Integrand size = 23, antiderivative size = 73 \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\cos (e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac {2 \cos (e+f x)}{3 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}} \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 198, 197} \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {2 \cos (e+f x)}{3 f (a+b)^2 \sqrt {a-b \cos ^2(e+f x)+b}}-\frac {\cos (e+f x)}{3 f (a+b) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 3265
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (a+b-b x^2\right )^{5/2}} \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {\cos (e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac {2 \text {Subst}\left (\int \frac {1}{\left (a+b-b x^2\right )^{3/2}} \, dx,x,\cos (e+f x)\right )}{3 (a+b) f} \\ & = -\frac {\cos (e+f x)}{3 (a+b) f \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}-\frac {2 \cos (e+f x)}{3 (a+b)^2 f \sqrt {a+b-b \cos ^2(e+f x)}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 \sqrt {2} \cos (e+f x) (-3 a-2 b+b \cos (2 (e+f x)))}{3 (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Time = 0.66 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {\left (2 b \left (\sin ^{2}\left (f x +e \right )\right )+3 a +b \right ) \cos \left (f x +e \right )}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (a^{2}+2 a b +b^{2}\right ) f}\) | \(55\) |
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Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (65) = 130\).
Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.84 \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {{\left (2 \, b \cos \left (f x + e\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{3 \, {\left ({\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} f\right )}} \]
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Timed out. \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\frac {2 \, \cos \left (f x + e\right )}{\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{2}} + \frac {\cos \left (f x + e\right )}{{\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )}}}{3 \, f} \]
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\[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 20.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.18 \[ \int \frac {\sin (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {4\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\sqrt {a+b\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,\left (b-6\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-4\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}{3\,f\,{\left (a+b\right )}^2\,{\left (b-4\,a\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-2\,b\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+b\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}^2} \]
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