Optimal. Leaf size=157 \[ -\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 157, 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,
90} \begin {gather*} -\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 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 {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (\cos (c+d x)+1)}{16 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))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^7(c+d x) \sec ^3(c+d x) \, dx\\ &=\frac {a^7 \text {Subst}\left (\int \frac {a^3}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{10} \text {Subst}\left (\int \frac {1}{(-a-x)^4 x^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{10} \text {Subst}\left (\int \left (-\frac {1}{16 a^7 (a-x)}-\frac {1}{a^5 x^3}+\frac {3}{a^6 x^2}-\frac {7}{a^7 x}+\frac {1}{2 a^4 (a+x)^4}+\frac {7}{4 a^5 (a+x)^3}+\frac {31}{8 a^6 (a+x)^2}+\frac {111}{16 a^7 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^6}{6 d (a-a \cos (c+d x))^3}-\frac {7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac {31 a^4}{8 d (a-a \cos (c+d x))}+\frac {111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {7 a^3 \log (\cos (c+d x))}{d}+\frac {a^3 \log (1+\cos (c+d x))}{16 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 = 0.71, size = 129, normalized size = 0.82 \begin {gather*} -\frac {a^3 (1+\cos (c+d x))^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (186 \csc ^2\left (\frac {1}{2} (c+d x)\right )+21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+2 \csc ^6\left (\frac {1}{2} (c+d x)\right )-12 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-56 \log (\cos (c+d x))+111 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \sec (c+d x)+4 \sec ^2(c+d x)\right )\right )}{768 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 85, normalized size = 0.54
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{2}+3 \sec \left (d x +c \right )-\frac {1}{6 \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {11}{8 \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {49}{8 \left (-1+\sec \left (d x +c \right )\right )}+\frac {111 \ln \left (-1+\sec \left (d x +c \right )\right )}{16}+\frac {\ln \left (1+\sec \left (d x +c \right )\right )}{16}\right )}{d}\) | \(85\) |
default | \(\frac {a^{3} \left (\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{2}+3 \sec \left (d x +c \right )-\frac {1}{6 \left (-1+\sec \left (d x +c \right )\right )^{3}}-\frac {11}{8 \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {49}{8 \left (-1+\sec \left (d x +c \right )\right )}+\frac {111 \ln \left (-1+\sec \left (d x +c \right )\right )}{16}+\frac {\ln \left (1+\sec \left (d x +c \right )\right )}{16}\right )}{d}\) | \(85\) |
norman | \(\frac {-\frac {a^{3}}{48 d}-\frac {23 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {91 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{48 d}-\frac {103 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {339 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {111 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {7 a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(172\) |
risch | \(\frac {a^{3} \left (165 \,{\mathrm e}^{9 i \left (d x +c \right )}-822 \,{\mathrm e}^{8 i \left (d x +c \right )}+1852 \,{\mathrm e}^{7 i \left (d x +c \right )}-2754 \,{\mathrm e}^{6 i \left (d x +c \right )}+3182 \,{\mathrm e}^{5 i \left (d x +c \right )}-2754 \,{\mathrm e}^{4 i \left (d x +c \right )}+1852 \,{\mathrm e}^{3 i \left (d x +c \right )}-822 \,{\mathrm e}^{2 i \left (d x +c \right )}+165 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}+\frac {111 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}-\frac {7 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(196\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 145, normalized size = 0.92 \begin {gather*} \frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 333 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - 336 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, {\left (165 \, a^{3} \cos \left (d x + c\right )^{4} - 411 \, a^{3} \cos \left (d x + c\right )^{3} + 298 \, a^{3} \cos \left (d x + c\right )^{2} - 36 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3}\right )}}{\cos \left (d x + c\right )^{5} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2}}}{48 \, 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. 297 vs.
\(2 (148) = 296\).
time = 4.61, size = 297, normalized size = 1.89 \begin {gather*} \frac {330 \, a^{3} \cos \left (d x + c\right )^{4} - 822 \, a^{3} \cos \left (d x + c\right )^{3} + 596 \, a^{3} \cos \left (d x + c\right )^{2} - 72 \, a^{3} \cos \left (d x + c\right ) - 24 \, a^{3} - 336 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) + 333 \, {\left (a^{3} \cos \left (d x + c\right )^{5} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 )}{48 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 243, normalized size = 1.55 \begin {gather*} \frac {666 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 672 \, a^{3} \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^{3} - \frac {27 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {234 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1221 \, a^{3} {\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 {48 \, {\left (33 \, a^{3} + \frac {50 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {21 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 151, normalized size = 0.96 \begin {gather*} \frac {111\,a^3\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{16\,d}+\frac {a^3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{16\,d}+\frac {-\frac {55\,a^3\,{\cos \left (c+d\,x\right )}^4}{8}+\frac {137\,a^3\,{\cos \left (c+d\,x\right )}^3}{8}-\frac {149\,a^3\,{\cos \left (c+d\,x\right )}^2}{12}+\frac {3\,a^3\,\cos \left (c+d\,x\right )}{2}+\frac {a^3}{2}}{d\,\left (-{\cos \left (c+d\,x\right )}^5+3\,{\cos \left (c+d\,x\right )}^4-3\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\right )}-\frac {7\,a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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