Optimal. Leaf size=267 \[ \frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{2 a \left (-7 a^2 b^2+5 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^6 d \sqrt{a^2-b^2}}-\frac{\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left (5 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 a b^3 d}-\frac{\left (20 a^2-13 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{x \left (-36 a^2 b^2+40 a^4+3 b^4\right )}{8 b^6}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b^2 d} \]
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Rubi [A] time = 0.758161, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2892, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{2 a \left (-7 a^2 b^2+5 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^6 d \sqrt{a^2-b^2}}-\frac{\left (a^2-b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left (5 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{3 a b^3 d}-\frac{\left (20 a^2-13 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^4 d}+\frac{x \left (-36 a^2 b^2+40 a^4+3 b^4\right )}{8 b^6}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b^2 d} \]
Antiderivative was successfully verified.
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Rule 2892
Rule 3049
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^2(c+d x) \left (15 a^2-8 b^2-a b \sin (c+d x)-4 \left (5 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a b^2}\\ &=\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin (c+d x) \left (-8 a \left (5 a^2-3 b^2\right )+5 a^2 b \sin (c+d x)+3 a \left (20 a^2-13 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a b^3}\\ &=-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{3 a^2 \left (20 a^2-13 b^2\right )-a b \left (20 a^2-9 b^2\right ) \sin (c+d x)-8 a^2 \left (15 a^2-11 b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{24 a b^4}\\ &=\frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{3 a^2 b \left (20 a^2-13 b^2\right )+3 a \left (40 a^4-36 a^2 b^2+3 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{24 a b^5}\\ &=\frac{\left (40 a^4-36 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac{\left (a \left (5 a^4-7 a^2 b^2+2 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^6}\\ &=\frac{\left (40 a^4-36 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac{\left (2 a \left (5 a^4-7 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^6 d}\\ &=\frac{\left (40 a^4-36 a^2 b^2+3 b^4\right ) x}{8 b^6}+\frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac{\left (4 a \left (5 a^4-7 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^6 d}\\ &=\frac{\left (40 a^4-36 a^2 b^2+3 b^4\right ) x}{8 b^6}-\frac{2 a \left (5 a^4-7 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^6 \sqrt{a^2-b^2} d}+\frac{a \left (15 a^2-11 b^2\right ) \cos (c+d x)}{3 b^5 d}-\frac{\left (20 a^2-13 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^4 d}+\frac{\left (5 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{3 a b^3 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b^2 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{a b^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 3.56595, size = 325, normalized size = 1.22 \[ \frac{\frac{240 a^3 b^2 \sin (2 (c+d x))-864 a^2 b^3 c \sin (c+d x)-864 a^2 b^3 d x \sin (c+d x)+24 b \left (-31 a^2 b^2+40 a^4+b^4\right ) \cos (c+d x)+\left (40 a^2 b^3-21 b^5\right ) \cos (3 (c+d x))-864 a^3 b^2 c-864 a^3 b^2 d x+960 a^4 b c \sin (c+d x)+960 a^4 b d x \sin (c+d x)+960 a^5 c+960 a^5 d x-176 a b^4 \sin (2 (c+d x))-10 a b^4 \sin (4 (c+d x))+72 a b^4 c+72 a b^4 d x+72 b^5 c \sin (c+d x)+72 b^5 d x \sin (c+d x)-3 b^5 \cos (5 (c+d x))}{a+b \sin (c+d x)}-\frac{384 a \left (-7 a^2 b^2+5 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}}{192 b^6 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.128, size = 938, normalized size = 3.5 \begin{align*} \text{result too large to display} \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 = 2.20357, size = 1391, normalized size = 5.21 \begin{align*} \left [-\frac{6 \, b^{5} \cos \left (d x + c\right )^{5} -{\left (20 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (40 \, a^{5} - 36 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x + 12 \,{\left (5 \, a^{4} - 2 \, a^{2} b^{2} +{\left (5 \, a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 3 \,{\left (40 \, a^{4} b - 36 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right ) +{\left (10 \, a b^{4} \cos \left (d x + c\right )^{3} - 3 \,{\left (40 \, a^{4} b - 36 \, a^{2} b^{3} + 3 \, b^{5}\right )} d x - 3 \,{\left (20 \, a^{3} b^{2} - 13 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}}, -\frac{6 \, b^{5} \cos \left (d x + c\right )^{5} -{\left (20 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left (40 \, a^{5} - 36 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - 24 \,{\left (5 \, a^{4} - 2 \, a^{2} b^{2} +{\left (5 \, a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 3 \,{\left (40 \, a^{4} b - 36 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right ) +{\left (10 \, a b^{4} \cos \left (d x + c\right )^{3} - 3 \,{\left (40 \, a^{4} b - 36 \, a^{2} b^{3} + 3 \, b^{5}\right )} d x - 3 \,{\left (20 \, a^{3} b^{2} - 13 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \,{\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}}\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 = 1.31052, size = 606, normalized size = 2.27 \begin{align*} \frac{\frac{3 \,{\left (40 \, a^{4} - 36 \, a^{2} b^{2} + 3 \, b^{4}\right )}{\left (d x + c\right )}}{b^{6}} - \frac{48 \,{\left (5 \, a^{5} - 7 \, a^{3} b^{2} + 2 \, a b^{4}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} b^{6}} + \frac{48 \,{\left (a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{4} - a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )} b^{5}} + \frac{2 \,{\left (36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 96 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 96 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 9 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 288 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 192 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 288 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 160 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 15 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 96 \, a^{3} - 64 \, a b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} b^{5}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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