Optimal. Leaf size=63 \[ -\frac {\csc ^2(c+d x)}{2 a d}-\frac {(a+b) \log (\sin (c+d x))}{a^2 d}+\frac {(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3273, 78}
\begin {gather*} \frac {(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d}-\frac {(a+b) \log (\sin (c+d x))}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3273
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1-x}{x^2 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{a x^2}+\frac {-a-b}{a^2 x}+\frac {b (a+b)}{a^2 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\csc ^2(c+d x)}{2 a d}-\frac {(a+b) \log (\sin (c+d x))}{a^2 d}+\frac {(a+b) \log \left (a+b \sin ^2(c+d x)\right )}{2 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 50, normalized size = 0.79 \begin {gather*} -\frac {a \csc ^2(c+d x)+(a+b) \left (2 \log (\sin (c+d x))-\log \left (a+b \sin ^2(c+d x)\right )\right )}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 101, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-a -b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}-\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{2}}+\frac {\left (a +b \right ) \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 a^{2}}}{d}\) | \(101\) |
default | \(\frac {\frac {1}{4 a \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-a -b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{2}}-\frac {1}{4 a \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{2}}+\frac {\left (a +b \right ) \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 a^{2}}}{d}\) | \(101\) |
risch | \(\frac {2 \,{\mathrm e}^{2 i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right ) b}{2 a^{2} d}\) | \(152\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 56, normalized size = 0.89 \begin {gather*} \frac {\frac {{\left (a + b\right )} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{2}} - \frac {{\left (a + b\right )} \log \left (\sin \left (d x + c\right )^{2}\right )}{a^{2}} - \frac {1}{a \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 = 0.44, size = 91, normalized size = 1.44 \begin {gather*} \frac {{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 2 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + a}{2 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{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 \frac {\cot ^{3}{\left (c + d x \right )}}{a + b \sin ^{2}{\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 = 0.55, size = 108, normalized size = 1.71 \begin {gather*} \frac {\frac {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}}{a} + \frac {4 \, {\left (a + b\right )} \log \left ({\left | -a {\left (\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 2 \, a + 4 \, b \right |}\right )}{a^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.46, size = 69, normalized size = 1.10 \begin {gather*} \frac {\ln \left (a+a\,{\mathrm {tan}\left (c+d\,x\right )}^2+b\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )\,\left (a+b\right )}{2\,a^2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^2}{2\,a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a+b\right )}{a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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