Optimal. Leaf size=116 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {5 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \csc (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
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Rubi [A] time = 0.14, antiderivative size = 116, 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, 88} \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {5 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \csc (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a^2}{(a-x)^3 x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{2 a^3 (a-x)^3}+\frac {5}{4 a^4 (a-x)^2}+\frac {17}{8 a^5 (a-x)}+\frac {1}{a^4 x^2}+\frac {2}{a^5 x}+\frac {1}{8 a^5 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2 \csc (c+d x)}{d}-\frac {17 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \log (1+\sin (c+d x))}{8 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {5 a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 74, normalized size = 0.64 \[ \frac {a^2 \left (-\frac {10}{\sin (c+d x)-1}+\frac {2}{(\sin (c+d x)-1)^2}-8 \csc (c+d x)-17 \log (1-\sin (c+d x))+16 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 240, normalized size = 2.07 \[ \frac {18 \, a^{2} \cos \left (d x + c\right )^{2} + 28 \, a^{2} \sin \left (d x + c\right ) - 26 \, a^{2} + 16 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, {\left (2 \, a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{8 \, {\left (2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{2} - 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 115, normalized size = 0.99 \[ \frac {2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 34 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 32 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {16 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )} + \frac {51 \, a^{2} \sin \left (d x + c\right )^{2} - 122 \, a^{2} \sin \left (d x + c\right ) + 75 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 176, normalized size = 1.52 \[ \frac {a^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {9 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2}}{2 d \cos \left (d x +c \right )^{4}}+\frac {a^{2}}{d \cos \left (d x +c \right )^{2}}+\frac {2 a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a^{2}}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a^{2}}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a^{2}}{8 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 104, normalized size = 0.90 \[ \frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 17 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (9 \, a^{2} \sin \left (d x + c\right )^{2} - 14 \, a^{2} \sin \left (d x + c\right ) + 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{3} - 2 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.05, size = 110, normalized size = 0.95 \[ \frac {a^2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{8\,d}-\frac {17\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{8\,d}+\frac {2\,a^2\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {9\,a^2\,{\sin \left (c+d\,x\right )}^2}{4}-\frac {7\,a^2\,\sin \left (c+d\,x\right )}{2}+a^2}{d\,\left ({\sin \left (c+d\,x\right )}^3-2\,{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\right )} \]
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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