Integrand size = 23, antiderivative size = 89 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {(a+b)^2 \log (\sin (c+d x))}{a^3 d}-\frac {(a+b)^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^3 d} \]
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Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3273, 90} \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {(a+b)^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^3 d}+\frac {(a+b)^2 \log (\sin (c+d x))}{a^3 d}+\frac {(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac {\csc ^4(c+d x)}{4 a d} \]
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Rule 90
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^3 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a x^3}+\frac {-2 a-b}{a^2 x^2}+\frac {(a+b)^2}{a^3 x}-\frac {b (a+b)^2}{a^3 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = \frac {(2 a+b) \csc ^2(c+d x)}{2 a^2 d}-\frac {\csc ^4(c+d x)}{4 a d}+\frac {(a+b)^2 \log (\sin (c+d x))}{a^3 d}-\frac {(a+b)^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 a^3 d} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.81 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {2 a (2 a+b) \csc ^2(c+d x)-a^2 \csc ^4(c+d x)+2 (a+b)^2 \left (2 \log (\sin (c+d x))-\log \left (a+b \sin ^2(c+d x)\right )\right )}{4 a^3 d} \]
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Time = 3.90 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.81
method | result | size |
derivativedivides | \(\frac {-\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a -4 b}{16 a^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 a^{3}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {7 a +4 b}{16 a^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(161\) |
default | \(\frac {-\frac {1}{16 a \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a -4 b}{16 a^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{2 a^{3}}-\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (a +b -b \left (\cos ^{2}\left (d x +c \right )\right )\right )}{2 a^{3}}-\frac {1}{16 a \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {7 a +4 b}{16 a^{2} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (a^{2}+2 a b +b^{2}\right ) \ln \left (\cos \left (d x +c \right )-1\right )}{2 a^{3}}}{d}\) | \(161\) |
risch | \(-\frac {2 \left (2 a \,{\mathrm e}^{6 i \left (d x +c \right )}+b \,{\mathrm e}^{6 i \left (d x +c \right )}-2 a \,{\mathrm e}^{4 i \left (d x +c \right )}-2 b \,{\mathrm e}^{4 i \left (d x +c \right )}+2 a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d a}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b}{a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{3} d}-\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right ) b}{a^{2} d}-\frac {\ln \left ({\mathrm e}^{4 i \left (d x +c \right )}-\frac {2 \left (2 a +b \right ) {\mathrm e}^{2 i \left (d x +c \right )}}{b}+1\right ) b^{2}}{2 a^{3} d}\) | \(277\) |
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (83) = 166\).
Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.22 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {2 \, {\left (2 \, a^{2} + a b\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 2 \, a b + 2 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )}} \]
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\[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{3}} - \frac {2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left (\sin \left (d x + c\right )^{2}\right )}{a^{3}} - \frac {2 \, {\left (2 \, a + b\right )} \sin \left (d x + c\right )^{2} - a}{a^{2} \sin \left (d x + c\right )^{4}}}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (83) = 166\).
Time = 0.43 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.30 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {\frac {a {\left (\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}^{2} + 12 \, a {\left (\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 8 \, b {\left (\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )}}{a^{2}} + \frac {32 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \log \left ({\left | -a {\left (\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1} + \frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1}\right )} + 2 \, a + 4 \, b \right |}\right )}{a^{3}}}{64 \, d} \]
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Time = 13.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+2\,a\,b+b^2\right )}{a^3\,d}-\frac {\ln \left (a+a\,{\mathrm {tan}\left (c+d\,x\right )}^2+b\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,a^3\,d}-\frac {\frac {1}{4\,a}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a+b\right )}{2\,a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^4} \]
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