Optimal. Leaf size=75 \[ \frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.13013, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2838, 2622, 288, 321, 207, 2620, 14} \[ \frac{a \tan (c+d x)}{d}-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2838
Rule 2622
Rule 288
Rule 321
Rule 207
Rule 2620
Rule 14
Rubi steps
\begin{align*} \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \csc ^2(c+d x) \sec ^2(c+d x) \, dx+a \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac{a \operatorname{Subst}\left (\int \frac{1+x^2}{x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{a \operatorname{Subst}\left (\int \left (1+\frac{1}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{a \tan (c+d x)}{d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x)}{d}+\frac{3 a \sec (c+d x)}{2 d}-\frac{a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac{a \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 1.48626, size = 172, normalized size = 2.29 \[ -\frac{2 a \cot (2 (c+d x))}{d}-\frac{a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{3 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{3 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{a \sin \left (\frac{1}{2} (c+d x)\right )}{d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}-\frac{a \sin \left (\frac{1}{2} (c+d x)\right )}{d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 93, normalized size = 1.2 \begin{align*}{\frac{a}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-2\,{\frac{a\cot \left ( dx+c \right ) }{d}}-{\frac{a}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }}+{\frac{3\,a}{2\,d\cos \left ( dx+c \right ) }}+{\frac{3\,a\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 = 1.13041, size = 113, normalized size = 1.51 \begin{align*} \frac{a{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, a{\left (\frac{1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.13302, size = 660, normalized size = 8.8 \begin{align*} \frac{8 \, a \cos \left (d x + c\right )^{3} + 6 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - 3 \,{\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (4 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) - 4 \, a}{4 \,{\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) -{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\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 = 1.29286, size = 138, normalized size = 1.84 \begin{align*} \frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{16 \, a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1} - \frac{18 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 4 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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