Optimal. Leaf size=157 \[ -\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (\cos (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.20, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 88} \[ -\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (\cos (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^7(c+d x) \sec ^3(c+d x) \, dx\\ &=\frac {a^7 \operatorname {Subst}\left (\int \frac {a^3}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{10} \operatorname {Subst}\left (\int \frac {1}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{10} \operatorname {Subst}\left (\int \left (-\frac {1}{16 a^7 (a-x)}-\frac {1}{a^5 x^3}+\frac {3}{a^6 x^2}-\frac {7}{a^7 x}+\frac {1}{2 a^4 (a+x)^4}+\frac {7}{4 a^5 (a+x)^3}+\frac {31}{8 a^6 (a+x)^2}+\frac {111}{16 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.06, size = 129, normalized size = 0.82 \[ -\frac {a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 \csc ^6\left (\frac {1}{2} (c+d x)\right )+21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+186 \csc ^2\left (\frac {1}{2} (c+d x)\right )-12 \left (4 \sec ^2(c+d x)+24 \sec (c+d x)+111 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-56 \log (\cos (c+d x))\right )\right )}{768 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 297, normalized size = 1.89 \[ \frac {330 \, a^{3} \cos \left (d x + c\right )^{4} - 822 \, a^{3} \cos \left (d x + c\right )^{3} + 596 \, a^{3} \cos \left (d x + c\right )^{2} - 72 \, a^{3} \cos \left (d x + c\right ) - 24 \, a^{3} - 336 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\cos \left (d x + c\right )\right ) + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 333 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 243, normalized size = 1.55 \[ \frac {666 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 672 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (2 \, a^{3} - \frac {27 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {234 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1221 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} + \frac {48 \, {\left (33 \, a^{3} + \frac {50 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 120, normalized size = 0.76 \[ \frac {a^{3} \left (\sec ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} \sec \left (d x +c \right )}{d}-\frac {a^{3}}{6 d \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {11 a^{3}}{8 d \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {49 a^{3}}{8 d \left (-1+\sec \left (d x +c \right )\right )}+\frac {111 a^{3} \ln \left (-1+\sec \left (d x +c \right )\right )}{16 d}+\frac {a^{3} \ln \left (1+\sec \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 0.92 \[ \frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 333 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 336 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (165 \, a^{3} \cos \left (d x + c\right )^{4} - 411 \, a^{3} \cos \left (d x + c\right )^{3} + 298 \, a^{3} \cos \left (d x + c\right )^{2} - 36 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 151, normalized size = 0.96 \[ \frac {111\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}+\frac {a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {-\frac {55\,a^3\,{\cos \left (c+d\,x\right )}^4}{8}+\frac {137\,a^3\,{\cos \left (c+d\,x\right )}^3}{8}-\frac {149\,a^3\,{\cos \left (c+d\,x\right )}^2}{12}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left (-{\cos \left (c+d\,x\right )}^5+3\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {7\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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