Optimal. Leaf size=1163 \[ \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^{7/3} b^{2/3}}+\frac {(-1)^{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 ) d^2}{54 a^{7/3} b^{2/3}}-\frac {\sqrt [3]{-1} \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^{7/3} b^{2/3}}-\frac {(-1)^{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 ) d^2}{54 a^{7/3} b^{2/3}}+\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^{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 {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) d^2}{54 a^{7/3} b^{2/3}}-\frac {\cos (c+d x) d}{18 b^2 x^5 \left (b x^3+a\right )}-\frac {\cos (c+d x) d}{18 a^2 b x^2}+\frac {\cos (c+d x) d}{18 a b^2 x^5}+\frac {4 \sqrt [3]{-1} \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}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 \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}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} \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}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 \sqrt [3]{-1} \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}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 \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}{27 a^{8/3} \sqrt [3]{b}}+\frac {4 (-1)^{2/3} \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}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\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 )}{3 a^3}-\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 )}{3 a^3}-\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 )}{3 a^3}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (b x^3+a\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (b x^3+a\right )^2}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\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 )}{3 a^3}-\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 )}{3 a^3}-\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 )}{3 a^3} \]
[Out]
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Rubi [A] time = 3.89, antiderivative size = 1163, normalized size of antiderivative = 1.00, number of steps used = 110, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346, 3334, 3344} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3334
Rule 3343
Rule 3344
Rule 3345
Rule 3346
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{x \left (a+b x^3\right )^3} \, dx &=-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}-\frac {\int \frac {\sin (c+d x)}{x^4 \left (a+b x^3\right )^2} \, dx}{2 b}+\frac {d \int \frac {\cos (c+d x)}{x^3 \left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \frac {\sin (c+d x)}{x^7 \left (a+b x^3\right )} \, dx}{b^2}-\frac {d \int \frac {\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{6 b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^6 \left (a+b x^3\right )} \, dx}{18 b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \left (\frac {\sin (c+d x)}{a x^7}-\frac {b \sin (c+d x)}{a^2 x^4}+\frac {b^2 \sin (c+d x)}{a^3 x}-\frac {b^3 x^2 \sin (c+d x)}{a^3 \left (a+b x^3\right )}\right ) \, dx}{b^2}-\frac {d \int \left (\frac {\cos (c+d x)}{a x^6}-\frac {b \cos (c+d x)}{a^2 x^3}+\frac {b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{6 b^2}-\frac {(5 d) \int \left (\frac {\cos (c+d x)}{a x^6}-\frac {b \cos (c+d x)}{a^2 x^3}+\frac {b^2 \cos (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 b^2}-\frac {d^2 \int \left (\frac {\sin (c+d x)}{a x^5}-\frac {b \sin (c+d x)}{a^2 x^2}+\frac {b^2 x \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^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\int \frac {\sin (c+d x)}{x} \, dx}{a^3}+\frac {\int \frac {\sin (c+d x)}{x^7} \, dx}{a b^2}-\frac {\int \frac {\sin (c+d x)}{x^4} \, dx}{a^2 b}-\frac {b \int \frac {x^2 \sin (c+d x)}{a+b x^3} \, dx}{a^3}-\frac {d \int \frac {\cos (c+d x)}{a+b x^3} \, dx}{6 a^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d \int \frac {\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac {(5 d) \int \frac {\cos (c+d x)}{x^6} \, dx}{18 a b^2}+\frac {d \int \frac {\cos (c+d x)}{x^3} \, dx}{6 a^2 b}+\frac {(5 d) \int \frac {\cos (c+d x)}{x^3} \, dx}{18 a^2 b}-\frac {d^2 \int \frac {x \sin (c+d x)}{a+b x^3} \, dx}{18 a^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{18 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{18 a^2 b}\\ &=\frac {4 d \cos (c+d x)}{45 a b^2 x^5}-\frac {2 d \cos (c+d x)}{9 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {d^2 \sin (c+d x)}{72 a b^2 x^4}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {d^2 \sin (c+d x)}{18 a^2 b x}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}-\frac {b \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}{a^3}-\frac {d \int \left (-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{6 a^2}-\frac {(5 d) \int \left (-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac {d \int \frac {\cos (c+d x)}{x^6} \, dx}{6 a b^2}-\frac {d \int \frac {\cos (c+d x)}{x^3} \, dx}{3 a^2 b}-\frac {d^2 \int \left (-\frac {\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \sin (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} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{18 a