Optimal. Leaf size=121 \[ \frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{13 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{a^3 d}+\frac{7 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec (c+d x)}{a^3 d} \]
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Rubi [A] time = 0.346745, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2875, 2873, 2607, 14, 2606, 270, 30, 194} \[ \frac{4 \tan ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{13 \sec ^7(c+d x)}{7 a^3 d}-\frac{3 \sec ^5(c+d x)}{a^3 d}+\frac{7 \sec ^3(c+d x)}{3 a^3 d}-\frac{\sec (c+d x)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2607
Rule 14
Rule 2606
Rule 270
Rule 30
Rule 194
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\int \sec ^4(c+d x) (a-a \sin (c+d x))^3 \tan ^6(c+d x) \, dx}{a^6}\\ &=\frac{\int \left (a^3 \sec ^4(c+d x) \tan ^6(c+d x)-3 a^3 \sec ^3(c+d x) \tan ^7(c+d x)+3 a^3 \sec ^2(c+d x) \tan ^8(c+d x)-a^3 \sec (c+d x) \tan ^9(c+d x)\right ) \, dx}{a^6}\\ &=\frac{\int \sec ^4(c+d x) \tan ^6(c+d x) \, dx}{a^3}-\frac{\int \sec (c+d x) \tan ^9(c+d x) \, dx}{a^3}-\frac{3 \int \sec ^3(c+d x) \tan ^7(c+d x) \, dx}{a^3}+\frac{3 \int \sec ^2(c+d x) \tan ^8(c+d x) \, dx}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac{\operatorname{Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac{3 \operatorname{Subst}\left (\int x^8 \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=\frac{\tan ^9(c+d x)}{3 a^3 d}-\frac{\operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac{\operatorname{Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac{3 \operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}\\ &=-\frac{\sec (c+d x)}{a^3 d}+\frac{7 \sec ^3(c+d x)}{3 a^3 d}-\frac{3 \sec ^5(c+d x)}{a^3 d}+\frac{13 \sec ^7(c+d x)}{7 a^3 d}-\frac{4 \sec ^9(c+d x)}{9 a^3 d}+\frac{\tan ^7(c+d x)}{7 a^3 d}+\frac{4 \tan ^9(c+d x)}{9 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.376476, size = 185, normalized size = 1.53 \[ \frac{-2304 \sin (c+d x)+27189 \sin (2 (c+d x))-16256 \sin (3 (c+d x))+12084 \sin (4 (c+d x))+384 \sin (5 (c+d x))-1007 \sin (6 (c+d x))+36252 \cos (c+d x)-12384 \cos (2 (c+d x))+2014 \cos (3 (c+d x))+4800 \cos (4 (c+d x))-6042 \cos (5 (c+d x))+608 \cos (6 (c+d x))-9408}{64512 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 190, normalized size = 1.6 \begin{align*} 128\,{\frac{1}{d{a}^{3}} \left ( -{\frac{1}{3072\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{3}}}-{\frac{1}{2048\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+{\frac{5}{4096\,\tan \left ( 1/2\,dx+c/2 \right ) -4096}}-{\frac{1}{144\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}+1/32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8}-{\frac{11}{224\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}+{\frac{5}{192\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}+{\frac{1}{256\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}-{\frac{1}{512\, \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}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-{\frac{5}{4096\,\tan \left ( 1/2\,dx+c/2 \right ) +4096}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20072, size = 489, normalized size = 4.04 \begin{align*} -\frac{32 \,{\left (\frac{6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{2 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{27 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + 1\right )}}{63 \,{\left (a^{3} + \frac{6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac{2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac{6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac{a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69633, size = 328, normalized size = 2.71 \begin{align*} -\frac{19 \, \cos \left (d x + c\right )^{6} + 9 \, \cos \left (d x + c\right )^{4} - 51 \, \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{2} + 7\right )} \sin \left (d x + c\right ) + 7}{63 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} +{\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\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 = 1.19524, size = 232, normalized size = 1.92 \begin{align*} \frac{\frac{21 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 17\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{315 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 3024 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 13020 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 32760 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 51282 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 43008 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 20988 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 5688 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 667}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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