Optimal. Leaf size=101 \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.110891, antiderivative size = 101, 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} \[ -\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d} \]
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 ^6(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^6(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^6(c+d x) \, dx+b \int \csc ^6(c+d x) \sec (c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{b \operatorname{Subst}\left (\int \frac{x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{b \operatorname{Subst}\left (\int \left (1+x^2+x^4+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{b \csc (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \cot (c+d x)}{d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot ^5(c+d x)}{5 d}-\frac{b \csc (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 0.0257432, size = 91, normalized size = 0.9 \[ -\frac{b \csc ^5(c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},\sin ^2(c+d x)\right )}{5 d}-\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 115, normalized size = 1.1 \begin{align*} -{\frac{8\,a\cot \left ( dx+c \right ) }{15\,d}}-{\frac{a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{4\,a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{15\,d}}-{\frac{b}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{b}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{b}{d\sin \left ( dx+c \right ) }}+{\frac{b\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 = 0.969416, size = 130, normalized size = 1.29 \begin{align*} -\frac{b{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac{2 \,{\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a}{\tan \left (d x + c\right )^{5}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7965, size = 473, normalized size = 4.68 \begin{align*} -\frac{16 \, a \cos \left (d x + c\right )^{5} + 30 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 70 \, b \cos \left (d x + c\right )^{2} - 15 \,{\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \,{\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right ) + 46 \, b}{30 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + 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 [B] time = 1.3389, size = 262, normalized size = 2.59 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 25 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 480 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 480 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 330 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{150 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 330 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 25 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 3 \, a + 3 \, b}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5}}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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