Optimal. Leaf size=102 \[ -\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d} \]
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Rubi [A]
time = 0.13, antiderivative size = 102, 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 {\cos ^5(c+d x)}{5 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{a^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^4-\frac {4 a^5}{a-x}+4 a^3 x+4 a^2 x^2+3 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d}\\ &=-\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.54, size = 73, normalized size = 0.72 \begin {gather*} \frac {3857-4920 \cos (c+d x)+1320 \cos (2 (c+d x))-380 \cos (3 (c+d x))+90 \cos (4 (c+d x))-12 \cos (5 (c+d x))+7680 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 79, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {-\frac {1}{5 \sec \left (d x +c \right )^{5}}+\frac {3}{4 \sec \left (d x +c \right )^{4}}-\frac {4}{3 \sec \left (d x +c \right )^{3}}+\frac {2}{\sec \left (d x +c \right )^{2}}-\frac {4}{\sec \left (d x +c \right )}-4 \ln \left (\sec \left (d x +c \right )\right )+4 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(79\) |
default | \(\frac {-\frac {1}{5 \sec \left (d x +c \right )^{5}}+\frac {3}{4 \sec \left (d x +c \right )^{4}}-\frac {4}{3 \sec \left (d x +c \right )^{3}}+\frac {2}{\sec \left (d x +c \right )^{2}}-\frac {4}{\sec \left (d x +c \right )}-4 \ln \left (\sec \left (d x +c \right )\right )+4 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(79\) |
norman | \(\frac {-\frac {32 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {166}{15 a d}-\frac {78 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {154 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {278 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2}}-\frac {4 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}\) | \(128\) |
risch | \(-\frac {4 i x}{a^{3}}-\frac {41 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{3} d}-\frac {41 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{3} d}-\frac {8 i c}{a^{3} d}+\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {3 \cos \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {19 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {11 \cos \left (2 d x +2 c \right )}{8 d \,a^{3}}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 73, normalized size = 0.72 \begin {gather*} -\frac {\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right )}{a^{3}} - \frac {240 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 70, normalized size = 0.69 \begin {gather*} -\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right ) - 240 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{60 \, a^{3} d} \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 = 172, normalized size = 1.69 \begin {gather*} -\frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {85 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {205 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 29}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 82, normalized size = 0.80 \begin {gather*} \frac {\frac {4\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3}-\frac {4\,\cos \left (c+d\,x\right )}{a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^2}{a^3}-\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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