Integrand size = 21, antiderivative size = 140 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2+3 b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^2(c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 b \left (a^2+2 b^2\right ) \log (\tan (c+d x))}{a^5 d}+\frac {2 b \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{a^5 d}-\frac {b \left (a^2+b^2\right )}{a^4 d (a+b \tan (c+d x))} \]
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Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3597, 908} \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \cot ^2(c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 b \left (a^2+2 b^2\right ) \log (\tan (c+d x))}{a^5 d}+\frac {2 b \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{a^5 d}-\frac {b \left (a^2+b^2\right )}{a^4 d (a+b \tan (c+d x))}-\frac {\left (a^2+3 b^2\right ) \cot (c+d x)}{a^4 d} \]
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Rule 908
Rule 3597
Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {b^2+x^2}{x^4 (a+x)^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (\frac {b^2}{a^2 x^4}-\frac {2 b^2}{a^3 x^3}+\frac {a^2+3 b^2}{a^4 x^2}-\frac {2 \left (a^2+2 b^2\right )}{a^5 x}+\frac {a^2+b^2}{a^4 (a+x)^2}+\frac {2 \left (a^2+2 b^2\right )}{a^5 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\left (a^2+3 b^2\right ) \cot (c+d x)}{a^4 d}+\frac {b \cot ^2(c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 b \left (a^2+2 b^2\right ) \log (\tan (c+d x))}{a^5 d}+\frac {2 b \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{a^5 d}-\frac {b \left (a^2+b^2\right )}{a^4 d (a+b \tan (c+d x))} \\ \end{align*}
Time = 4.10 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.74 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {-\cot ^2(c+d x) \left (2 a^4+9 a^2 b^2+a^4 \csc ^2(c+d x)\right )+3 b^2 \left (a^2+b^2+a^2 \csc ^2(c+d x)-2 \left (a^2+2 b^2\right ) \log (\sin (c+d x))+2 a^2 \log (a \cos (c+d x)+b \sin (c+d x))+4 b^2 \log (a \cos (c+d x)+b \sin (c+d x))\right )+a b \cot (c+d x) \left (-2 a^2-9 b^2+2 a^2 \csc ^2(c+d x)-6 \left (a^2+2 b^2\right ) \log (\sin (c+d x))+6 a^2 \log (a \cos (c+d x)+b \sin (c+d x))+12 b^2 \log (a \cos (c+d x)+b \sin (c+d x))\right )}{3 a^5 d (b+a \cot (c+d x))} \]
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Time = 1.72 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {-\frac {\left (a^{2}+b^{2}\right ) b}{a^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5}}-\frac {1}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+3 b^{2}}{a^{4} \tan \left (d x +c \right )}+\frac {b}{a^{3} \tan \left (d x +c \right )^{2}}-\frac {2 b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(127\) |
default | \(\frac {-\frac {\left (a^{2}+b^{2}\right ) b}{a^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \left (a^{2}+2 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{5}}-\frac {1}{3 a^{2} \tan \left (d x +c \right )^{3}}-\frac {a^{2}+3 b^{2}}{a^{4} \tan \left (d x +c \right )}+\frac {b}{a^{3} \tan \left (d x +c \right )^{2}}-\frac {2 b \left (a^{2}+2 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{5}}}{d}\) | \(127\) |
risch | \(-\frac {4 i \left (-12 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+6 b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-18 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+i a^{3}+6 i a \,b^{2}+18 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-a^{2} b +6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) a^{4} d}-\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{5} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{5} d}\) | \(353\) |
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Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (138) = 276\).
Time = 0.29 (sec) , antiderivative size = 442, normalized size of antiderivative = 3.16 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 \, {\left (a^{4} + 6 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 6 \, a^{2} b^{2} - 3 \, {\left (a^{4} + 6 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{2} b^{2} + 2 \, b^{4} - 2 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 3 \, {\left ({\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{2} b^{2} + 2 \, b^{4} - 2 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) - 2 \, {\left (6 \, a b^{3} \cos \left (d x + c\right ) - {\left (a^{3} b + 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{5} b d \cos \left (d x + c\right )^{4} - 2 \, a^{5} b d \cos \left (d x + c\right )^{2} + a^{5} b d - {\left (a^{6} d \cos \left (d x + c\right )^{3} - a^{6} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, a^{2} b \tan \left (d x + c\right ) - 6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (d x + c\right )^{3} - a^{3} - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{4} b \tan \left (d x + c\right )^{4} + a^{5} \tan \left (d x + c\right )^{3}} + \frac {6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5}} - \frac {6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{3 \, d} \]
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Time = 0.50 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {6 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}} - \frac {6 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5} b} + \frac {3 \, {\left (2 \, a^{2} b^{2} \tan \left (d x + c\right ) + 4 \, b^{4} \tan \left (d x + c\right ) + 3 \, a^{3} b + 5 \, a b^{3}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )} a^{5}} - \frac {11 \, a^{2} b \tan \left (d x + c\right )^{3} + 22 \, b^{3} \tan \left (d x + c\right )^{3} - 3 \, a^{3} \tan \left (d x + c\right )^{2} - 9 \, a b^{2} \tan \left (d x + c\right )^{2} + 3 \, a^{2} b \tan \left (d x + c\right ) - a^{3}}{a^{5} \tan \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 4.31 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,\left (a^2+2\,b^2\right )\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (2\,a^2\,b+4\,b^3\right )}\right )\,\left (a^2+2\,b^2\right )}{a^5\,d}-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2+2\,b^2\right )}{a^3}-\frac {2\,b\,\mathrm {tan}\left (c+d\,x\right )}{3\,a^2}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (a^2+2\,b^2\right )}{a^4}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4+a\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \]
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