Optimal. Leaf size=69 \[ -\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.102516, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3872, 2838, 2621, 302, 207, 3767} \[ -\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2838
Rule 2621
Rule 302
Rule 207
Rule 3767
Rubi steps
\begin{align*} \int \csc ^4(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^4(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^4(c+d x) \, dx+a \int \csc ^4(c+d x) \sec (c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 0.0285499, size = 69, normalized size = 1. \[ -\frac{a \csc ^3(c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},\sin ^2(c+d x)\right )}{3 d}-\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.104, size = 81, normalized size = 1.2 \begin{align*} -{\frac{2\,a\cot \left ( dx+c \right ) }{3\,d}}-{\frac{a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{a}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{a}{d\sin \left ( dx+c \right ) }}+{\frac{a\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00723, size = 103, normalized size = 1.49 \begin{align*} -\frac{a{\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 )} + \frac{2 \,{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71284, size = 282, normalized size = 4.09 \begin{align*} -\frac{4 \, a \cos \left (d x + c\right )^{2} - 3 \,{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \,{\left (a \cos \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 2 \, a \cos \left (d x + c\right ) - 8 \, a}{6 \,{\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\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 = 1.63253, size = 107, normalized size = 1.55 \begin{align*} \frac{12 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - 3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{12 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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