Optimal. Leaf size=50 \[ \frac {4 \log (1+\sin (c+d x))}{a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {\sin ^2(c+d x)}{2 a^3 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45}
\begin {gather*} \frac {\sin ^2(c+d x)}{2 a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {4 \log (\sin (c+d x)+1)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {(a-x)^2}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \left (-3 a+x+\frac {4 a^2}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {4 \log (1+\sin (c+d x))}{a^3 d}-\frac {3 \sin (c+d x)}{a^3 d}+\frac {\sin ^2(c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.76 \begin {gather*} \frac {8 \log (1+\sin (c+d x))-6 \sin (c+d x)+\sin ^2(c+d x)}{2 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 38, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \sin \left (d x +c \right )+4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(38\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-3 \sin \left (d x +c \right )+4 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(38\) |
risch | \(-\frac {4 i x}{a^{3}}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}-\frac {8 i c}{a^{3} d}+\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a^{3} d}-\frac {\cos \left (2 d x +2 c \right )}{4 a^{3} d}\) | \(93\) |
norman | \(\frac {-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {6 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {28 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {28 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {74 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {74 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {154 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {154 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {256 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {256 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {412 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {412 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {350 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {350 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {8 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3} d}-\frac {4 \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3} d}\) | \(341\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 41, normalized size = 0.82 \begin {gather*} \frac {\frac {\sin \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right )}{a^{3}} + \frac {8 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 0.72 \begin {gather*} -\frac {\cos \left (d x + c\right )^{2} - 8 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 6 \, \sin \left (d x + c\right )}{2 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 564 vs.
\(2 (44) = 88\).
time = 27.73, size = 564, normalized size = 11.28 \begin {gather*} \begin {cases} \frac {8 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {16 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {8 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {4 \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {8 \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {4 \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {6 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} + \frac {2 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} - \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{5}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs.
\(2 (48) = 96\).
time = 7.70, size = 115, normalized size = 2.30 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{a^{3}} - \frac {4 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.56, size = 36, normalized size = 0.72 \begin {gather*} \frac {8\,\ln \left (\sin \left (c+d\,x\right )+1\right )-6\,\sin \left (c+d\,x\right )+{\sin \left (c+d\,x\right )}^2}{2\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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