Optimal. Leaf size=1161 \[ -\frac {\text {Ci}\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 {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 ) d^2}{54 a^2 b}-\frac {\text {Ci}\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 {Ci}\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 {Ci}\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 {Ci}\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 {Ci}\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 {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 {5 (-1)^{2/3} \text {Ci}\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}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.37, 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 = {3331, 3343, 3345, 3297, 3303, 3299, 3302, 3333, 3346, 3344} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3331
Rule 3333
Rule 3343
Rule 3344
Rule 3345
Rule 3346
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] time = 0.43, size = 675, normalized size = 0.58 \[ \frac {-\frac {i \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-i \text {$\#$1}^2 d^2 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-\text {$\#$1}^2 d^2 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-i \text {$\#$1}^2 d^2 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-6 \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+10 i \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-6 i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-10 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+6 i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+10 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-6 \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+10 i \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]}{b}+\frac {i \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {i \text {$\#$1}^2 d^2 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1}^2 d^2 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-\text {$\#$1}^2 d^2 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+i \text {$\#$1}^2 d^2 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-6 \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-10 i \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+6 i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-10 \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-6 i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+10 \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-6 \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-10 i \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]}{b}-\frac {6 x \cos (d x) \left (d x \cos (c) \left (a+b x^3\right )-\sin (c) \left (8 a+5 b x^3\right )\right )}{\left (a+b x^3\right )^2}+\frac {6 x \sin (d x) \left (d x \sin (c) \left (a+b x^3\right )+\cos (c) \left (8 a+5 b x^3\right )\right )}{\left (a+b x^3\right )^2}}{108 a^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.96, size = 1223, normalized size = 1.05 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.06, size = 392, normalized size = 0.34 \[ d^{8} \left (\frac {\sin \left (d x +c \right ) \left (5 \left (d x +c \right )^{4} b -20 c \left (d x +c \right )^{3} b +30 c^{2} \left (d x +c \right )^{2} b +8 \left (d x +c \right ) a \,d^{3}-20 \left (d x +c \right ) b \,c^{3}-8 a c \,d^{3}+5 b \,c^{4}\right )}{18 a^{2} d^{6} \left (\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) \left (\left (d x +c \right )^{2}-2 \left (d x +c \right ) c +c^{2}\right )}{18 a^{2} d^{6} \left (\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}\right )}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}-10\right ) \left (-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{54 a^{2} d^{6} b}-\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\Si \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\Ci \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1} -c}}{9 a^{2} d^{6} b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________