
derivativedivides | \(d^{2} \left (-\frac {b \,d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )}{a}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}}{a^{2}}\right )\) | \(388\) |
default | \(d^{2} \left (-\frac {b \,d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{a \,d^{3}-b \,c^{3}+3 b \,c^{2} \left (d x +c \right )-3 b c \left (d x +c \right )^{2}+b \left (d x +c \right )^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}+\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\sinIntegral \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\cosineIntegral \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{-\textit {\_RR1} +c}}{9 a \,d^{3} b}\right )}{a}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\sinIntegral \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\cosineIntegral \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a^{2}}+\frac {-\frac {\sin \left (d x +c \right )}{2 d^{2} x^{2}}-\frac {\cos \left (d x +c \right )}{2 d x}-\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (d x \right ) \sin \left (c \right )}{2}}{a^{2}}\right )\) | \(388\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains complex when optimal does not.
time = 0.44, size = 908, normalized size = 1.14 \begin {gather*} -\frac {{\left ({\left (b^{2} x^{5} + a b x^{2} - \sqrt {3} {\left (i \, b^{2} x^{5} + i \, a b x^{2}\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2} + \sqrt {3} {\left (i \, b^{2} x^{5} + i \, a b x^{2}\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 ({\left (b^{2} x^{5} + a b x^{2} - \sqrt {3} {\left (i \, b^{2} x^{5} + i \, a b x^{2}\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2} + \sqrt {3} {\left (i \, b^{2} x^{5} + i \, a b x^{2}\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 ({\left (b^{2} x^{5} + a b x^{2} - \sqrt {3} {\left (-i \, b^{2} x^{5} - i \, a b x^{2}\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2} + \sqrt {3} {\left (-i \, b^{2} x^{5} - i \, a b x^{2}\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 ({\left (b^{2} x^{5} + a b x^{2} - \sqrt {3} {\left (-i \, b^{2} x^{5} - i \, a b x^{2}\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2} + \sqrt {3} {\left (-i \, b^{2} x^{5} - i \, a b x^{2}\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 )} + 9 \, {\left (-i \, a b d^{3} x^{5} - i \, a^{2} d^{3} x^{2}\right )} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} + 9 \, {\left (i \, a b d^{3} x^{5} + i \, a^{2} d^{3} x^{2}\right )} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - 2 \, {\left ({\left (b^{2} x^{5} + a b x^{2}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2}\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 )} - 2 \, {\left ({\left (b^{2} x^{5} + a b x^{2}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 5 \, {\left (b^{2} x^{5} + a b x^{2}\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 )} + 18 \, {\left (a b d^{2} x^{4} + a^{2} d^{2} x\right )} \cos \left (d x + c\right ) + 6 \, {\left (5 \, a b d x^{3} + 3 \, a^{2} d\right )} \sin \left (d x + c\right )}{36 \, {\left (a^{3} b d x^{5} + a^{4} d x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (b\,x^3+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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