Optimal. Leaf size=106 \[ -\frac {\tanh ^{-1}(\cos (c+d x))}{16 a d}-\frac {\cot (c+d x) \csc (c+d x)}{16 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}+\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac {\csc ^6(c+d x)}{6 a d} \]
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Rubi [A]
time = 0.13, 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 = {3957, 2914,
2686, 30, 2691, 3853, 3855} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2691
Rule 2914
Rule 3853
Rule 3855
Rule 3957
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 {\text {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 = 0.32, size = 122, normalized size = 1.15 \begin {gather*} -\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3 \csc ^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 )+3 \sec ^4\left (\frac {1}{2} (c+d x)\right )+2 \sec ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x)}{192 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 79, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {-\frac {1}{32 \left (-1+\cos \left (d x +c \right )\right )^{2}}+\frac {1}{-16+16 \cos \left (d x +c \right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{32}-\frac {1}{24 \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {1}{32 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{32}}{d a}\) | \(79\) |
default | \(\frac {-\frac {1}{32 \left (-1+\cos \left (d x +c \right )\right )^{2}}+\frac {1}{-16+16 \cos \left (d x +c \right )}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{32}-\frac {1}{24 \left (1+\cos \left (d x +c \right )\right )^{3}}-\frac {1}{32 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{32}}{d a}\) | \(79\) |
norman | \(\frac {-\frac {1}{128 a d}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a d}-\frac {3 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{192 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}\) | \(117\) |
risch | \(\frac {3 \,{\mathrm e}^{9 i \left (d x +c \right )}+6 \,{\mathrm e}^{8 i \left (d x +c \right )}-8 \,{\mathrm e}^{7 i \left (d x +c \right )}-22 \,{\mathrm e}^{6 i \left (d x +c \right )}-150 \,{\mathrm e}^{5 i \left (d x +c \right )}-22 \,{\mathrm e}^{4 i \left (d x +c \right )}-8 \,{\mathrm e}^{3 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}{24 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 a d}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 130, normalized size = 1.23 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 217 vs.
\(2 (96) = 192\).
time = 2.67, size = 217, normalized size = 2.05 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\csc ^{5}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 182, normalized size = 1.72 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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