Optimal. Leaf size=84 \[ -\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {2 a b \sin (c+d x)}{d}-\frac {b^2 \sin ^2(c+d x)}{2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 84, 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 = {2800, 908}
\begin {gather*} -\frac {\left (a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {2 a b \sin (c+d x)}{d}-\frac {2 a b \csc (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 908
Rule 2800
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )}{x^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-2 a+\frac {a^2 b^2}{x^3}+\frac {2 a b^2}{x^2}+\frac {-a^2+b^2}{x}-x\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (a^2-b^2\right ) \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 = 0.16, size = 70, normalized size = 0.83 \begin {gather*} -\frac {4 a b \csc (c+d x)+a^2 \csc ^2(c+d x)+2 \left (a^2-b^2\right ) \log (\sin (c+d x))+4 a b \sin (c+d x)+b^2 \sin ^2(c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 93, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )+b^{2} \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(93\) |
default | \(\frac {a^{2} \left (-\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 a b \left (-\frac {\cos ^{4}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )+b^{2} \left (\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\) | \(93\) |
risch | \(i a^{2} x -i b^{2} x +\frac {b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}+\frac {i a b \,{\mathrm e}^{i \left (d x +c \right )}}{d}-\frac {i a b \,{\mathrm e}^{-i \left (d x +c \right )}}{d}+\frac {b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {2 i a^{2} c}{d}-\frac {2 i b^{2} c}{d}-\frac {2 i a \left (i a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(201\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 69, normalized size = 0.82 \begin {gather*} -\frac {b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) + 2 \, {\left (a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + \frac {4 \, a b \sin \left (d x + c\right ) + a^{2}}{\sin \left (d x + c\right )^{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 = 115, normalized size = 1.37 \begin {gather*} \frac {2 \, b^{2} \cos \left (d x + c\right )^{4} - 3 \, b^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} + b^{2} - 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 8 \, {\left (a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \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 ^{3}{\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 = 9.05, size = 99, normalized size = 1.18 \begin {gather*} -\frac {b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) + 2 \, {\left (a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {3 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} - 4 \, a b \sin \left (d x + c\right ) - a^{2}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.39, size = 221, normalized size = 2.63 \begin {gather*} \frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (a^2-b^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+8\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a^2}{2}+24\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+20\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2-b^2\right )}{d}-\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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