b^2}-\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{12 a^2 b}-\frac {\left (5 d^2\right ) \int \frac {\sin (c+d x)}{x^2} \, dx}{36 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{72 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x} \, dx}{18 a^2 b}+\frac {\cos (c) \int \frac {\sin (d x)}{x} \, dx}{a^3}+\frac {\sin (c) \int \frac {\cos (d x)}{x} \, dx}{a^3}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}+\frac {d^3 \cos (c+d x)}{216 a b^2 x^3}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\sin (c+d x)}{6 a b^2 x^6}-\frac {d^2 \sin (c+d x)}{120 a b^2 x^4}+\frac {\sin (c+d x)}{3 a^2 b x^3}+\frac {d^2 \sin (c+d x)}{6 a^2 b x}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac {\sqrt [3]{b} \int \frac {\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{18 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}+\frac {(5 d) \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{8/3}}-\frac {d^2 \int \frac {\sin (c+d x)}{x^5} \, dx}{30 a b^2}+\frac {d^2 \int \frac {\sin (c+d x)}{x^2} \, dx}{6 a^2 b}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{-1} d^2\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac {\left ((-1)^{2/3} d^2\right ) \int \frac {\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{120 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{72 a b^2}-\frac {d^3 \int \frac {\cos (c+d x)}{x} \, dx}{12 a^2 b}-\frac {\left (5 d^3\right ) \int \frac {\cos (c+d x)}{x} \, dx}{36 a^2 b}+\frac {d^4 \int \frac {\sin (c+d x)}{x^3} \, dx}{216 a b^2}+\frac {\left (d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{18 a^2 b}-\frac {\left (d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{18 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d^3 \cos (c+d x)}{360 a b^2 x^3}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {d^3 \cos (c) \text {Ci}(d x)}{18 a^2 b}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {d^4 \sin (c+d x)}{432 a b^2 x^2}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {d^3 \sin (c) \text {Si}(d x)}{18 a^2 b}-\frac {d^3 \int \frac {\cos (c+d x)}{x^4} \, dx}{120 a b^2}+\frac {d^3 \int \frac {\cos (c+d x)}{x} \, dx}{6 a^2 b}-\frac {d^4 \int \frac {\sin (c+d x)}{x^3} \, dx}{360 a b^2}-\frac {d^4 \int \frac {\sin (c+d x)}{x^3} \, dx}{216 a b^2}+\frac {d^5 \int \frac {\cos (c+d x)}{x^2} \, dx}{432 a b^2}-\frac {\left (d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{12 a^2 b}-\frac {\left (5 d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{36 a^2 b}-\frac {\left (\sqrt [3]{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}{3 a^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}{18 a^{8/3}}+\frac {\left (5 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}{54 a^{8/3}}+\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^{7/3} \sqrt [3]{b}}+\frac {\left (\sqrt [3]{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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac {\left (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}{18 a^{8/3}}+\frac {\left (5 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}{54 a^{8/3}}+\frac {\left (\sqrt [3]{-1} 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]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac {\left (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}{18 a^{8/3}}+\frac {\left (5 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}{54 a^{8/3}}+\frac {\left ((-1)^{2/3} 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 )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}+\frac {\left (d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{12 a^2 b}+\frac {\left (5 d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{36 a^2 b}-\frac {\left (\sqrt [3]{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}{3 a^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}{18 a^{8/3}}-\frac {\left (5 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}{54 a^{8/3}}+\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^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}+\frac {\left (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}{18 a^{8/3}}+\frac {\left (5 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}{54 a^{8/3}}-\frac {\left (\sqrt [3]{-1} 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]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^3}-\frac {\left (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}{18 a^{8/3}}-\frac {\left (5 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}{54 a^{8/3}}+\frac {\left ((-1)^{2/3} 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 )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{7/3} \sqrt [3]{b}}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d^5 \cos (c+d x)}{432 a b^2 x}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}-\frac {d^3 \cos (c) \text {Ci}(d x)}{6 a^2 b}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}-\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 )}{3 a^3}+\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}+\frac {d^4 \sin (c+d x)}{720 a b^2 x^2}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\frac {d^3 \sin (c) \text {Si}(d x)}{6 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 )}{3 a^3}-\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^4 \int \frac {\sin (c+d x)}{x^3} \, dx}{360 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x^2} \, dx}{720 a b^2}-\frac {d^5 \int \frac {\cos (c+d x)}{x^2} \, dx}{432 a b^2}-\frac {d^6 \int \frac {\sin (c+d x)}{x} \, dx}{432 a b^2}+\frac {\left (d^3 \cos (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{6 a^2 b}-\frac {\left (d^3 \sin (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{6 a^2 b}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}+\frac {d^5 \cos (c+d x)}{720 a b^2 x}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}-\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 )}{3 a^3}+\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\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 )}{3 a^3}-\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {d^5 \int \frac {\cos (c+d x)}{x^2} \, dx}{720 a b^2}+\frac {d^6 \int \frac {\sin (c+d x)}{x} \, dx}{720 a b^2}+\frac {d^6 \int \frac {\sin (c+d x)}{x} \, dx}{432 a b^2}-\frac {\left (d^6 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{432 a b^2}-\frac {\left (d^6 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{432 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {d^6 \text {Ci}(d x) \sin (c)}{432 a b^2}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}-\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 )}{3 a^3}+\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {d^6 \cos (c) \text {Si}(d x)}{432 a b^2}+\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 )}{3 a^3}-\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {d^6 \int \frac {\sin (c+d x)}{x} \, dx}{720 a b^2}+\frac {\left (d^6 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{720 a b^2}+\frac {\left (d^6 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{432 a b^2}+\frac {\left (d^6 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{720 a b^2}+\frac {\left (d^6 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{432 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}+\frac {d^6 \text {Ci}(d x) \sin (c)}{720 a b^2}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}-\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 )}{3 a^3}+\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\frac {d^6 \cos (c) \text {Si}(d x)}{720 a b^2}+\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 )}{3 a^3}-\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\left (d^6 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx}{720 a b^2}-\frac {\left (d^6 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx}{720 a b^2}\\ &=\frac {d \cos (c+d x)}{18 a b^2 x^5}-\frac {d \cos (c+d x)}{18 a^2 b x^2}-\frac {d \cos (c+d x)}{18 b^2 x^5 \left (a+b x^3\right )}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}-\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 )}{3 a^3}+\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}-\frac {\sin (c+d x)}{6 a b^2 x^6}+\frac {\sin (c+d x)}{3 a^2 b x^3}-\frac {\sin (c+d x)}{6 b x^3 \left (a+b x^3\right )^2}+\frac {\sin (c+d x)}{6 b^2 x^6 \left (a+b x^3\right )}+\frac {\cos (c) \text {Si}(d x)}{a^3}+\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 )}{3 a^3}-\frac {(-1)^{2/3} 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^{7/3} b^{2/3}}+\frac {4 \sqrt [3]{-1} 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}+\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^{7/3} b^{2/3}}+\frac {4 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 )}{27 a^{8/3} \sqrt [3]{b}}-\frac {\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 )}{3 a^3}-\frac {\sqrt [3]{-1} 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^{7/3} b^{2/3}}+\frac {4 (-1)^{2/3} 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 )}{27 a^{8/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [C] time = 1.15, size = 2109, normalized size = 1.81 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 1.14, size = 1117, normalized size = 0.96 \[ \text {result too large to display} \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 363, normalized size = 0.31 \[ \frac {\sin \left (d x +c \right ) d^{3} \left (2 \left (d x +c \right )^{3} b -6 c \left (d x +c \right )^{2} b +6 \left (d x +c \right ) b \,c^{2}+3 a \,d^{3}-2 b \,c^{3}\right )}{6 a^{2} \left (\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}\right )^{2}}-\frac {\cos \left (d x +c \right ) d^{4} x}{18 \left (\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}\right ) a^{2}}+\frac {\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )}{a^{3}}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\left (a \,d^{3}+18 \textit {\_R1} b -18 c b \right ) \left (-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )\right )}{\textit {\_R1} -c}}{54 b \,a^{3}}-\frac {4 d^{3} \left (\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\Si \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\Ci \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1}^{2}-2 \textit {\_RR1} c +c^{2}}\right )}{27 a^{2} b} \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3} x}\,{d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )}{x\,{\left (b\,x^3+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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