Optimal. Leaf size=111 \[ \frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}+\frac{6 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.147573, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {2836, 12, 44} \[ \frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{3 a^4}{d (a-a \sin (c+d x))}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{3 a^3 \csc (c+d x)}{d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}+\frac{6 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 44
Rubi steps
\begin{align*} \int \csc ^3(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{a^3}{(a-x)^3 x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^8 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 (a-x)^3}+\frac{3}{a^4 (a-x)^2}+\frac{6}{a^5 (a-x)}+\frac{1}{a^3 x^3}+\frac{3}{a^4 x^2}+\frac{6}{a^5 x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \csc (c+d x)}{d}-\frac{a^3 \csc ^2(c+d x)}{2 d}-\frac{6 a^3 \log (1-\sin (c+d x))}{d}+\frac{6 a^3 \log (\sin (c+d x))}{d}+\frac{a^5}{2 d (a-a \sin (c+d x))^2}+\frac{3 a^4}{d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.77874, size = 73, normalized size = 0.66 \[ -\frac{a^3 \left (\frac{6}{\sin (c+d x)-1}-\frac{1}{(\sin (c+d x)-1)^2}+\csc ^2(c+d x)+6 \csc (c+d x)+12 \log (1-\sin (c+d x))-12 \log (\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.149, size = 241, normalized size = 2.2 \begin{align*}{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+6\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+6\,{\frac{{a}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3}}{4\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{15\,{a}^{3}}{8\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{45\,{a}^{3}}{8\,d\sin \left ( dx+c \right ) }}+{\frac{{a}^{3}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,{a}^{3}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{3}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14212, size = 139, normalized size = 1.25 \begin{align*} -\frac{12 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + \frac{12 \, a^{3} \sin \left (d x + c\right )^{3} - 18 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53028, size = 567, normalized size = 5.11 \begin{align*} -\frac{18 \, a^{3} \cos \left (d x + c\right )^{2} - 17 \, a^{3} - 12 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \,{\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} \, \sin \left (d x + c\right )\right ) + 12 \,{\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \,{\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 4 \,{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{2} + 2 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) + 2 \, 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 = 1.27533, size = 267, normalized size = 2.41 \begin{align*} -\frac{a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 96 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - 48 \, 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{72 \, 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}} - \frac{8 \,{\left (25 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 92 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 136 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 92 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 25 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{4}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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