Optimal. Leaf size=163 \[ -\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 163, 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 = {3957, 2915, 12,
90} \begin {gather*} -\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^3}{32 d (a \cos (c+d x)+a)^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {3 a^2}{16 d (a \cos (c+d x)+a)}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (\cos (c+d x)+1)}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=\frac {a^7 \text {Subst}\left (\int \frac {a}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^8 \text {Subst}\left (\int \frac {1}{(-a-x)^4 x (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^8 \text {Subst}\left (\int \left (-\frac {1}{16 a^5 (a-x)^3}-\frac {3}{16 a^6 (a-x)^2}-\frac {11}{32 a^7 (a-x)}-\frac {1}{a^7 x}+\frac {1}{8 a^4 (a+x)^4}+\frac {5}{16 a^5 (a+x)^3}+\frac {1}{2 a^6 (a+x)^2}+\frac {21}{32 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^4}{24 d (a-a \cos (c+d x))^3}-\frac {5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac {a^2}{2 d (a-a \cos (c+d x))}-\frac {a^3}{32 d (a+a \cos (c+d x))^2}-\frac {3 a^2}{16 d (a+a \cos (c+d x))}+\frac {21 a \log (1-\cos (c+d x))}{32 d}-\frac {a \log (\cos (c+d x))}{d}+\frac {11 a \log (1+\cos (c+d x))}{32 d}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 165, normalized size = 1.01 \begin {gather*} -\frac {a \left (30 \csc ^2\left (\frac {1}{2} (c+d x)\right )+6 \csc ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right )+192 \csc ^2(c+d x)+96 \csc ^4(c+d x)+64 \csc ^6(c+d x)+120 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+384 \log (\cos (c+d x))-120 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-384 \log (\sin (c+d x))-30 \sec ^2\left (\frac {1}{2} (c+d x)\right )-6 \sec ^4\left (\frac {1}{2} (c+d x)\right )-\sec ^6\left (\frac {1}{2} (c+d x)\right )\right )}{384 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 89, normalized size = 0.55
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {1}{24 \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {9}{32 \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {15}{16 \left (-1+\sec \left (d x +c \right )\right )}+\frac {21 \ln \left (-1+\sec \left (d x +c \right )\right )}{32}-\frac {1}{32 \left (1+\sec \left (d x +c \right )\right )^{2}}+\frac {1}{4+4 \sec \left (d x +c \right )}+\frac {11 \ln \left (1+\sec \left (d x +c \right )\right )}{32}\right )}{d}\) | \(89\) |
default | \(\frac {a \left (-\frac {1}{24 \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {9}{32 \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {15}{16 \left (-1+\sec \left (d x +c \right )\right )}+\frac {21 \ln \left (-1+\sec \left (d x +c \right )\right )}{32}-\frac {1}{32 \left (1+\sec \left (d x +c \right )\right )^{2}}+\frac {1}{4+4 \sec \left (d x +c \right )}+\frac {11 \ln \left (1+\sec \left (d x +c \right )\right )}{32}\right )}{d}\) | \(89\) |
norman | \(\frac {-\frac {a}{192 d}-\frac {7 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}-\frac {11 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {7 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {21 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(141\) |
risch | \(\frac {a \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}+18 \,{\mathrm e}^{8 i \left (d x +c \right )}-136 \,{\mathrm e}^{7 i \left (d x +c \right )}-34 \,{\mathrm e}^{6 i \left (d x +c \right )}+402 \,{\mathrm e}^{5 i \left (d x +c \right )}-34 \,{\mathrm e}^{4 i \left (d x +c \right )}-136 \,{\mathrm e}^{3 i \left (d x +c \right )}+18 \,{\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{24 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4}}+\frac {21 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(188\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 136, normalized size = 0.83 \begin {gather*} \frac {33 \, a \log \left (\cos \left (d x + c\right ) + 1\right ) + 63 \, a \log \left (\cos \left (d x + c\right ) - 1\right ) - 96 \, a \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (15 \, a \cos \left (d x + c\right )^{4} + 9 \, a \cos \left (d x + c\right )^{3} - 49 \, a \cos \left (d x + c\right )^{2} - 11 \, a \cos \left (d x + c\right ) + 44 \, a\right )}}{\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}}{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. 307 vs.
\(2 (152) = 304\).
time = 5.72, size = 307, normalized size = 1.88 \begin {gather*} \frac {30 \, a \cos \left (d x + c\right )^{4} + 18 \, a \cos \left (d x + c\right )^{3} - 98 \, a \cos \left (d x + c\right )^{2} - 22 \, a \cos \left (d x + c\right ) - 96 \, {\left (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\right )} \log \left (-\cos \left (d x + c\right )\right ) + 33 \, {\left (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\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 63 \, {\left (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\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 88 \, a}{96 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 196, normalized size = 1.20 \begin {gather*} \frac {252 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, a \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 - \frac {21 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {132 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {462 \, a {\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 {42 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 142, normalized size = 0.87 \begin {gather*} \frac {\frac {5\,a\,{\cos \left (c+d\,x\right )}^4}{16}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^3}{16}-\frac {49\,a\,{\cos \left (c+d\,x\right )}^2}{48}-\frac {11\,a\,\cos \left (c+d\,x\right )}{48}+\frac {11\,a}{12}}{d\,\left ({\cos \left (c+d\,x\right )}^5-{\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+2\,{\cos \left (c+d\,x\right )}^2+\cos \left (c+d\,x\right )-1\right )}-\frac {a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d}+\frac {21\,a\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{32\,d}+\frac {11\,a\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{32\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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