Optimal. Leaf size=58 \[ \frac{3 \sec (c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d} \]
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Rubi [A] time = 0.0959269, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3175, 2622, 288, 321, 207} \[ \frac{3 \sec (c+d x)}{2 a d}-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2622
Rule 288
Rule 321
Rule 207
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac{\int \csc ^3(c+d x) \sec ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a d}\\ &=\frac{3 \sec (c+d x)}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 a d}\\ &=-\frac{3 \tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{3 \sec (c+d x)}{2 a d}-\frac{\csc ^2(c+d x) \sec (c+d x)}{2 a d}\\ \end{align*}
Mathematica [B] time = 0.264649, size = 146, normalized size = 2.52 \[ \frac{\csc ^4(c+d x) \left (-6 \cos (2 (c+d x))+2 \cos (3 (c+d x))+3 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-3 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+\cos (c+d x) \left (3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2\right )+2\right )}{2 a d \left (\csc ^2\left (\frac{1}{2} (c+d x)\right )-\sec ^2\left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.077, size = 87, normalized size = 1.5 \begin{align*}{\frac{1}{4\,da \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{4\,da}}+{\frac{1}{4\,da \left ( 1+\cos \left ( dx+c \right ) \right ) }}-{\frac{3\,\ln \left ( 1+\cos \left ( dx+c \right ) \right ) }{4\,da}}+{\frac{1}{da\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968064, size = 95, normalized size = 1.64 \begin{align*} \frac{\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )} - \frac{3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac{3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84009, size = 266, normalized size = 4.59 \begin{align*} \frac{6 \, \cos \left (d x + c\right )^{2} - 3 \,{\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 4}{4 \,{\left (a d \cos \left (d x + c\right )^{3} - a 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] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\csc ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19287, size = 201, normalized size = 3.47 \begin{align*} \frac{\frac{6 \, \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac{\frac{14 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1}{a{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}} - \frac{\cos \left (d x + c\right ) - 1}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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