Optimal. Leaf size=91 \[ \frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{4 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.156614, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2711, 2607, 14, 2606, 270, 30} \[ \frac{2 \tan ^7(c+d x)}{7 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{4 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 2711
Rule 2607
Rule 14
Rule 2606
Rule 270
Rule 30
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac{\int \left (a^2 \sec ^4(c+d x) \tan ^4(c+d x)-2 a^2 \sec ^3(c+d x) \tan ^5(c+d x)+a^2 \sec ^2(c+d x) \tan ^6(c+d x)\right ) \, dx}{a^4}\\ &=\frac{\int \sec ^4(c+d x) \tan ^4(c+d x) \, dx}{a^2}+\frac{\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^2}-\frac{2 \int \sec ^3(c+d x) \tan ^5(c+d x) \, dx}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac{\operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{\tan ^7(c+d x)}{7 a^2 d}+\frac{\operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac{2 \operatorname{Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac{2 \sec ^3(c+d x)}{3 a^2 d}+\frac{4 \sec ^5(c+d x)}{5 a^2 d}-\frac{2 \sec ^7(c+d x)}{7 a^2 d}+\frac{\tan ^5(c+d x)}{5 a^2 d}+\frac{2 \tan ^7(c+d x)}{7 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.247231, size = 126, normalized size = 1.38 \[ -\frac{\sec ^3(c+d x) (-1232 \sin (c+d x)-824 \sin (2 (c+d x))+1896 \sin (3 (c+d x))-412 \sin (4 (c+d x))-72 \sin (5 (c+d x))-1442 \cos (c+d x)+1664 \cos (2 (c+d x))-309 \cos (3 (c+d x))-288 \cos (4 (c+d x))+103 \cos (5 (c+d x))+672)}{13440 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 160, normalized size = 1.8 \begin{align*} 32\,{\frac{1}{d{a}^{2}} \left ( -{\frac{1}{384\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{256\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+{\frac{1}{256\,\tan \left ( 1/2\,dx+c/2 \right ) -256}}-{\frac{1}{56\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+1/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-6}-{\frac{3}{40\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}+1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}+{\frac{1}{384\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-{\frac{1}{256\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{1}{256\,\tan \left ( 1/2\,dx+c/2 \right ) +256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08625, size = 427, normalized size = 4.69 \begin{align*} -\frac{32 \,{\left (\frac{4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{14 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 1\right )}}{105 \,{\left (a^{2} + \frac{4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63311, size = 267, normalized size = 2.93 \begin{align*} -\frac{18 \, \cos \left (d x + c\right )^{4} - 44 \, \cos \left (d x + c\right )^{2} +{\left (9 \, \cos \left (d x + c\right )^{4} - 66 \, \cos \left (d x + c\right )^{2} + 25\right )} \sin \left (d x + c\right ) + 10}{105 \,{\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\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 = 1.1964, size = 197, normalized size = 2.16 \begin{align*} \frac{\frac{35 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{105 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 735 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2030 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2030 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1701 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 707 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 116}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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