Optimal. Leaf size=165 \[ -\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.160082, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ -\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
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))^2 \, dx &=\int \left (a^2 \csc ^5(c+d x)+2 a b \csc ^4(c+d x) \sec (c+d x)+b^2 \csc ^3(c+d x) \sec ^2(c+d x)\right ) \, dx\\ &=a^2 \int \csc ^5(c+d x) \, dx+(2 a b) \int \csc ^4(c+d x) \sec (c+d x) \, dx+b^2 \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx\\ &=-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{1}{4} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{1}{8} \left (3 a^2\right ) \int \csc (c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{2 a b \csc (c+d x)}{d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a b \csc (c+d x)}{d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 6.19963, size = 994, normalized size = 6.02 \[ -\frac{a^2 \cos ^2(c+d x) (a+b \tan (c+d x))^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\left (-3 a^2-4 b^2\right ) \cos ^2(c+d x) (a+b \tan (c+d x))^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{32 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a b \cos ^2(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )}{12 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{7 a b \cos ^2(c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{6 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a b \cos ^2(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))^2}{12 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{a^2 \cos ^2(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{64 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{b^2 \cos ^2(c+d x) (a+b \tan (c+d x))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\left (3 a^2+4 b^2\right ) \cos ^2(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{32 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{7 a b \cos ^2(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{6 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{3 \left (a^2+4 b^2\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2}{8 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{2 a b \cos ^2(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))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{3 \left (a^2+4 b^2\right ) \cos ^2(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2}{8 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{2 a b \cos ^2(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))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{b^2 \cos ^2(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{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))^2}-\frac{b^2 \cos ^2(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^2}{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))^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.054, size = 183, normalized size = 1.1 \begin{align*} -{\frac{{b}^{2}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }}+{\frac{3\,{b}^{2}}{2\,d\cos \left ( dx+c \right ) }}+{\frac{3\,{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}-{\frac{2\,ab}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{ab}{d\sin \left ( dx+c \right ) }}+2\,{\frac{ab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{3\,{a}^{2}\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}\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 = 1.07451, size = 252, normalized size = 1.53 \begin{align*} \frac{3 \, a^{2}{\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 \, 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 )} - 16 \, a 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 )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.97415, size = 856, normalized size = 5.19 \begin{align*} \frac{18 \,{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 30 \,{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 48 \, b^{2} - 9 \,{\left ({\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 2 \,{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 9 \,{\left ({\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 2 \,{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 48 \,{\left (a b \cos \left (d x + c\right )^{5} - 2 \, a b \cos \left (d x + c\right )^{3} + a b \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \,{\left (a b \cos \left (d x + c\right )^{5} - 2 \, a b \cos \left (d x + c\right )^{3} + a b \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32 \,{\left (3 \, a b \cos \left (d x + c\right )^{3} - 4 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \,{\left (d \cos \left (d x + c\right )^{5} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \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 = 1.6524, size = 363, normalized size = 2.2 \begin{align*} \frac{3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 16 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 384 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 384 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 72 \,{\left (a^{2} + 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{384 \, b^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} - \frac{150 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 600 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 240 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 24 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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