Optimal. Leaf size=643 \[ -\frac{6 i a^2 f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3 \sqrt{a^2-b^2}}+\frac{6 i a^2 f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3 \sqrt{a^2-b^2}}-\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{6 a^2 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{6 a^2 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d \sqrt{a^2-b^2}}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d \sqrt{a^2-b^2}}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac{6 f^3 \sin (c+d x)}{b d^4}-\frac{(e+f x)^3 \cos (c+d x)}{b d} \]
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Rubi [A] time = 1.17592, antiderivative size = 643, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.393, Rules used = {4515, 3296, 2637, 32, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{6 i a^2 f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^3 \sqrt{a^2-b^2}}+\frac{6 i a^2 f^2 (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^3 \sqrt{a^2-b^2}}-\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{3 a^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^2 \sqrt{a^2-b^2}}+\frac{6 a^2 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{6 a^2 f^3 \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d^4 \sqrt{a^2-b^2}}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 d \sqrt{a^2-b^2}}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b^2 d \sqrt{a^2-b^2}}-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac{6 f^3 \sin (c+d x)}{b d^4}-\frac{(e+f x)^3 \cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 4515
Rule 3296
Rule 2637
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 ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \sin (c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{a \int (e+f x)^3 \, dx}{b^2}+\frac{a^2 \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b^2}+\frac{(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{b d}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}-\frac{(e+f x)^3 \cos (c+d x)}{b d}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{\left (2 a^2\right ) \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^2}-\frac{\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b d^2}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac{\left (2 i a^2\right ) \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}{b \sqrt{a^2-b^2}}+\frac{\left (2 i a^2\right ) \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}{b \sqrt{a^2-b^2}}-\frac{\left (6 f^3\right ) \int \cos (c+d x) \, dx}{b d^3}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{\left (3 i a^2 f\right ) \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^2 \sqrt{a^2-b^2} d}-\frac{\left (3 i a^2 f\right ) \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^2 \sqrt{a^2-b^2} d}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}-\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}+\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{\left (6 a^2 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^2 \sqrt{a^2-b^2} d^2}-\frac{\left (6 a^2 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^2 \sqrt{a^2-b^2} d^2}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}-\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}+\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}-\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}+\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{\left (6 i a^2 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^2 \sqrt{a^2-b^2} d^3}-\frac{\left (6 i a^2 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^2 \sqrt{a^2-b^2} d^3}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}-\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}+\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}-\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}+\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac{\left (6 a^2 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^2 \sqrt{a^2-b^2} d^4}-\frac{\left (6 a^2 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^2 \sqrt{a^2-b^2} d^4}\\ &=-\frac{a (e+f x)^4}{4 b^2 f}+\frac{6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac{(e+f x)^3 \cos (c+d x)}{b d}-\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}+\frac{i a^2 (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d}-\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}+\frac{3 a^2 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^2 \sqrt{a^2-b^2} d^2}-\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}+\frac{6 i a^2 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^2 \sqrt{a^2-b^2} d^3}+\frac{6 a^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d^4}-\frac{6 a^2 f^3 \text{Li}_4\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b^2 \sqrt{a^2-b^2} d^4}-\frac{6 f^3 \sin (c+d x)}{b d^4}+\frac{3 f (e+f x)^2 \sin (c+d x)}{b d^2}\\ \end{align*}
Mathematica [A] time = 6.67317, size = 1020, normalized size = 1.59 \[ \frac{-a x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right ) d^4-4 b (e+f x) \left (d^2 (e+f x)^2-6 f^2\right ) \cos (c+d x) d+\frac{4 a^2 \left (2 \sqrt{b^2-a^2} e^3 \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right ) d^3+\sqrt{a^2-b^2} f^3 x^3 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3+3 \sqrt{a^2-b^2} e^2 f x \log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^3-\sqrt{a^2-b^2} f^3 x^3 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e f^2 x^2 \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 \sqrt{a^2-b^2} e^2 f x \log \left (\frac{e^{i (c+d x)} b}{i a+\sqrt{b^2-a^2}}+1\right ) d^3-3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d^2+3 i \sqrt{a^2-b^2} f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d^2+6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d+6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right ) d-6 \sqrt{a^2-b^2} e f^2 \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d-6 \sqrt{a^2-b^2} f^3 x \text{PolyLog}\left (3,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right ) d+6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-6 i \sqrt{a^2-b^2} f^3 \text{PolyLog}\left (4,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )\right )}{\sqrt{-\left (a^2-b^2\right )^2}}+12 b f \left (d^2 (e+f x)^2-2 f^2\right ) \sin (c+d x)}{4 b^2 d^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.894, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{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 = 4.40277, size = 6095, normalized size = 9.48 \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 )^{2}}{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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