Optimal. Leaf size=129 \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \csc (c+d x)}{d}-\frac {23 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {7 a \log (\sin (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.12, antiderivative size = 129, 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} \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \csc (c+d x)}{d}-\frac {23 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {7 a \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a^2}{(a-x)^3 x^2 (a+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 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^7 \operatorname {Subst}\left (\int \left (\frac {1}{4 a^4 (a-x)^3}+\frac {3}{4 a^5 (a-x)^2}+\frac {23}{16 a^6 (a-x)}+\frac {1}{a^5 x^2}+\frac {1}{a^6 x}+\frac {1}{8 a^5 (a+x)^2}+\frac {7}{16 a^6 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}-\frac {23 a \log (1-\sin (c+d x))}{16 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {7 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {3 a^2}{4 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 = 0.19, size = 76, normalized size = 0.59 \[ -\frac {a \csc (c+d x) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\sin ^2(c+d x)\right )}{d}-\frac {a \left (-\sec ^4(c+d x)-2 \sec ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 229, normalized size = 1.78 \[ -\frac {22 \, a \cos \left (d x + c\right )^{2} - 16 \, {\left (a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 7 \, {\left (a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 23 \, {\left (a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - 6 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 121, normalized size = 0.94 \[ \frac {14 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 46 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 32 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {23 \, a \sin \left (d x + c\right )^{2} + 59 \, a \sin \left (d x + c\right ) + 32 \, a}{\sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )} + \frac {69 \, a \sin \left (d x + c\right )^{2} - 162 \, a \sin \left (d x + c\right ) + 97 \, a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 120, normalized size = 0.93 \[ \frac {a}{4 d \cos \left (d x +c \right )^{4}}+\frac {a}{2 d \cos \left (d x +c \right )^{2}}+\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {a}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a}{8 d \sin \left (d x +c \right )}+\frac {15 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 114, normalized size = 0.88 \[ \frac {7 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 23 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (15 \, a \sin \left (d x + c\right )^{3} - 11 \, a \sin \left (d x + c\right )^{2} - 14 \, a \sin \left (d x + c\right ) + 8 \, a\right )}}{\sin \left (d x + c\right )^{4} - \sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 118, normalized size = 0.91 \[ \frac {a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {15\,a\,{\sin \left (c+d\,x\right )}^3}{8}-\frac {11\,a\,{\sin \left (c+d\,x\right )}^2}{8}-\frac {7\,a\,\sin \left (c+d\,x\right )}{4}+a}{d\,\left ({\sin \left (c+d\,x\right )}^4-{\sin \left (c+d\,x\right )}^3-{\sin \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\right )}-\frac {23\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{16\,d}+\frac {7\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{16\,d} \]
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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