Optimal. Leaf size=111 \[ -\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Rubi [A]
time = 0.12, antiderivative size = 111, 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 = {3957, 2915, 12,
46} \begin {gather*} -\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{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 {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 46
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^5(c+d x) \sec ^3(c+d x) \, dx\\ &=\frac {a^5 \text {Subst}\left (\int \frac {a^3}{(-a-x)^3 x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^8 \text {Subst}\left (\int \frac {1}{(-a-x)^3 x^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^8 \text {Subst}\left (\int \left (-\frac {1}{a^3 x^3}+\frac {3}{a^4 x^2}-\frac {6}{a^5 x}+\frac {1}{a^3 (a+x)^3}+\frac {3}{a^4 (a+x)^2}+\frac {6}{a^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^5}{2 d (a-a \cos (c+d x))^2}-\frac {3 a^4}{d (a-a \cos (c+d x))}+\frac {6 a^3 \log (1-\cos (c+d x))}{d}-\frac {6 a^3 \log (\cos (c+d x))}{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 = 0.63, size = 100, normalized size = 0.90 \begin {gather*} -\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (12 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right )+48 \left (\log (\cos (c+d x))-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-24 \sec (c+d x)-4 \sec ^2(c+d x)\right )}{64 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 63, normalized size = 0.57
method | result | size |
derivativedivides | \(-\frac {a^{3} \left (-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\frac {4}{-1+\sec \left (d x +c \right )}-6 \ln \left (-1+\sec \left (d x +c \right )\right )+\frac {1}{2 \left (-1+\sec \left (d x +c \right )\right )^{2}}\right )}{d}\) | \(63\) |
default | \(-\frac {a^{3} \left (-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\frac {4}{-1+\sec \left (d x +c \right )}-6 \ln \left (-1+\sec \left (d x +c \right )\right )+\frac {1}{2 \left (-1+\sec \left (d x +c \right )\right )^{2}}\right )}{d}\) | \(63\) |
norman | \(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {23 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {75 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {12 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {6 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(153\) |
risch | \(\frac {4 a^{3} \left (3 \,{\mathrm e}^{7 i \left (d x +c \right )}-9 \,{\mathrm e}^{6 i \left (d x +c \right )}+13 \,{\mathrm e}^{5 i \left (d x +c \right )}-16 \,{\mathrm e}^{4 i \left (d x +c \right )}+13 \,{\mathrm e}^{3 i \left (d x +c \right )}-9 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {12 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {6 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 103, normalized size = 0.93 \begin {gather*} \frac {12 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.17, size = 177, normalized size = 1.59 \begin {gather*} \frac {12 \, a^{3} \cos \left (d x + c\right )^{3} - 18 \, a^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{3} \cos \left (d x + c\right ) + a^{3} - 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, 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 ) + 12 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, 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 )}{2 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\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 = 0.54, size = 186, normalized size = 1.68 \begin {gather*} \frac {48 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 48 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {a^{3} - \frac {12 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {75 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {46 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 96, normalized size = 0.86 \begin {gather*} \frac {6\,a^3\,{\cos \left (c+d\,x\right )}^3-9\,a^3\,{\cos \left (c+d\,x\right )}^2+2\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{d\,\left ({\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {12\,a^3\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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