Optimal. Leaf size=47 \[ \frac {\log (\sin (c+d x))}{a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\sin ^2(c+d x)}{2 a^2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2915, 12, 45}
\begin {gather*} \frac {\sin ^2(c+d x)}{2 a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\log (\sin (c+d x))}{a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2915
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {a (a-x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {(a-x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\text {Subst}\left (\int \left (-2 a+\frac {a^2}{x}+x\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\log (\sin (c+d x))}{a^2 d}-\frac {2 \sin (c+d x)}{a^2 d}+\frac {\sin ^2(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.77 \begin {gather*} \frac {2 \log (\sin (c+d x))-4 \sin (c+d x)+\sin ^2(c+d x)}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 34, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-2 \sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-2 \sin \left (d x +c \right )+\ln \left (\sin \left (d x +c \right )\right )}{d \,a^{2}}\) | \(34\) |
risch | \(-\frac {i x}{a^{2}}-\frac {{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {2 i c}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}-\frac {2 \sin \left (d x +c \right )}{a^{2} d}\) | \(86\) |
norman | \(\frac {-\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {10 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {10 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {18 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {18 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {30 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {30 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {34 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {34 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}\) | \(262\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 39, normalized size = 0.83 \begin {gather*} \frac {\frac {\sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right )}{a^{2}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 36, normalized size = 0.77 \begin {gather*} -\frac {\cos \left (d x + c\right )^{2} - 2 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, \sin \left (d x + c\right )}{2 \, a^{2} 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 = 0.44, size = 47, normalized size = 1.00 \begin {gather*} \frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{2} - 4 \, a^{2} \sin \left (d x + c\right )}{a^{4}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.11, size = 120, normalized size = 2.55 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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