Optimal. Leaf size=106 \[ -\frac {\csc ^6(c+d x)}{6 a d}-\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc (c+d x)}{16 a d} \]
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Rubi [A] time = 0.17, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2835, 2606, 30, 2611, 3768, 3770} \[ -\frac {\csc ^6(c+d x)}{6 a d}-\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc (c+d x)}{16 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^4(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^6(c+d x) \, dx}{a}\\ &=\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}+\frac {\int \csc ^5(c+d x) \, dx}{6 a}-\frac {\operatorname {Subst}\left (\int x^5 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {\int \csc ^3(c+d x) \, dx}{8 a}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{16 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\csc ^6(c+d x)}{6 a d}+\frac {\int \csc (c+d x) \, dx}{16 a}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot (c+d x) \csc (c+d x)}{16 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\csc ^6(c+d x)}{6 a d}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 122, normalized size = 1.15 \[ -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (3 \csc ^4\left (\frac {1}{2} (c+d x)\right )+12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+2 \sec ^6\left (\frac {1}{2} (c+d x)\right )+3 \sec ^4\left (\frac {1}{2} (c+d x)\right )+24 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{192 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 217, normalized size = 2.05 \[ \frac {6 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} - 3 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 10 \, \cos \left (d x + c\right ) - 16}{96 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 182, normalized size = 1.72 \[ \frac {\frac {3 \, {\left (\frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} + \frac {12 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a} + \frac {\frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {9 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 108, normalized size = 1.02 \[ -\frac {1}{32 a d \left (-1+\cos \left (d x +c \right )\right )^{2}}+\frac {1}{16 a d \left (-1+\cos \left (d x +c \right )\right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{32 a d}-\frac {1}{24 a d \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {1}{32 a d \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{32 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 130, normalized size = 1.23 \[ \frac {\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) - 8\right )}}{a \cos \left (d x + c\right )^{5} + a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) + a} - \frac {3 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a} + \frac {3 \, \log \left (\cos \left (d x + c\right ) - 1\right )}{a}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 115, normalized size = 1.08 \[ -\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{16\,a\,d}-\frac {-\frac {{\cos \left (c+d\,x\right )}^4}{16}-\frac {{\cos \left (c+d\,x\right )}^3}{16}+\frac {5\,{\cos \left (c+d\,x\right )}^2}{48}+\frac {5\,\cos \left (c+d\,x\right )}{48}+\frac {1}{6}}{d\,\left (a\,{\cos \left (c+d\,x\right )}^5+a\,{\cos \left (c+d\,x\right )}^4-2\,a\,{\cos \left (c+d\,x\right )}^3-2\,a\,{\cos \left (c+d\,x\right )}^2+a\,\cos \left (c+d\,x\right )+a\right )} \]
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 \[ \frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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