Integrand size = 23, antiderivative size = 131 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {(4 a+b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a+b)^{5/2} d}-\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {(2 a-b) \cos (c+d x) \sin (c+d x)}{8 a (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )} \]
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Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3252, 12, 3260, 211} \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {(4 a+b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} d (a+b)^{5/2}}-\frac {(2 a-b) \sin (c+d x) \cos (c+d x)}{8 a d (a+b)^2 \left (a+b \sin ^2(c+d x)\right )}-\frac {\sin (c+d x) \cos (c+d x)}{4 d (a+b) \left (a+b \sin ^2(c+d x)\right )^2} \]
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Rule 12
Rule 211
Rule 3252
Rule 3260
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {\int \frac {a+2 a \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx}{4 a (a+b)} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {(2 a-b) \cos (c+d x) \sin (c+d x)}{8 a (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )}+\frac {\int \frac {a (4 a+b)}{a+b \sin ^2(c+d x)} \, dx}{8 a^2 (a+b)^2} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {(2 a-b) \cos (c+d x) \sin (c+d x)}{8 a (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )}+\frac {(4 a+b) \int \frac {1}{a+b \sin ^2(c+d x)} \, dx}{8 a (a+b)^2} \\ & = -\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {(2 a-b) \cos (c+d x) \sin (c+d x)}{8 a (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )}+\frac {(4 a+b) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a (a+b)^2 d} \\ & = \frac {(4 a+b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a+b)^{5/2} d}-\frac {\cos (c+d x) \sin (c+d x)}{4 (a+b) d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {(2 a-b) \cos (c+d x) \sin (c+d x)}{8 a (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )} \\ \end{align*}
Time = 11.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {\frac {(4 a+b) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{a^{3/2} (a+b)^{5/2}}-\frac {\left (8 a^2+4 a b-b^2+b (-2 a+b) \cos (2 (c+d x))\right ) \sin (2 (c+d x))}{a (a+b)^2 (2 a+b-b \cos (2 (c+d x)))^2}}{8 d} \]
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Time = 1.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (4 a -b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{8 a \left (a +b \right )}-\frac {\left (4 a +b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {\left (4 a +b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) a \sqrt {a \left (a +b \right )}}}{d}\) | \(131\) |
default | \(\frac {\frac {-\frac {\left (4 a -b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{8 a \left (a +b \right )}-\frac {\left (4 a +b \right ) \tan \left (d x +c \right )}{8 \left (a^{2}+2 a b +b^{2}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{2}}+\frac {\left (4 a +b \right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 \left (a^{2}+2 a b +b^{2}\right ) a \sqrt {a \left (a +b \right )}}}{d}\) | \(131\) |
risch | \(\frac {i \left (4 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+16 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+8 a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-16 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a \,b^{2}-b^{3}\right )}{4 b a \left (a +b \right )^{2} d \left (-b \,{\mathrm e}^{4 i \left (d x +c \right )}+4 a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right ) b}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right ) b}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}\) | \(575\) |
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (117) = 234\).
Time = 0.32 (sec) , antiderivative size = 771, normalized size of antiderivative = 5.89 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\left [-\frac {{\left ({\left (4 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} + 4 \, a^{3} + 9 \, a^{2} b + 6 \, a b^{2} + b^{3} - 2 \, {\left (4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{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 ) - 4 \, {\left ({\left (2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{4} + 7 \, a^{3} b + 2 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{32 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} d\right )}}, -\frac {{\left ({\left (4 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{4} + 4 \, a^{3} + 9 \, a^{2} b + 6 \, a b^{2} + b^{3} - 2 \, {\left (4 \, a^{2} b + 5 \, a b^{2} + b^{3}\right )} \cos \left (d x + c\right )^{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 ) - 2 \, {\left ({\left (2 \, a^{3} b + a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )^{3} - {\left (4 \, a^{4} + 7 \, a^{3} b + 2 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right )} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.46 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {\frac {{\left (4 \, a + b\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {{\left (4 \, a^{2} + 3 \, a b - b^{2}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, a^{2} + a b\right )} \tan \left (d x + c\right )}{a^{5} + 2 \, a^{4} b + a^{3} b^{2} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{2}}}{8 \, d} \]
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Time = 0.38 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.46 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )} {\left (4 \, a + b\right )}}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \sqrt {a^{2} + a b}} - \frac {4 \, a^{2} \tan \left (d x + c\right )^{3} + 3 \, a b \tan \left (d x + c\right )^{3} - b^{2} \tan \left (d x + c\right )^{3} + 4 \, a^{2} \tan \left (d x + c\right ) + a b \tan \left (d x + c\right )}{{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}}}{8 \, d} \]
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Time = 14.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^2+2\,a\,b+b^2\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{5/2}}\right )\,\left (4\,a+b\right )}{8\,a^{3/2}\,d\,{\left (a+b\right )}^{5/2}}-\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a+b\right )}{8\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (4\,a-b\right )}{8\,a\,\left (a+b\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2+2\,a\,b+b^2\right )+a^2+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+2\,b\,a\right )\right )} \]
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