Optimal. Leaf size=544 \[ \frac{6 i a f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \sqrt{a^2-b^2}}-\frac{6 i a f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \sqrt{a^2-b^2}}+\frac{3 a f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{3 a f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{6 a f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \sqrt{a^2-b^2}}+\frac{6 a f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \sqrt{a^2-b^2}}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d \sqrt{a^2-b^2}}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d \sqrt{a^2-b^2}}+\frac{(e+f x)^4}{4 b f} \]
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Rubi [A] time = 0.967603, antiderivative size = 544, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {4515, 32, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{6 i a f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \sqrt{a^2-b^2}}-\frac{6 i a f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \sqrt{a^2-b^2}}+\frac{3 a f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{3 a f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{6 a f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^4 \sqrt{a^2-b^2}}+\frac{6 a f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^4 \sqrt{a^2-b^2}}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d \sqrt{a^2-b^2}}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d \sqrt{a^2-b^2}}+\frac{(e+f x)^4}{4 b f} \]
Antiderivative was successfully verified.
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Rule 4515
Rule 32
Rule 3323
Rule 2264
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \, dx}{b}-\frac{a \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b}\\ &=\frac{(e+f x)^4}{4 b f}-\frac{(2 a) \int \frac{e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\sqrt{a^2-b^2}}-\frac{(2 i a) \int \frac{e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\sqrt{a^2-b^2}}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{(3 i a f) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d}+\frac{(3 i a f) \int (e+f x)^2 \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}+\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{\left (6 a f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^2}+\frac{\left (6 a f^2\right ) \int (e+f x) \text{Li}_2\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^2}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}+\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{\left (6 i a f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^3}+\frac{\left (6 i a f^3\right ) \int \text{Li}_3\left (\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^3}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}+\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{\left (6 a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \sqrt{a^2-b^2} d^4}+\frac{\left (6 a f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \sqrt{a^2-b^2} d^4}\\ &=\frac{(e+f x)^4}{4 b f}+\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}-\frac{i a (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d}+\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{3 a f (e+f x)^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{6 i a f^2 (e+f x) \text{Li}_3\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{6 a f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^4}+\frac{6 a f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^4}\\ \end{align*}
Mathematica [A] time = 3.29527, size = 956, normalized size = 1.76 \[ \frac{x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right )}{4 b}-\frac{a \left (2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d^2+6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d+6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d-6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d-6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d+6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )\right )}{b \sqrt{-\left (a^2-b^2\right )^2} d^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.843, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\sin \left ( dx+c \right ) }{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.31947, size = 5493, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \sin \left (d x + c\right )}{b \sin \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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