Optimal. Leaf size=800 \[ -\frac {\text {Ci}(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 {Ci}\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 {Ci}\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 {Ci}\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 {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 )}{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 (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}} \]
[Out]
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Rubi [A] time = 1.79, 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 = {3343, 3345, 3297, 3303, 3299, 3302, 3333, 3346} \[ -\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}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3333
Rule 3343
Rule 3345
Rule 3346
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] time = 1.20, size = 470, normalized size = 0.59 \[ \frac {\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-5 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-5 i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+5 i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-5 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]+\text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-5 \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+5 i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-5 i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-5 \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\& \right ]-\frac {3 \left (3 d^2 x^2 \sin (c) \left (a+b x^3\right ) \text {Ci}(d x)+3 d^2 x^2 \cos (c) \left (a+b x^3\right ) \text {Si}(d x)+3 a \sin (c+d x)+3 a d x \cos (c+d x)+3 b d x^4 \cos (c+d x)+5 b x^3 \sin (c+d x)\right )}{x^2 \left (a+b x^3\right )}}{18 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.96, size = 916, normalized size = 1.14 \[ \text {result too large to display} \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 388, normalized size = 0.48 \[ d^{2} \left (\frac {-\frac {\sin \left (d x +c \right )}{2 x^{2} d^{2}}-\frac {\cos \left (d x +c \right )}{2 x d}-\frac {\Si \left (d x \right ) \cos \relax (c )}{2}-\frac {\Ci \left (d x \right ) \sin \relax (c )}{2}}{a^{2}}-\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{3 a^{2}}-\frac {b \,d^{3} \left (\frac {\sin \left (d x +c \right ) \left (\frac {d x +c}{3 a \,d^{3}}-\frac {c}{3 a \,d^{3}}\right )}{\left (d x +c \right )^{3} b -3 c \left (d x +c \right )^{2} b +3 \left (d x +c \right ) b \,c^{2}+a \,d^{3}-b \,c^{3}}+\frac {2 \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {-\Si \left (-d x +\textit {\_R1} -c \right ) \cos \left (\textit {\_R1} \right )+\Ci \left (d x -\textit {\_R1} +c \right ) \sin \left (\textit {\_R1} \right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{9 a \,d^{3} b}-\frac {\munderset {\textit {\_RR1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\Si \left (-d x +\textit {\_RR1} -c \right ) \sin \left (\textit {\_RR1} \right )+\Ci \left (d x -\textit {\_RR1} +c \right ) \cos \left (\textit {\_RR1} \right )}{\textit {\_RR1} -c}}{9 a \,d^{3} b}\right )}{a}\right ) \]
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 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2} x^{3}}\,{d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )}{x^3\,{\left (b\,x^3+a\right )}^2} \,d x \]
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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