Optimal. Leaf size=80 \[ -\frac{3 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]
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Rubi [A] time = 0.156333, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2872, 3770, 3767, 8, 3768, 2648} \[ -\frac{3 a^3 \cot (c+d x)}{d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2872
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2648
Rubi steps
\begin{align*} \int \csc ^3(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^2 \int \left (4 a \csc (c+d x)+3 a \csc ^2(c+d x)+a \csc ^3(c+d x)-\frac{4 a}{-1+\sin (c+d x)}\right ) \, dx\\ &=a^3 \int \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (4 a^3\right ) \int \csc (c+d x) \, dx-\left (4 a^3\right ) \int \frac{1}{-1+\sin (c+d x)} \, dx\\ &=-\frac{4 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac{1}{2} a^3 \int \csc (c+d x) \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac{9 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a^3 \cot (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac{4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.13107, size = 124, normalized size = 1.55 \[ \frac{a^3 \left (12 \tan \left (\frac{1}{2} (c+d x)\right )-12 \cot \left (\frac{1}{2} (c+d x)\right )-\csc ^2\left (\frac{1}{2} (c+d x)\right )+\sec ^2\left (\frac{1}{2} (c+d x)\right )+36 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-36 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{64 \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.119, size = 117, normalized size = 1.5 \begin{align*}{\frac{{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{9\,{a}^{3}}{2\,d\cos \left ( dx+c \right ) }}+{\frac{9\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{{a}^{3}}{d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-6\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04362, size = 182, normalized size = 2.28 \begin{align*} \frac{a^{3}{\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 )} + 6 \, a^{3}{\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 )} - 12 \, a^{3}{\left (\frac{1}{\tan \left (d x + c\right )} - \tan \left (d x + c\right )\right )} + 4 \, a^{3} \tan \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.15538, size = 721, normalized size = 9.01 \begin{align*} \frac{28 \, a^{3} \cos \left (d x + c\right )^{3} + 18 \, a^{3} \cos \left (d x + c\right )^{2} - 26 \, a^{3} \cos \left (d x + c\right ) - 16 \, a^{3} - 9 \,{\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3} -{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 9 \,{\left (a^{3} \cos \left (d x + c\right )^{3} + a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - a^{3} -{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 2 \,{\left (14 \, a^{3} \cos \left (d x + c\right )^{2} + 5 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3}\right )} \sin \left (d x + c\right )}{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 = 1.30554, size = 157, normalized size = 1.96 \begin{align*} \frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{64 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1} - \frac{54 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{\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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