Optimal. Leaf size=156 \[ -\frac{b x \left (-6 a^2 d^2+6 a b c d+b^2 \left (-\left (2 c^2+d^2\right )\right )\right )}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{2 (b c-a d)^3 \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{d^3 f \sqrt{c^2-d^2}} \]
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Rubi [A] time = 0.37833, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2793, 3023, 2735, 2660, 618, 204} \[ -\frac{b x \left (-6 a^2 d^2+6 a b c d+b^2 \left (-\left (2 c^2+d^2\right )\right )\right )}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{2 (b c-a d)^3 \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{d^3 f \sqrt{c^2-d^2}} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{(a+b \sin (e+f x))^3}{c+d \sin (e+f x)} \, dx &=-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}+\frac{\int \frac{b^3 c+2 a^3 d-b \left (a b c-6 a^2 d-b^2 d\right ) \sin (e+f x)-b^2 (2 b c-5 a d) \sin ^2(e+f x)}{c+d \sin (e+f x)} \, dx}{2 d}\\ &=\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}+\frac{\int \frac{d \left (b^3 c+2 a^3 d\right )-b \left (6 a b c d-6 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{c+d \sin (e+f x)} \, dx}{2 d^2}\\ &=-\frac{b \left (6 a b c d-6 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) x}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{(b c-a d)^3 \int \frac{1}{c+d \sin (e+f x)} \, dx}{d^3}\\ &=-\frac{b \left (6 a b c d-6 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) x}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}-\frac{\left (2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{d^3 f}\\ &=-\frac{b \left (6 a b c d-6 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) x}{2 d^3}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}+\frac{\left (4 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac{1}{2} (e+f x)\right )\right )}{d^3 f}\\ &=-\frac{b \left (6 a b c d-6 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) x}{2 d^3}-\frac{2 (b c-a d)^3 \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{d^3 \sqrt{c^2-d^2} f}+\frac{b^2 (2 b c-5 a d) \cos (e+f x)}{2 d^2 f}-\frac{b^2 \cos (e+f x) (a+b \sin (e+f x))}{2 d f}\\ \end{align*}
Mathematica [A] time = 0.360915, size = 137, normalized size = 0.88 \[ \frac{2 b (e+f x) \left (6 a^2 d^2-6 a b c d+b^2 \left (2 c^2+d^2\right )\right )+4 b^2 d (b c-3 a d) \cos (e+f x)-\frac{8 (b c-a d)^3 \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{\sqrt{c^2-d^2}}+b^3 \left (-d^2\right ) \sin (2 (e+f x))}{4 d^3 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 506, normalized size = 3.2 \begin{align*} 2\,{\frac{{a}^{3}}{f\sqrt{{c}^{2}-{d}^{2}}}\arctan \left ( 1/2\,{\frac{2\,c\tan \left ( 1/2\,fx+e/2 \right ) +2\,d}{\sqrt{{c}^{2}-{d}^{2}}}} \right ) }-6\,{\frac{{a}^{2}bc}{df\sqrt{{c}^{2}-{d}^{2}}}\arctan \left ( 1/2\,{\frac{2\,c\tan \left ( 1/2\,fx+e/2 \right ) +2\,d}{\sqrt{{c}^{2}-{d}^{2}}}} \right ) }+6\,{\frac{a{b}^{2}{c}^{2}}{f{d}^{2}\sqrt{{c}^{2}-{d}^{2}}}\arctan \left ( 1/2\,{\frac{2\,c\tan \left ( 1/2\,fx+e/2 \right ) +2\,d}{\sqrt{{c}^{2}-{d}^{2}}}} \right ) }-2\,{\frac{{b}^{3}{c}^{3}}{f{d}^{3}\sqrt{{c}^{2}-{d}^{2}}}\arctan \left ( 1/2\,{\frac{2\,c\tan \left ( 1/2\,fx+e/2 \right ) +2\,d}{\sqrt{{c}^{2}-{d}^{2}}}} \right ) }+{\frac{{b}^{3}}{df} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-6\,{\frac{{b}^{2} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}a}{df \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{{b}^{3} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}c}{f{d}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{{b}^{3}}{df}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-2}}-6\,{\frac{a{b}^{2}}{df \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{{b}^{3}c}{f{d}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{2}}}+6\,{\frac{b\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ){a}^{2}}{df}}-6\,{\frac{{b}^{2}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ac}{f{d}^{2}}}+2\,{\frac{{b}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ){c}^{2}}{f{d}^{3}}}+{\frac{{b}^{3}}{df}\arctan \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \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 [A] time = 1.87498, size = 1208, normalized size = 7.74 \begin{align*} \left [\frac{{\left (2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} +{\left (6 \, a^{2} b - b^{3}\right )} c^{2} d^{2} -{\left (6 \, a^{2} b + b^{3}\right )} d^{4}\right )} f x -{\left (b^{3} c^{2} d^{2} - b^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-c^{2} + d^{2}} \log \left (\frac{{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \,{\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt{-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right )}{2 \,{\left (c^{2} d^{3} - d^{5}\right )} f}, \frac{{\left (2 \, b^{3} c^{4} - 6 \, a b^{2} c^{3} d + 6 \, a b^{2} c d^{3} +{\left (6 \, a^{2} b - b^{3}\right )} c^{2} d^{2} -{\left (6 \, a^{2} b + b^{3}\right )} d^{4}\right )} f x -{\left (b^{3} c^{2} d^{2} - b^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{c^{2} - d^{2}} \arctan \left (-\frac{c \sin \left (f x + e\right ) + d}{\sqrt{c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + 2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} - b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} \cos \left (f x + e\right )}{2 \,{\left (c^{2} d^{3} - d^{5}\right )} f}\right ] \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 [A] time = 1.57045, size = 340, normalized size = 2.18 \begin{align*} \frac{\frac{{\left (2 \, b^{3} c^{2} - 6 \, a b^{2} c d + 6 \, a^{2} b d^{2} + b^{3} d^{2}\right )}{\left (f x + e\right )}}{d^{3}} - \frac{4 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (\pi \left \lfloor \frac{f x + e}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (c\right ) + \arctan \left (\frac{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + d}{\sqrt{c^{2} - d^{2}}}\right )\right )}}{\sqrt{c^{2} - d^{2}} d^{3}} + \frac{2 \,{\left (b^{3} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 2 \, b^{3} c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 6 \, a b^{2} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - b^{3} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2 \, b^{3} c - 6 \, a b^{2} d\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{2} d^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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