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 = 0.15, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 12
Rule 44
Rule 2836
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*}
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Mathematica [A] time = 0.79, 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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fricas [B] time = 0.47, size = 235, normalized size = 2.12 \[ -\frac {18 \, a^{3} \cos \left (d x + c\right )^{2} - 17 \, a^{3} - 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 4 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{2} + 2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 198, normalized size = 1.78 \[ -\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - 48 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {8 \, {\left (25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 92 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 136 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 92 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.69, size = 241, normalized size = 2.17 \[ \frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {6 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3}}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3}}{2 d \cos \left (d x +c \right )^{2}}+\frac {6 a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3}}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {15 a^{3}}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {45 a^{3}}{8 d \sin \left (d x +c \right )}+\frac {a^{3}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3 a^{3}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3 a^{3}}{2 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 103, normalized size = 0.93 \[ -\frac {12 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + \frac {12 \, a^{3} \sin \left (d x + c\right )^{3} - 18 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 97, normalized size = 0.87 \[ \frac {12\,a^3\,\mathrm {atanh}\left (2\,\sin \left (c+d\,x\right )-1\right )}{d}-\frac {6\,a^3\,{\sin \left (c+d\,x\right )}^3-9\,a^3\,{\sin \left (c+d\,x\right )}^2+2\,a^3\,\sin \left (c+d\,x\right )+\frac {a^3}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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