Optimal. Leaf size=141 \[ -\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.134904, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3517, 3768, 3770, 2621, 321, 207, 2622} \[ -\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3517
Rule 3768
Rule 3770
Rule 2621
Rule 321
Rule 207
Rule 2622
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+b \tan (c+d x))^3 \, dx &=\int \left (a^3 \csc ^3(c+d x)+3 a^2 b \csc ^2(c+d x) \sec (c+d x)+3 a b^2 \csc (c+d x) \sec ^2(c+d x)+b^3 \sec ^3(c+d x)\right ) \, dx\\ &=a^3 \int \csc ^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^2(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx+b^3 \int \sec ^3(c+d x) \, dx\\ &=-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{b^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} a^3 \int \csc (c+d x) \, dx+\frac{1}{2} b^3 \int \sec (c+d x) \, dx-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec (c+d x) \tan (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 (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 6.18345, size = 897, normalized size = 6.36 \[ -\frac{3 a^2 b \cos ^3(c+d x) \tan \left (\frac{1}{2} (c+d x)\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) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^3 \cos ^3(c+d x) \csc ^2\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 ^2\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{3 a^2 b \cos ^3(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left (-a^3-6 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}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left (-b^3-6 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{\left (a^3+6 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}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left (b^3+6 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{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} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 170, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{a{b}^{2}}{d\cos \left ( dx+c \right ) }}+3\,{\frac{a{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-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 ) \csc \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15234, size = 231, normalized size = 1.64 \begin{align*} \frac{a^{3}{\left (\frac{2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - b^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a b^{2}{\left (\frac{2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{2} b{\left (\frac{2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.73967, size = 720, normalized size = 5.11 \begin{align*} -\frac{12 \, a b^{2} \cos \left (d x + c\right ) - 2 \,{\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} +{\left ({\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} -{\left (a^{3} + 6 \, 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 ) -{\left ({\left (a^{3} + 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} -{\left (a^{3} + 6 \, 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 ) -{\left ({\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} -{\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left ({\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4} -{\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (b^{3} -{\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left (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 [B] time = 1.98513, size = 410, normalized size = 2.91 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \,{\left (6 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 4 \,{\left (6 \, a^{2} b + b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 4 \,{\left (a^{3} + 6 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 8 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 24 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 24 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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