Optimal. Leaf size=253 \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{8 d (a \sin (c+d x)+a)^3}+\frac {13 a}{128 d (a-a \sin (c+d x))^2}-\frac {41 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {5 \csc (c+d x)}{a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (\sin (c+d x)+1)}{256 a d} \]
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Rubi [A] time = 0.26, antiderivative size = 253, 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} \[ -\frac {a^3}{64 d (a \sin (c+d x)+a)^4}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}-\frac {a^2}{8 d (a \sin (c+d x)+a)^3}+\frac {13 a}{128 d (a-a \sin (c+d x))^2}-\frac {41 a}{64 d (a \sin (c+d x)+a)^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {105}{32 d (a \sin (c+d x)+a)}-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {5 \csc (c+d x)}{a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (\sin (c+d x)+1)}{256 a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x) \sec ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {a^7 \operatorname {Subst}\left (\int \frac {a^4}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{11} \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 x^4 (a+x)^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^{11} \operatorname {Subst}\left (\int \left (\frac {1}{32 a^9 (a-x)^4}+\frac {13}{64 a^{10} (a-x)^3}+\frac {95}{128 a^{11} (a-x)^2}+\frac {515}{256 a^{12} (a-x)}+\frac {1}{a^9 x^4}-\frac {1}{a^{10} x^3}+\frac {5}{a^{11} x^2}-\frac {5}{a^{12} x}+\frac {1}{16 a^8 (a+x)^5}+\frac {3}{8 a^9 (a+x)^4}+\frac {41}{32 a^{10} (a+x)^3}+\frac {105}{32 a^{11} (a+x)^2}+\frac {1795}{256 a^{12} (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {5 \csc (c+d x)}{a d}+\frac {\csc ^2(c+d x)}{2 a d}-\frac {\csc ^3(c+d x)}{3 a d}-\frac {515 \log (1-\sin (c+d x))}{256 a d}-\frac {5 \log (\sin (c+d x))}{a d}+\frac {1795 \log (1+\sin (c+d x))}{256 a d}+\frac {a^2}{96 d (a-a \sin (c+d x))^3}+\frac {13 a}{128 d (a-a \sin (c+d x))^2}+\frac {95}{128 d (a-a \sin (c+d x))}-\frac {a^3}{64 d (a+a \sin (c+d x))^4}-\frac {a^2}{8 d (a+a \sin (c+d x))^3}-\frac {41 a}{64 d (a+a \sin (c+d x))^2}-\frac {105}{32 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.14, size = 231, normalized size = 0.91 \[ \frac {a^{11} \left (-\frac {\csc ^3(c+d x)}{3 a^{12}}+\frac {\csc ^2(c+d x)}{2 a^{12}}-\frac {5 \csc (c+d x)}{a^{12}}-\frac {515 \log (1-\sin (c+d x))}{256 a^{12}}-\frac {5 \log (\sin (c+d x))}{a^{12}}+\frac {1795 \log (\sin (c+d x)+1)}{256 a^{12}}+\frac {95}{128 a^{11} (a-a \sin (c+d x))}-\frac {105}{32 a^{11} (a \sin (c+d x)+a)}+\frac {13}{128 a^{10} (a-a \sin (c+d x))^2}-\frac {41}{64 a^{10} (a \sin (c+d x)+a)^2}+\frac {1}{96 a^9 (a-a \sin (c+d x))^3}-\frac {1}{8 a^9 (a \sin (c+d x)+a)^3}-\frac {1}{64 a^8 (a \sin (c+d x)+a)^4}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 360, normalized size = 1.42 \[ \frac {5010 \, \cos \left (d x + c\right )^{8} - 6360 \, \cos \left (d x + c\right )^{6} + 746 \, \cos \left (d x + c\right )^{4} + 236 \, \cos \left (d x + c\right )^{2} - 3840 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 5385 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 1545 \, {\left (\cos \left (d x + c\right )^{10} - 2 \, \cos \left (d x + c\right )^{8} + \cos \left (d x + c\right )^{6} - {\left (\cos \left (d x + c\right )^{8} - \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3465 \, \cos \left (d x + c\right )^{8} - 3660 \, \cos \left (d x + c\right )^{6} + 213 \, \cos \left (d x + c\right )^{4} + 38 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 112}{768 \, {\left (a d \cos \left (d x + c\right )^{10} - 2 \, a d \cos \left (d x + c\right )^{8} + a d \cos \left (d x + c\right )^{6} - {\left (a d \cos \left (d x + c\right )^{8} - a d \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 187, normalized size = 0.74 \[ \frac {\frac {21540 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {6180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} - \frac {15360 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {19745 \, \sin \left (d x + c\right )^{6} - 76875 \, \sin \left (d x + c\right )^{5} + 111723 \, \sin \left (d x + c\right )^{4} - 74081 \, \sin \left (d x + c\right )^{3} + 23040 \, \sin \left (d x + c\right )^{2} - 4608 \, \sin \left (d x + c\right ) + 1024}{{\left (\sin \left (d x + c\right )^{2} - \sin \left (d x + c\right )\right )}^{3} a} - \frac {44875 \, \sin \left (d x + c\right )^{4} + 189580 \, \sin \left (d x + c\right )^{3} + 301458 \, \sin \left (d x + c\right )^{2} + 214060 \, \sin \left (d x + c\right ) + 57355}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 225, normalized size = 0.89 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}+\frac {13}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {95}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {515 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{3 a d \sin \left (d x +c \right )^{3}}+\frac {1}{2 a d \sin \left (d x +c \right )^{2}}-\frac {5}{d a \sin \left (d x +c \right )}-\frac {5 \ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{8 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {41}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {105}{32 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {1795 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 227, normalized size = 0.90 \[ -\frac {\frac {2 \, {\left (3465 \, \sin \left (d x + c\right )^{9} + 2505 \, \sin \left (d x + c\right )^{8} - 10200 \, \sin \left (d x + c\right )^{7} - 6840 \, \sin \left (d x + c\right )^{6} + 10023 \, \sin \left (d x + c\right )^{5} + 5863 \, \sin \left (d x + c\right )^{4} - 3344 \, \sin \left (d x + c\right )^{3} - 1344 \, \sin \left (d x + c\right )^{2} + 64 \, \sin \left (d x + c\right ) - 128\right )}}{a \sin \left (d x + c\right )^{10} + a \sin \left (d x + c\right )^{9} - 3 \, a \sin \left (d x + c\right )^{8} - 3 \, a \sin \left (d x + c\right )^{7} + 3 \, a \sin \left (d x + c\right )^{6} + 3 \, a \sin \left (d x + c\right )^{5} - a \sin \left (d x + c\right )^{4} - a \sin \left (d x + c\right )^{3}} - \frac {5385 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {1545 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a} + \frac {3840 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.26, size = 233, normalized size = 0.92 \[ \frac {\frac {1155\,{\sin \left (c+d\,x\right )}^9}{128}+\frac {835\,{\sin \left (c+d\,x\right )}^8}{128}-\frac {425\,{\sin \left (c+d\,x\right )}^7}{16}-\frac {285\,{\sin \left (c+d\,x\right )}^6}{16}+\frac {3341\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {5863\,{\sin \left (c+d\,x\right )}^4}{384}-\frac {209\,{\sin \left (c+d\,x\right )}^3}{24}-\frac {7\,{\sin \left (c+d\,x\right )}^2}{2}+\frac {\sin \left (c+d\,x\right )}{6}-\frac {1}{3}}{d\,\left (-a\,{\sin \left (c+d\,x\right )}^{10}-a\,{\sin \left (c+d\,x\right )}^9+3\,a\,{\sin \left (c+d\,x\right )}^8+3\,a\,{\sin \left (c+d\,x\right )}^7-3\,a\,{\sin \left (c+d\,x\right )}^6-3\,a\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4+a\,{\sin \left (c+d\,x\right )}^3\right )}-\frac {515\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{256\,a\,d}+\frac {1795\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{256\,a\,d}-\frac {5\,\ln \left (\sin \left (c+d\,x\right )\right )}{a\,d} \]
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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