Optimal. Leaf size=101 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{4 d (a-a \sin (c+d x))}-\frac {7 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
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Rubi [A] time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 72} \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}+\frac {3 a^3}{4 d (a-a \sin (c+d x))}-\frac {7 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 72
Rule 2836
Rubi steps
\begin {align*} \int \csc (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}{(a-x)^3 x (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 x (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \left (\frac {1}{2 a^2 (a-x)^3}+\frac {3}{4 a^3 (a-x)^2}+\frac {7}{8 a^4 (a-x)}+\frac {1}{a^4 x}-\frac {1}{8 a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {7 a^2 \log (1-\sin (c+d x))}{8 d}+\frac {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 {3 a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 66, normalized size = 0.65 \[ -\frac {a^2 \left (\frac {6}{\sin (c+d x)-1}-\frac {2}{(\sin (c+d x)-1)^2}+7 \log (1-\sin (c+d x))-8 \log (\sin (c+d x))+\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 166, normalized size = 1.64 \[ \frac {6 \, a^{2} \sin \left (d x + c\right ) - 8 \, a^{2} + 8 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 7 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 91, normalized size = 0.90 \[ -\frac {2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 14 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 16 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {21 \, a^{2} \sin \left (d x + c\right )^{2} - 54 \, a^{2} \sin \left (d x + c\right ) + 37 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 112, normalized size = 1.11 \[ \frac {a^{2}}{2 d \cos \left (d x +c \right )^{4}}+\frac {a^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{4 d}+\frac {3 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2}}{2 d \cos \left (d x +c \right )^{2}}+\frac {a^{2} \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 84, normalized size = 0.83 \[ -\frac {a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 7 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - 8 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, a^{2} \sin \left (d x + c\right ) - 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 91, normalized size = 0.90 \[ \frac {a^2\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {a^2\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{8\,d}-\frac {\frac {3\,a^2\,\sin \left (c+d\,x\right )}{4}-a^2}{d\,\left ({\sin \left (c+d\,x\right )}^2-2\,\sin \left (c+d\,x\right )+1\right )}-\frac {7\,a^2\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{8\,d} \]
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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