Optimal. Leaf size=691 \[ -\frac {(-1)^{2/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{4/3} b^{2/3}}-\frac {\sin \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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \sin \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 )}{9 a^{4/3} b^{2/3}}+\frac {(-1)^{2/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{4/3} 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 )}{9 a^{4/3} 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 )}{9 a^{4/3} b^{2/3}}-\frac {d \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a b}-\frac {d \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 )}{9 a b}-\frac {d \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 )}{9 a b}-\frac {d \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a b}+\frac {d \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 )}{9 a b}+\frac {d \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 )}{9 a b}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\sin (c+d x)}{3 a b x} \]
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Rubi [A] time = 1.30, antiderivative size = 691, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3343, 3345, 3297, 3303, 3299, 3302, 3346} \[ -\frac {(-1)^{2/3} \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{4/3} b^{2/3}}-\frac {\sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{4/3} b^{2/3}}+\frac {(-1)^{2/3} \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{4/3} 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 )}{9 a^{4/3} 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 )}{9 a^{4/3} b^{2/3}}-\frac {d \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a b}-\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a b}-\frac {d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a b}-\frac {d \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a b}+\frac {d \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 )}{9 a b}+\frac {d \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 )}{9 a b}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\sin (c+d x)}{3 a b x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3343
Rule 3345
Rule 3346
Rubi steps
\begin {align*} \int \frac {x \sin (c+d x)}{\left (a+b x^3\right )^2} \, dx &=-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {\int \frac {\sin (c+d x)}{x^2 \left (a+b x^3\right )} \, dx}{3 b}+\frac {d \int \frac {\cos (c+d x)}{x \left (a+b x^3\right )} \, dx}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {\int \left (\frac {\sin (c+d x)}{a x^2}-\frac {b x \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{3 b}+\frac {d \int \left (\frac {\cos (c+d x)}{a x}-\frac {b x^2 \cos (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\int \frac {x \sin (c+d x)}{a+b x^3} \, dx}{3 a}-\frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{3 a b}-\frac {d \int \frac {x^2 \cos (c+d x)}{a+b x^3} \, dx}{3 a}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{3 a b}\\ &=\frac {\sin (c+d x)}{3 a b x}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}+\frac {\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}{3 a}-\frac {d \int \left (\frac {\cos (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\cos (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{3 a}-\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{3 a b}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{3 a b}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{3 a b}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{3 a b}+\frac {\sin (c+d x)}{3 a b x}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {d \sin (c) \text {Si}(d x)}{3 a b}-\frac {\int \frac {\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}+\frac {\sqrt [3]{-1} \int \frac {\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}-\frac {(-1)^{2/3} \int \frac {\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{4/3} \sqrt [3]{b}}-\frac {d \int \frac {\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {d \int \frac {\cos (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{3 a b}+\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{3 a b}\\ &=\frac {\sin (c+d x)}{3 a b x}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}-\frac {\cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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^{4/3} \sqrt [3]{b}}-\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}{9 a b^{2/3}}-\frac {\left (\sqrt [3]{-1} \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^{4/3} \sqrt [3]{b}}-\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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {\left ((-1)^{2/3} \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^{4/3} \sqrt [3]{b}}-\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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {\sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\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^{4/3} \sqrt [3]{b}}+\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}{9 a b^{2/3}}+\frac {\left (\sqrt [3]{-1} \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^{4/3} \sqrt [3]{b}}-\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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}-\frac {\left ((-1)^{2/3} \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^{4/3} \sqrt [3]{b}}+\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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a b^{2/3}}\\ &=-\frac {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 b}-\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 b}-\frac {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 b}-\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 )}{9 a^{4/3} 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 )}{9 a^{4/3} b^{2/3}}+\frac {\sqrt [3]{-1} \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^{4/3} b^{2/3}}+\frac {\sin (c+d x)}{3 a b x}-\frac {\sin (c+d x)}{3 b x \left (a+b x^3\right )}+\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 )}{9 a^{4/3} b^{2/3}}-\frac {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 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 )}{9 a^{4/3} b^{2/3}}+\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 b}+\frac {\sqrt [3]{-1} \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^{4/3} b^{2/3}}+\frac {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 b}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 408, normalized size = 0.59 \[ -\frac {\left (a+b x^3\right ) \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-\sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]+\left (a+b x^3\right ) \text {RootSum}\left [\text {$\#$1}^3 b+a\& ,\frac {-\sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i \text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+i \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))+\text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Ci}(d (x-\text {$\#$1}))-i \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\text {$\#$1} d \sin (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))-\cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))+i \text {$\#$1} d \cos (\text {$\#$1} d+c) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}}\& \right ]-6 b x^2 \sin (c+d x)}{18 a b \left (a+b x^3\right )} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.71, size = 661, normalized size = 0.96 \[ \frac {12 \, a b d^{2} x^{2} \sin \left (d x + c\right ) - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (-i \, b^{2} x^{3} - i \, a b - \sqrt {3} {\left (b^{2} x^{3} + a b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (i \, b^{2} x^{3} + i \, a b + \sqrt {3} {\left (b^{2} x^{3} + a b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (-i \, b^{2} x^{3} - i \, a b + \sqrt {3} {\left (b^{2} x^{3} + a b\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (i \, b^{2} x^{3} + i \, a b - \sqrt {3} {\left (b^{2} x^{3} + a b\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (-2 i \, b^{2} x^{3} - 2 i \, a b\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - {\left (2 \, a b d^{3} x^{3} + 2 \, a^{2} d^{3} - {\left (2 i \, b^{2} x^{3} + 2 i \, a b\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}}\right )} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )}}{36 \, {\left (a^{2} b^{2} d^{2} x^{3} + a^{3} b d^{2}\right )}} \]
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 \[ \int \frac {x \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 508, normalized size = 0.74 \[ \frac {\frac {\sin \left (d x +c \right ) \left (\frac {d^{3} \left (d x +c \right )^{2}}{3 a}-\frac {c \,d^{3} \left (d x +c \right )}{3 a}\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 {d^{3} \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 {\left (\textit {\_R1} +c \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}}\right )}{9 a b}-\frac {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 {\textit {\_RR1} \left (\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 )\right )}{\textit {\_RR1} -c}\right )}{9 a b}-c \,d^{6} \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 )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \]
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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