Optimal. Leaf size=229 \[ -\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.208038, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3517, 3768, 3770, 2621, 302, 207, 2622, 288, 321} \[ -\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3517
Rule 3768
Rule 3770
Rule 2621
Rule 302
Rule 207
Rule 2622
Rule 288
Rule 321
Rubi steps
\begin{align*} \int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx &=\int \left (a^3 \csc ^5(c+d x)+3 a^2 b \csc ^4(c+d x) \sec (c+d x)+3 a b^2 \csc ^3(c+d x) \sec ^2(c+d x)+b^3 \csc ^2(c+d x) \sec ^3(c+d x)\right ) \, dx\\ &=a^3 \int \csc ^5(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^4(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc ^2(c+d x) \sec ^3(c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{4} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}+\frac{1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (9 a b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{3 b^3 \csc (c+d x)}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (9 a b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{3 b^3 \csc (c+d x)}{2 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 6.20045, size = 1229, normalized size = 5.37 \[ -\frac{a^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left (a^3+4 b^2 a\right ) \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^2 b \cos ^3(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left (-2 \cos \left (\frac{1}{2} (c+d x)\right ) b^3-7 a^2 \cos \left (\frac{1}{2} (c+d x)\right ) b\right ) \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc \left (\frac{1}{2} (c+d x)\right )}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^2 b \cos ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \tan \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a^3 \cos ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left (a^3+4 b^2 a\right ) \cos ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left (a^3+12 b^2 a\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left (b^3+2 a^2 b\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left (a^3+12 b^2 a\right ) \cos ^3(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left (b^3+2 a^2 b\right ) \cos ^3(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\cos ^3(c+d x) \sec \left (\frac{1}{2} (c+d x)\right ) \left (-2 \sin \left (\frac{1}{2} (c+d x)\right ) b^3-7 a^2 \sin \left (\frac{1}{2} (c+d x)\right ) b\right ) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a b^2 \cos ^3(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.068, size = 254, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}}{2\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{b}^{3}}{2\,d\sin \left ( dx+c \right ) }}+{\frac{3\,{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{3\,a{b}^{2}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }}+{\frac{9\,a{b}^{2}}{2\,d\cos \left ( dx+c \right ) }}+{\frac{9\,a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{b{a}^{2}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-3\,{\frac{b{a}^{2}}{d\sin \left ( dx+c \right ) }}+3\,{\frac{b{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{3\,{a}^{3}\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1056, size = 338, normalized size = 1.48 \begin{align*} \frac{a^{3}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, a b^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, b^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 8 \, a^{2} b{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.95534, size = 1060, normalized size = 4.63 \begin{align*} \frac{6 \,{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 48 \, a b^{2} \cos \left (d x + c\right ) - 10 \,{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \,{\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left ({\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \,{\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \,{\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \,{\left (3 \,{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} + b^{3} - 4 \,{\left (2 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\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 = 2.14968, size = 504, normalized size = 2.2 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 120 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 32 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 96 \,{\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 96 \,{\left (2 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 24 \,{\left (a^{3} + 12 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{64 \,{\left (b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, a b^{2}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac{50 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 600 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 120 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 32 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 8 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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