Optimal. Leaf size=93 \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {2 a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a^3 \log (1-\sin (c+d x))}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.16, antiderivative size = 93, 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} \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {2 a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a^3 \log (1-\sin (c+d x))}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2836
Rubi steps
\begin {align*} \int \csc ^2(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^2}{(a-x)^3 x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 (a-x)^3}+\frac {2}{a^3 (a-x)^2}+\frac {3}{a^4 (a-x)}+\frac {1}{a^3 x^2}+\frac {3}{a^4 x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a^3 \log (1-\sin (c+d x))}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^5}{2 d (a-a \sin (c+d x))^2}+\frac {2 a^4}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 63, normalized size = 0.68 \[ \frac {a^3 \left (-\frac {4}{\sin (c+d x)-1}+\frac {1}{(\sin (c+d x)-1)^2}-2 \csc (c+d x)-6 \log (1-\sin (c+d x))+6 \log (\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 185, normalized size = 1.99 \[ \frac {6 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) - 8 \, a^{3} + 6 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 6 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3} - {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left (2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 166, normalized size = 1.78 \[ -\frac {12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - 6 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 130 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 176, normalized size = 1.89 \[ \frac {a^{3}}{d \cos \left (d x +c \right )^{4}}+\frac {3 a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {9 a^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3}}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a^{3}}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a^{3}}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a^{3}}{8 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 90, normalized size = 0.97 \[ -\frac {6 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - 6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3} - 2 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 80, normalized size = 0.86 \[ \frac {6\,a^3\,\mathrm {atanh}\left (2\,\sin \left (c+d\,x\right )-1\right )}{d}-\frac {3\,a^3\,{\sin \left (c+d\,x\right )}^2-\frac {9\,a^3\,\sin \left (c+d\,x\right )}{2}+a^3}{d\,\left ({\sin \left (c+d\,x\right )}^3-2\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\right )} \]
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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