Integrand size = 23, antiderivative size = 196 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a+b)^{5/2} d}-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )} \]
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Time = 0.18 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3266, 479, 591, 464, 211} \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 d (a+b)^2}+\frac {b \cot (c+d x) \left ((4 a+b) \tan ^2(c+d x)+4 a+5 b\right )}{8 a^2 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}-\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d (a+b)^{5/2}}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2} \]
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Rule 211
Rule 464
Rule 479
Rule 591
Rule 3266
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^2 \left (a+(a+b) x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \left (-4 a-5 b+(-4 a-b) x^2\right )}{x^2 \left (a+(a+b) x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 a (a+b) d} \\ & = \frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {(2 a+3 b) (4 a+5 b)+(2 a+b) (4 a+b) x^2}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{8 a^2 (a+b)^2 d} \\ & = -\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}-\frac {\left (3 b \left (8 a^2+12 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a^3 (a+b)^2 d} \\ & = -\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a+b)^{5/2} d}-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )} \\ \end{align*}
Time = 2.82 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.09 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {(-2 a-b+b \cos (2 (c+d x))) \csc ^6(c+d x) \left (\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right ) (2 a+b-b \cos (2 (c+d x)))^2}{(a+b)^{5/2}}+8 \sqrt {a} (2 a+b-b \cos (2 (c+d x)))^2 \cot (c+d x)+\frac {4 a^{3/2} b^2 \sin (2 (c+d x))}{a+b}+\frac {\sqrt {a} b^2 (10 a+7 b) (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{(a+b)^2}\right )}{64 a^{7/2} d \left (b+a \csc ^2(c+d x)\right )^3} \]
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Time = 1.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {-\frac {b \left (\frac {\frac {\left (12 a +7 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 a +8 b}+\frac {3 a b \left (4 a +3 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {3 \left (8 a^{2}+12 a b +5 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(160\) |
default | \(\frac {-\frac {b \left (\frac {\frac {\left (12 a +7 b \right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{8 a +8 b}+\frac {3 a b \left (4 a +3 b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {3 \left (8 a^{2}+12 a b +5 b^{2}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}\right )}{a^{3}}-\frac {1}{a^{3} \tan \left (d x +c \right )}}{d}\) | \(160\) |
risch | \(-\frac {i \left (24 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+36 a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+15 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-144 a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}-312 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-234 a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+128 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+464 a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+632 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+386 a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+90 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-64 a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-224 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-214 a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-60 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+8 a^{2} b^{2}+26 a \,b^{3}+15 b^{4}\right )}{4 a^{3} \left (a +b \right )^{2} d \left (-b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {3 \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 ) b}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {9 \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 ) b^{2}}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}+\frac {15 b^{3} \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 )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{3}}-\frac {3 \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 ) b}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}-\frac {9 \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 ) b^{2}}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{2}}-\frac {15 b^{3} \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 )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d \,a^{3}}\) | \(912\) |
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Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (180) = 360\).
Time = 0.36 (sec) , antiderivative size = 1003, normalized size of antiderivative = 5.12 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
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Time = 0.48 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.38 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {{\left (8 \, a^{4} + 32 \, a^{3} b + 60 \, a^{2} b^{2} + 51 \, a b^{3} + 15 \, b^{4}\right )} \tan \left (d x + c\right )^{4} + 8 \, a^{4} + 16 \, a^{3} b + 8 \, a^{2} b^{2} + {\left (16 \, a^{4} + 48 \, a^{3} b + 60 \, a^{2} b^{2} + 25 \, a b^{3}\right )} \tan \left (d x + c\right )^{2}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (d x + c\right )^{5} + 2 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \tan \left (d x + c\right )}}{8 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.18 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {\frac {3 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a^{2} + a b}} + \frac {12 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 19 \, a b^{3} \tan \left (d x + c\right )^{3} + 7 \, b^{4} \tan \left (d x + c\right )^{3} + 12 \, a^{2} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{3} \tan \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}} + \frac {8}{a^{3} \tan \left (d x + c\right )}}{8 \, d} \]
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Time = 15.80 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.28 \[ \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=-\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (8\,a^3+24\,a^2\,b+36\,a\,b^2+15\,b^3\right )}{8\,a^3\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (16\,a^3+48\,a^2\,b+60\,a\,b^2+25\,b^3\right )}{8\,a^2\,\left (a^2+2\,a\,b+b^2\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^2+2\,a\,b+b^2\right )+a^2\,\mathrm {tan}\left (c+d\,x\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a^2+2\,b\,a\right )\right )}-\frac {3\,b\,\mathrm {atan}\left (\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^5+2\,a^4\,b+a^3\,b^2\right )\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{a^{7/2}\,{\left (a+b\right )}^{3/2}\,\left (24\,a^2\,b+36\,a\,b^2+15\,b^3\right )}\right )\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{8\,a^{7/2}\,d\,{\left (a+b\right )}^{5/2}} \]
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