Optimal. Leaf size=138 \[ \frac{4 A \cot (c+d x)}{a^3 d}+\frac{164 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac{29 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{19 A \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d} \]
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Rubi [A] time = 0.22469, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {2966, 3770, 3767, 8, 3768, 2650, 2648} \[ \frac{4 A \cot (c+d x)}{a^3 d}+\frac{164 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}+\frac{29 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{19 A \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2966
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (\frac{9 A \csc (c+d x)}{a^3}-\frac{4 A \csc ^2(c+d x)}{a^3}+\frac{A \csc ^3(c+d x)}{a^3}-\frac{2 A}{a^3 (1+\sin (c+d x))^3}-\frac{5 A}{a^3 (1+\sin (c+d x))^2}-\frac{9 A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=\frac{A \int \csc ^3(c+d x) \, dx}{a^3}-\frac{(2 A) \int \frac{1}{(1+\sin (c+d x))^3} \, dx}{a^3}-\frac{(4 A) \int \csc ^2(c+d x) \, dx}{a^3}-\frac{(5 A) \int \frac{1}{(1+\sin (c+d x))^2} \, dx}{a^3}+\frac{(9 A) \int \csc (c+d x) \, dx}{a^3}-\frac{(9 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=-\frac{9 A \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{5 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}+\frac{9 A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}+\frac{A \int \csc (c+d x) \, dx}{2 a^3}-\frac{(4 A) \int \frac{1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}-\frac{(5 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{3 a^3}+\frac{(4 A) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}\\ &=-\frac{19 A \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{4 A \cot (c+d x)}{a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{29 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac{32 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}-\frac{(4 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=-\frac{19 A \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac{4 A \cot (c+d x)}{a^3 d}-\frac{A \cot (c+d x) \csc (c+d x)}{2 a^3 d}+\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{29 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac{164 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 4.23625, size = 245, normalized size = 1.78 \[ \frac{A \left (-240 \tan \left (\frac{1}{2} (c+d x)\right )+240 \cot \left (\frac{1}{2} (c+d x)\right )-15 \csc ^2\left (\frac{1}{2} (c+d x)\right )+15 \sec ^2\left (\frac{1}{2} (c+d x)\right )+1140 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1140 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\frac{2624 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{232}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{464 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{48}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}-\frac{96 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}\right )}{120 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.197, size = 209, normalized size = 1.5 \begin{align*}{\frac{A}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3}}}+{\frac{16\,A}{5\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}-8\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}+{\frac{52\,A}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-18\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+32\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}-{\frac{A}{8\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+2\,{\frac{A}{d{a}^{3}\tan \left ( 1/2\,dx+c/2 \right ) }}+{\frac{19\,A}{2\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02398, size = 840, normalized size = 6.09 \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 [B] time = 2.04622, size = 1327, normalized size = 9.62 \begin{align*} \frac{896 \, A \cos \left (d x + c\right )^{5} - 1222 \, A \cos \left (d x + c\right )^{4} - 3218 \, A \cos \left (d x + c\right )^{3} + 1168 \, A \cos \left (d x + c\right )^{2} + 2292 \, A \cos \left (d x + c\right ) - 285 \,{\left (A \cos \left (d x + c\right )^{5} + 3 \, A \cos \left (d x + c\right )^{4} - 3 \, A \cos \left (d x + c\right )^{3} - 7 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) +{\left (A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} - 5 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + 4 \, A\right )} \sin \left (d x + c\right ) + 4 \, A\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 285 \,{\left (A \cos \left (d x + c\right )^{5} + 3 \, A \cos \left (d x + c\right )^{4} - 3 \, A \cos \left (d x + c\right )^{3} - 7 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) +{\left (A \cos \left (d x + c\right )^{4} - 2 \, A \cos \left (d x + c\right )^{3} - 5 \, A \cos \left (d x + c\right )^{2} + 2 \, A \cos \left (d x + c\right ) + 4 \, A\right )} \sin \left (d x + c\right ) + 4 \, A\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 2 \,{\left (448 \, A \cos \left (d x + c\right )^{4} + 1059 \, A \cos \left (d x + c\right )^{3} - 550 \, A \cos \left (d x + c\right )^{2} - 1134 \, A \cos \left (d x + c\right ) + 12 \, A\right )} \sin \left (d x + c\right ) + 24 \, A}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{3} - 7 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d +{\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 5 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d\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 = 1.17636, size = 243, normalized size = 1.76 \begin{align*} \frac{\frac{1140 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac{15 \,{\left (114 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A\right )}}{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{15 \,{\left (A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 16 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a^{6}} + \frac{16 \,{\left (240 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 825 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 1165 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 755 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 199 \, A\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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