Optimal. Leaf size=118 \[ -\frac {a^3}{8 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^2}{8 d (a \cos (c+d x)+a)}+\frac {11 a \log (1-\cos (c+d x))}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log (\cos (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.12, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2836, 12, 88} \[ -\frac {a^3}{8 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^2}{8 d (a \cos (c+d x)+a)}+\frac {11 a \log (1-\cos (c+d x))}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log (\cos (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^5(c+d x) \sec (c+d x) \, dx\\ &=\frac {a^5 \operatorname {Subst}\left (\int \frac {a}{(-a-x)^3 x (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{(-a-x)^3 x (-a+x)^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^6 \operatorname {Subst}\left (\int \left (-\frac {1}{8 a^4 (a-x)^2}-\frac {5}{16 a^5 (a-x)}-\frac {1}{a^5 x}+\frac {1}{4 a^3 (a+x)^3}+\frac {1}{2 a^4 (a+x)^2}+\frac {11}{16 a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^3}{8 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^2}{8 d (a+a \cos (c+d x))}+\frac {11 a \log (1-\cos (c+d x))}{16 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {5 a \log (1+\cos (c+d x))}{16 d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 164, normalized size = 1.39 \[ -\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {a \left (\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 193, normalized size = 1.64 \[ \frac {6 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - 16 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\cos \left (d x + c\right )\right ) + 5 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 11 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 149, normalized size = 1.26 \[ \frac {22 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 32 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {{\left (a - \frac {10 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {33 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 80, normalized size = 0.68 \[ -\frac {a}{8 d \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {3 a}{4 d \left (-1+\sec \left (d x +c \right )\right )}+\frac {11 a \ln \left (-1+\sec \left (d x +c \right )\right )}{16 d}+\frac {a}{8 d \left (1+\sec \left (d x +c \right )\right )}+\frac {5 a \ln \left (1+\sec \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 95, normalized size = 0.81 \[ \frac {5 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 11 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 16 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 99, normalized size = 0.84 \[ \frac {11\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {5\,a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {\frac {3\,a\,{\cos \left (c+d\,x\right )}^2}{8}+\frac {a\,\cos \left (c+d\,x\right )}{8}-\frac {3\,a}{4}}{d\,\left ({\cos \left (c+d\,x\right )}^3-\cos \left (c+d\,x\right )+{\sin \left (c+d\,x\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \csc ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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