Optimal. Leaf size=46 \[ \frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2800, 45}
\begin {gather*} \frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2800
Rubi steps
\begin {align*} \int \cot (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^2}{x} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (2 a+\frac {a^2}{x}+x\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 46, normalized size = 1.00 \begin {gather*} \frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 40, normalized size = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right ) b^{2}}{2}+2 a b \sin \left (d x +c \right )+a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(40\) |
default | \(\frac {\frac {\left (\sin ^{2}\left (d x +c \right )\right ) b^{2}}{2}+2 a b \sin \left (d x +c \right )+a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}\) | \(40\) |
risch | \(-i a^{2} x -\frac {b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 i a^{2} c}{d}+\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {2 a b \sin \left (d x +c \right )}{d}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 40, normalized size = 0.87 \begin {gather*} \frac {b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 0.91 \begin {gather*} -\frac {b^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 13.32, size = 41, normalized size = 0.89 \begin {gather*} \frac {b^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 4 \, a b \sin \left (d x + c\right )}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.44, size = 117, normalized size = 2.54 \begin {gather*} \frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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