Optimal. Leaf size=134 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {7 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {4 a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
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Rubi [A] time = 0.15, antiderivative size = 134, 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} \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {7 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a^2 \csc (c+d x)}{d}-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {4 a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \csc ^3(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^3}{(a-x)^3 x^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^8 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^8 \operatorname {Subst}\left (\int \left (\frac {1}{2 a^4 (a-x)^3}+\frac {7}{4 a^5 (a-x)^2}+\frac {31}{8 a^6 (a-x)}+\frac {1}{a^4 x^3}+\frac {2}{a^5 x^2}+\frac {4}{a^6 x}-\frac {1}{8 a^6 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {4 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 {7 a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.13, size = 84, normalized size = 0.63 \[ -\frac {a^2 \left (\frac {14}{\sin (c+d x)-1}-\frac {2}{(\sin (c+d x)-1)^2}+4 \csc ^2(c+d x)+16 \csc (c+d x)+31 \log (1-\sin (c+d x))-32 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 302, normalized size = 2.25 \[ -\frac {44 \, a^{2} \cos \left (d x + c\right )^{2} - 40 \, a^{2} - 32 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 31 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + 2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, a^{2} \cos \left (d x + c\right )^{2} - 19 \, a^{2}\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{4} - 3 \, d \cos \left (d x + c\right )^{2} + 2 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 125, normalized size = 0.93 \[ -\frac {4 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 124 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 128 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {3 \, a^{2} \sin \left (d x + c\right )^{4} + 114 \, a^{2} \sin \left (d x + c\right )^{3} - 173 \, a^{2} \sin \left (d x + c\right )^{2} + 32 \, a^{2} \sin \left (d x + c\right ) + 16 \, a^{2}}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 199, normalized size = 1.49 \[ \frac {a^{2}}{4 d \cos \left (d x +c \right )^{4}}+\frac {a^{2}}{2 d \cos \left (d x +c \right )^{2}}+\frac {4 a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a^{2}}{2 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a^{2}}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a^{2}}{4 d \sin \left (d x +c \right )}+\frac {15 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3 a^{2}}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3 a^{2}}{2 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 119, normalized size = 0.89 \[ -\frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 31 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - 32 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a^{2} \sin \left (d x + c\right )^{3} - 22 \, a^{2} \sin \left (d x + c\right )^{2} + 4 \, a^{2} \sin \left (d x + c\right ) + 2 \, a^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 126, normalized size = 0.94 \[ \frac {4\,a^2\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {a^2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{8\,d}-\frac {\frac {15\,a^2\,{\sin \left (c+d\,x\right )}^3}{4}-\frac {11\,a^2\,{\sin \left (c+d\,x\right )}^2}{2}+a^2\,\sin \left (c+d\,x\right )+\frac {a^2}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2\right )}-\frac {31\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{8\,d} \]
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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