Optimal. Leaf size=161 \[ \frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.157433, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3517, 3768, 3770, 2621, 321, 207, 2622, 2606, 30} \[ \frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3517
Rule 3768
Rule 3770
Rule 2621
Rule 321
Rule 207
Rule 2622
Rule 2606
Rule 30
Rubi steps
\begin{align*} \int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx &=\int \left (a^4 \csc ^3(c+d x)+4 a^3 b \csc ^2(c+d x) \sec (c+d x)+6 a^2 b^2 \csc (c+d x) \sec ^2(c+d x)+4 a b^3 \sec ^3(c+d x)+b^4 \sec ^3(c+d x) \tan (c+d x)\right ) \, dx\\ &=a^4 \int \csc ^3(c+d x) \, dx+\left (4 a^3 b\right ) \int \csc ^2(c+d x) \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \csc (c+d x) \sec ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \sec ^3(c+d x) \, dx+b^4 \int \sec ^3(c+d x) \tan (c+d x) \, dx\\ &=-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{2 a b^3 \sec (c+d x) \tan (c+d x)}{d}+\frac{1}{2} a^4 \int \csc (c+d x) \, dx+\left (2 a b^3\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^4 \operatorname{Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}+\frac{2 a b^3 \sec (c+d x) \tan (c+d x)}{d}-\frac{\left (4 a^3 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (6 a^2 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \csc (c+d x)}{d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}+\frac{2 a b^3 \sec (c+d x) \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 6.19325, size = 1128, normalized size = 7.01 \[ -\frac{2 a^3 b \cos ^4(c+d x) \tan \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^2 \left (36 a^2+b^2\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a^4 \cos ^4(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a^4 \cos ^4(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 a^3 b \cos ^4(c+d x) \cot \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (-a^4-12 b^2 a^2\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \left (2 b a^3+b^3 a\right ) \cos ^4(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))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left (a^4+12 b^2 a^2\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \left (2 b a^3+b^3 a\right ) \cos ^4(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))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left (-\sin \left (\frac{1}{2} (c+d x)\right ) b^4-36 a^2 \sin \left (\frac{1}{2} (c+d x)\right ) b^2\right ) (a+b \tan (c+d x))^4}{6 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))^4}+\frac{\cos ^4(c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right ) b^4+36 a^2 \sin \left (\frac{1}{2} (c+d x)\right ) b^2\right ) (a+b \tan (c+d x))^4}{6 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))^4}+\frac{\left (b^4+12 a b^3\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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))^4}+\frac{\left (b^4-12 a b^3\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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))^4}-\frac{b^4 \cos ^4(c+d x) \sin \left (\frac{1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 192, normalized size = 1.2 \begin{align*}{\frac{{b}^{4}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+2\,{\frac{{b}^{3}a\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+2\,{\frac{{b}^{3}a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2}}{d\cos \left ( dx+c \right ) }}+6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{b{a}^{3}}{d\sin \left ( dx+c \right ) }}+4\,{\frac{b{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{4}\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{4}\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 = 1.17389, size = 254, normalized size = 1.58 \begin{align*} \frac{3 \, a^{4}{\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 )} - 12 \, a 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 )} + 36 \, a^{2} 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 )} - 24 \, a^{3} 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 )} + \frac{4 \, b^{4}}{\cos \left (d x + c\right )^{3}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.03541, size = 818, normalized size = 5.08 \begin{align*} \frac{6 \,{\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} - 4 \, b^{4} - 4 \,{\left (18 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left ({\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} -{\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left ({\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} -{\left (a^{4} + 12 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 12 \,{\left ({\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} -{\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \,{\left ({\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} -{\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 24 \,{\left (a b^{3} \cos \left (d x + c\right ) -{\left (2 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{12 \,{\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{3}\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 = 2.76962, size = 405, normalized size = 2.52 \begin{align*} \frac{3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 48 \,{\left (2 \, a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 48 \,{\left (2 \, a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 12 \,{\left (a^{4} + 12 \, a^{2} b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{3 \,{\left (6 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 72 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 16 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{4}\right )}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{16 \,{\left (6 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 18 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 36 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 18 \, a^{2} b^{2} - b^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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