Optimal. Leaf size=143 \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc (c+d x)}{d}-\frac {39 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {9 a \log (\sin (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.13, antiderivative size = 143, 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, 88} \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc (c+d x)}{d}-\frac {39 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {9 a \log (\sin (c+d x)+1)}{16 d} \]
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)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a^3}{(a-x)^3 x^3 (a+x)^2} \, 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)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^8 \operatorname {Subst}\left (\int \left (\frac {1}{4 a^5 (a-x)^3}+\frac {1}{a^6 (a-x)^2}+\frac {39}{16 a^7 (a-x)}+\frac {1}{a^5 x^3}+\frac {1}{a^6 x^2}+\frac {3}{a^7 x}-\frac {1}{8 a^6 (a+x)^2}-\frac {9}{16 a^7 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {39 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}-\frac {9 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {a^2}{d (a-a \sin (c+d x))}+\frac {a^2}{8 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 86, normalized size = 0.60 \[ -\frac {a \csc (c+d x) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\sin ^2(c+d x)\right )}{d}-\frac {a \left (2 \csc ^2(c+d x)-\sec ^4(c+d x)-4 \sec ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 294, normalized size = 2.06 \[ \frac {30 \, a \cos \left (d x + c\right )^{4} - 16 \, a \cos \left (d x + c\right )^{2} + 48 \, {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 39 \, {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) - 6 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 125, normalized size = 0.87 \[ -\frac {36 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 156 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - 192 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {4 \, {\left (9 \, a \sin \left (d x + c\right ) + 11 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {27 \, a \sin \left (d x + c\right )^{4} + 74 \, a \sin \left (d x + c\right )^{3} - 141 \, a \sin \left (d x + c\right )^{2} + 32 \, a}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{2}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 151, normalized size = 1.06 \[ \frac {a}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a}{8 d \sin \left (d x +c \right )}+\frac {15 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3 a}{4 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3 a}{2 d \sin \left (d x +c \right )^{2}}+\frac {3 a \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 127, normalized size = 0.89 \[ -\frac {9 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) + 39 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - 48 \, a \log \left (\sin \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a \sin \left (d x + c\right )^{4} - 3 \, a \sin \left (d x + c\right )^{3} - 22 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 4 \, a\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} - \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.31, size = 134, normalized size = 0.94 \[ \frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {39\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{16\,d}-\frac {9\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{16\,d}-\frac {\frac {15\,a\,{\sin \left (c+d\,x\right )}^4}{8}-\frac {3\,a\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {11\,a\,{\sin \left (c+d\,x\right )}^2}{4}+\frac {a\,\sin \left (c+d\,x\right )}{2}+\frac {a}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^5-{\sin \left (c+d\,x\right )}^4-{\sin \left (c+d\,x\right )}^3+{\sin \left (c+d\,x\right )}^2\right )} \]
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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