Optimal. Leaf size=145 \[ -\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}-\frac{20 \tan ^9(c+d x)}{9 a^4 d}-\frac{16 \tan ^7(c+d x)}{7 a^4 d}-\frac{4 \tan ^5(c+d x)}{5 a^4 d}+\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac{16 \sec ^9(c+d x)}{9 a^4 d}+\frac{9 \sec ^7(c+d x)}{7 a^4 d}-\frac{\sec ^5(c+d x)}{5 a^4 d} \]
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Rubi [A] time = 0.40922, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2875, 2873, 2606, 14, 2607, 270} \[ -\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}-\frac{20 \tan ^9(c+d x)}{9 a^4 d}-\frac{16 \tan ^7(c+d x)}{7 a^4 d}-\frac{4 \tan ^5(c+d x)}{5 a^4 d}+\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac{16 \sec ^9(c+d x)}{9 a^4 d}+\frac{9 \sec ^7(c+d x)}{7 a^4 d}-\frac{\sec ^5(c+d x)}{5 a^4 d} \]
Antiderivative was successfully verified.
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Rule 2875
Rule 2873
Rule 2606
Rule 14
Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\int \sec ^9(c+d x) (a-a \sin (c+d x))^4 \tan ^3(c+d x) \, dx}{a^8}\\ &=\frac{\int \left (a^4 \sec ^9(c+d x) \tan ^3(c+d x)-4 a^4 \sec ^8(c+d x) \tan ^4(c+d x)+6 a^4 \sec ^7(c+d x) \tan ^5(c+d x)-4 a^4 \sec ^6(c+d x) \tan ^6(c+d x)+a^4 \sec ^5(c+d x) \tan ^7(c+d x)\right ) \, dx}{a^8}\\ &=\frac{\int \sec ^9(c+d x) \tan ^3(c+d x) \, dx}{a^4}+\frac{\int \sec ^5(c+d x) \tan ^7(c+d x) \, dx}{a^4}-\frac{4 \int \sec ^8(c+d x) \tan ^4(c+d x) \, dx}{a^4}-\frac{4 \int \sec ^6(c+d x) \tan ^6(c+d x) \, dx}{a^4}+\frac{6 \int \sec ^7(c+d x) \tan ^5(c+d x) \, dx}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac{\operatorname{Subst}\left (\int x^4 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac{4 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac{4 \operatorname{Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac{6 \operatorname{Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-x^4+3 x^6-3 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac{\operatorname{Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac{4 \operatorname{Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac{4 \operatorname{Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac{6 \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac{\sec ^5(c+d x)}{5 a^4 d}+\frac{9 \sec ^7(c+d x)}{7 a^4 d}-\frac{16 \sec ^9(c+d x)}{9 a^4 d}+\frac{8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac{4 \tan ^5(c+d x)}{5 a^4 d}-\frac{16 \tan ^7(c+d x)}{7 a^4 d}-\frac{20 \tan ^9(c+d x)}{9 a^4 d}-\frac{8 \tan ^{11}(c+d x)}{11 a^4 d}\\ \end{align*}
Mathematica [A] time = 0.457422, size = 166, normalized size = 1.14 \[ \frac{\sec ^3(c+d x) (844800 \sin (c+d x)-191752 \sin (2 (c+d x))+11264 \sin (3 (c+d x))-69728 \sin (4 (c+d x))+25600 \sin (5 (c+d x))+17432 \sin (6 (c+d x))-1024 \sin (7 (c+d x))-215721 \cos (c+d x)-619520 \cos (2 (c+d x))+23969 \cos (3 (c+d x))+32768 \cos (4 (c+d x))+54475 \cos (5 (c+d x))-8192 \cos (6 (c+d x))-2179 \cos (7 (c+d x))+844800)}{7096320 a^4 d (\sin (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 220, normalized size = 1.5 \begin{align*} 16\,{\frac{1}{d{a}^{4}} \left ( -{\frac{1}{768\, \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{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512}}+1/11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-11}-1/2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-10}+{\frac{23}{18\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{9}}}-2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-8}+{\frac{235}{112\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{7}}}-{\frac{145}{96\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{6}}}+{\frac{29}{40\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{5}}}-{\frac{13}{64\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{13}{768\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}+{\frac{3}{512\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15319, size = 686, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74186, size = 402, normalized size = 2.77 \begin{align*} -\frac{128 \, \cos \left (d x + c\right )^{6} - 320 \, \cos \left (d x + c\right )^{4} + 805 \, \cos \left (d x + c\right )^{2} + 4 \,{\left (8 \, \cos \left (d x + c\right )^{6} - 60 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{2} - 105\right )} \sin \left (d x + c\right ) - 735}{3465 \,{\left (a^{4} d \cos \left (d x + c\right )^{7} - 8 \, a^{4} d \cos \left (d x + c\right )^{5} + 8 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \,{\left (a^{4} d \cos \left (d x + c\right )^{5} - 2 \, a^{4} 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 = 1.33721, size = 267, normalized size = 1.84 \begin{align*} -\frac{\frac{1155 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2\right )}}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} - \frac{3465 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 45045 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 279510 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 669900 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1205358 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1334718 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1144440 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 627660 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 257345 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 57013 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5498}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{11}}}{110880 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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