Integrand size = 23, antiderivative size = 77 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b} d}-\frac {(a-b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \]
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Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3266, 472, 211} \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {b^2 \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d \sqrt {a+b}}-\frac {(a-b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \]
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Rule 211
Rule 472
Rule 3266
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a x^4}+\frac {a-b}{a^2 x^2}+\frac {b^2}{a^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {(a-b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d} \\ & = \frac {b^2 \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{5/2} \sqrt {a+b} d}-\frac {(a-b) \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.55 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=-\frac {(2 a+b-b \cos (2 (c+d x))) \csc ^2(c+d x) \left (-3 b^2 \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \sqrt {a+b} \cot (c+d x) \left (2 a-3 b+a \csc ^2(c+d x)\right )\right )}{6 a^{5/2} \sqrt {a+b} d \left (b+a \csc ^2(c+d x)\right )} \]
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Time = 0.84 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {a -b}{a^{2} \tan \left (d x +c \right )}+\frac {b^{2} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{d}\) | \(69\) |
default | \(\frac {-\frac {1}{3 a \tan \left (d x +c \right )^{3}}-\frac {a -b}{a^{2} \tan \left (d x +c \right )}+\frac {b^{2} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{a^{2} \sqrt {a \left (a +b \right )}}}{d}\) | \(69\) |
risch | \(\frac {2 i \left (3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+6 a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 a +3 b \right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, d \,a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, d \,a^{2}}\) | \(251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (67) = 134\).
Time = 0.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 5.86 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\left [-\frac {4 \, {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sqrt {-a^{2} - a b} \log \left (\frac {{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} - {\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \, {\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) - 12 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )}{12 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + a^{3} b\right )} d\right )} \sin \left (d x + c\right )}, -\frac {2 \, {\left (2 \, a^{3} - a^{2} b - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (b^{2} \cos \left (d x + c\right )^{2} - b^{2}\right )} \sqrt {a^{2} + a b} \arctan \left (\frac {{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt {a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 6 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )}{6 \, {\left ({\left (a^{4} + a^{3} b\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{4} + a^{3} b\right )} d\right )} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\int \frac {\csc ^{4}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, b^{2} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{\sqrt {{\left (a + b\right )} a} a^{2}} - \frac {3 \, {\left (a - b\right )} \tan \left (d x + c\right )^{2} + a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.44 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {\frac {3 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} b^{2}}{\sqrt {a^{2} + a b} a^{2}} - \frac {3 \, a \tan \left (d x + c\right )^{2} - 3 \, b \tan \left (d x + c\right )^{2} + a}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \]
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Time = 13.89 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {\csc ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx=\frac {b^2\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\sqrt {a+b}}{\sqrt {a}}\right )}{a^{5/2}\,d\,\sqrt {a+b}}-\frac {\frac {1}{3\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a-b\right )}{a^2}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \]
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