Optimal. Leaf size=171 \[ \frac{a^2}{40 d (a \sin (c+d x)+a)^5}-\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{5}{128 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{7 a}{64 d (a \sin (c+d x)+a)^4}+\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.123351, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2707, 88, 206} \[ \frac{a^2}{40 d (a \sin (c+d x)+a)^5}-\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{5}{128 d \left (a^3 \sin (c+d x)+a^3\right )}+\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{7 a}{64 d (a \sin (c+d x)+a)^4}+\frac{1}{6 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a-x)^3 (a+x)^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{64 a (a-x)^3}-\frac{1}{32 a^2 (a-x)^2}-\frac{a^2}{8 (a+x)^6}+\frac{7 a}{16 (a+x)^5}-\frac{1}{2 (a+x)^4}+\frac{5}{32 a (a+x)^3}+\frac{5}{128 a^2 (a+x)^2}+\frac{1}{128 a^2 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{1}{128 a d (a-a \sin (c+d x))^2}+\frac{a^2}{40 d (a+a \sin (c+d x))^5}-\frac{7 a}{64 d (a+a \sin (c+d x))^4}+\frac{1}{6 d (a+a \sin (c+d x))^3}-\frac{5}{64 a d (a+a \sin (c+d x))^2}-\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{5}{128 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a^2 d}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{128 a^3 d}+\frac{1}{128 a d (a-a \sin (c+d x))^2}+\frac{a^2}{40 d (a+a \sin (c+d x))^5}-\frac{7 a}{64 d (a+a \sin (c+d x))^4}+\frac{1}{6 d (a+a \sin (c+d x))^3}-\frac{5}{64 a d (a+a \sin (c+d x))^2}-\frac{1}{32 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac{5}{128 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.925661, size = 102, normalized size = 0.6 \[ \frac{15 \tanh ^{-1}(\sin (c+d x))-\frac{15 \sin ^6(c+d x)+45 \sin ^5(c+d x)-620 \sin ^4(c+d x)-540 \sin ^3(c+d x)+157 \sin ^2(c+d x)+351 \sin (c+d x)+112}{(\sin (c+d x)-1)^2 (\sin (c+d x)+1)^5}}{1920 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.106, size = 162, normalized size = 1. \begin{align*}{\frac{1}{128\,d{a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}+{\frac{1}{32\,d{a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{256\,d{a}^{3}}}+{\frac{1}{40\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7}{64\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{6\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{5}{64\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5}{128\,d{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{256\,d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.96124, size = 254, normalized size = 1.49 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{6} + 45 \, \sin \left (d x + c\right )^{5} - 620 \, \sin \left (d x + c\right )^{4} - 540 \, \sin \left (d x + c\right )^{3} + 157 \, \sin \left (d x + c\right )^{2} + 351 \, \sin \left (d x + c\right ) + 112\right )}}{a^{3} \sin \left (d x + c\right )^{7} + 3 \, a^{3} \sin \left (d x + c\right )^{6} + a^{3} \sin \left (d x + c\right )^{5} - 5 \, a^{3} \sin \left (d x + c\right )^{4} - 5 \, a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2} + 3 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac{15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3}}}{3840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67077, size = 657, normalized size = 3.84 \begin{align*} -\frac{30 \, \cos \left (d x + c\right )^{6} + 1150 \, \cos \left (d x + c\right )^{4} - 2076 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} +{\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} +{\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 18 \,{\left (5 \, \cos \left (d x + c\right )^{4} + 50 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 672}{3840 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4} +{\left (a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\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 = 4.82376, size = 184, normalized size = 1.08 \begin{align*} \frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3}} + \frac{30 \,{\left (3 \, \sin \left (d x + c\right )^{2} + 10 \, \sin \left (d x + c\right ) - 9\right )}}{a^{3}{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac{137 \, \sin \left (d x + c\right )^{5} + 1285 \, \sin \left (d x + c\right )^{4} + 4970 \, \sin \left (d x + c\right )^{3} + 6010 \, \sin \left (d x + c\right )^{2} + 3245 \, \sin \left (d x + c\right ) + 673}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{15360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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