Integrand size = 25, antiderivative size = 137 \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{b^{5/2} f}+\frac {a (3 a+5 b) \cos (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}+\frac {a \cos (e+f x) \sin ^2(e+f x)}{3 b (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3265, 424, 393, 223, 209} \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{b^{5/2} f}+\frac {a (3 a+5 b) \cos (e+f x)}{3 b^2 f (a+b)^2 \sqrt {a-b \cos ^2(e+f x)+b}}+\frac {a \sin ^2(e+f x) \cos (e+f x)}{3 b f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rule 209
Rule 223
Rule 393
Rule 424
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-b x^2\right )^{5/2}} \, dx,x,\cos (e+f x)\right )}{f} \\ & = \frac {a \cos (e+f x) \sin ^2(e+f x)}{3 b (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-a-3 b+3 (a+b) x^2}{\left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{3 b (a+b) f} \\ & = \frac {a (3 a+5 b) \cos (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}+\frac {a \cos (e+f x) \sin ^2(e+f x)}{3 b (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{b^2 f} \\ & = \frac {a (3 a+5 b) \cos (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}+\frac {a \cos (e+f x) \sin ^2(e+f x)}{3 b (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{b^2 f} \\ & = -\frac {\arctan \left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{b^{5/2} f}+\frac {a (3 a+5 b) \cos (e+f x)}{3 b^2 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}}+\frac {a \cos (e+f x) \sin ^2(e+f x)}{3 b (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.97 \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {\frac {2 \sqrt {2} a \cos (e+f x) \left (3 a^2+7 a b+3 b^2-b (2 a+3 b) \cos (2 (e+f x))\right )}{(a+b)^2 (2 a+b-b \cos (2 (e+f x)))^{3/2}}-\frac {3 \log \left (\sqrt {2} \sqrt {-b} \cos (e+f x)+\sqrt {2 a+b-b \cos (2 (e+f x))}\right )}{\sqrt {-b}}}{3 b^2 f} \]
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Time = 1.45 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.77
method | result | size |
default | \(\frac {\sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\frac {\arctan \left (\frac {\sqrt {b}\, \left (\sin ^{2}\left (f x +e \right )-\frac {-a +b}{2 b}\right )}{\sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{2 b^{\frac {5}{2}}}-\frac {a^{2} \left (2 b \left (\sin ^{2}\left (f x +e \right )\right )+3 a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}{3 b^{2} \sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {2 a \left (\cos ^{2}\left (f x +e \right )\right )}{b^{2} \left (a +b \right ) \sqrt {-\left (-b \left (\sin ^{2}\left (f x +e \right )\right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(243\) |
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (123) = 246\).
Time = 0.75 (sec) , antiderivative size = 885, normalized size of antiderivative = 6.46 \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{6} + 160 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{5} + 10 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}\right ) + 8 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{24 \, {\left ({\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right )} f\right )}}, \frac {3 \, {\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{5} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{12 \, {\left ({\left (a^{2} b^{5} + 2 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b^{4} + 3 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b^{3} + 4 \, a^{3} b^{4} + 6 \, a^{2} b^{5} + 4 \, a b^{6} + b^{7}\right )} f\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (123) = 246\).
Time = 0.37 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.07 \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {{\left (\frac {3 \, \cos \left (f x + e\right )^{2}}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} b} - \frac {2 \, a}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} b^{2}} - \frac {2}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} b}\right )} \cos \left (f x + e\right ) + \frac {3 \, \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{2}} + \frac {\cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}} - \frac {3 \, \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b^{2}} + \frac {2 \, a \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} b^{2}} - \frac {2 \, \cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} b} + \frac {4 \, \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} b}}{3 \, f} \]
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\[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{5}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^5(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^5}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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