Integrand size = 27, antiderivative size = 392 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=-\frac {2 d^2 \cos (e+f x)}{3 (b c-3 d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-12 c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (7 b c^2-12 c d-3 b d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 d \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 (b c-3 d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) (b c-3 d)^2 f \sqrt {c+d \sin (e+f x)}} \]
[Out]
Time = 1.06 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {2881, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {2 b^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{f (a+b) (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 d^2 \left (-4 a c d+7 b c^2-3 b d^2\right ) \cos (e+f x)}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt {c+d \sin (e+f x)}}-\frac {2 d^2 \cos (e+f x)}{3 f \left (c^2-d^2\right ) (b c-a d) (c+d \sin (e+f x))^{3/2}}+\frac {2 d \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{3 f \left (c^2-d^2\right ) (b c-a d) \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (-4 a c d+7 b c^2-3 b d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 f \left (c^2-d^2\right )^2 (b c-a d)^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2881
Rule 2884
Rule 2886
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}+\frac {2 \int \frac {-\frac {3}{2} \left (a c d-b \left (c^2-d^2\right )\right )-\frac {1}{2} d (3 b c-a d) \sin (e+f x)+\frac {1}{2} b d^2 \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}} \, dx}{3 (b c-a d) \left (c^2-d^2\right )} \\ & = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {4 \int \frac {\frac {1}{4} \left (3 b^2 \left (c^2-d^2\right )^2-2 a b c d \left (3 c^2-d^2\right )+a^2 d^2 \left (3 c^2+d^2\right )\right )+\frac {1}{2} d \left (2 a^2 c d^2-2 a b d \left (c^2-d^2\right )-b^2 \left (3 c^3-c d^2\right )\right ) \sin (e+f x)-\frac {1}{4} b d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 (b c-a d)^2 \left (c^2-d^2\right )^2} \\ & = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 \int \frac {-\frac {1}{4} b d \left (c^2-d^2\right ) \left (a b c d-a^2 d^2+3 b^2 \left (c^2-d^2\right )\right )-\frac {1}{4} b^2 d^2 (b c-a d) \left (c^2-d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b d (b c-a d)^2 \left (c^2-d^2\right )^2}-\frac {\left (d \left (7 b c^2-4 a c d-3 b d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 (b c-a d)^2 \left (c^2-d^2\right )^2} \\ & = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}+\frac {b^2 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{(b c-a d)^2}+\frac {d \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 (b c-a d) \left (c^2-d^2\right )}-\frac {\left (d \left (7 b c^2-4 a c d-3 b d^2\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \\ & = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (7 b c^2-4 a c d-3 b d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (b^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{(b c-a d)^2 \sqrt {c+d \sin (e+f x)}}+\frac {\left (d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 (b c-a d) \left (c^2-d^2\right ) \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 d^2 \cos (e+f x)}{3 (b c-a d) \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{3/2}}-\frac {2 d^2 \left (7 b c^2-4 a c d-3 b d^2\right ) \cos (e+f x)}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {c+d \sin (e+f x)}}-\frac {2 d \left (7 b c^2-4 a c d-3 b d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 (b c-a d)^2 \left (c^2-d^2\right )^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 d \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 (b c-a d) \left (c^2-d^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {2 b^2 \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) (b c-a d)^2 f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.61 (sec) , antiderivative size = 1049, normalized size of antiderivative = 2.68 \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {c+d \sin (e+f x)} \left (-\frac {2 d^2 \cos (e+f x)}{3 (b c-3 d) \left (c^2-d^2\right ) (c+d \sin (e+f x))^2}-\frac {2 \left (7 b c^2 d^2 \cos (e+f x)-12 c d^3 \cos (e+f x)-3 b d^4 \cos (e+f x)\right )}{3 (b c-3 d)^2 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))}\right )}{f}-\frac {-\frac {2 \left (-6 b^2 c^4+36 b c^3 d-54 c^2 d^2+19 b^2 c^2 d^2-24 b c d^3-18 d^4-9 b^2 d^4\right ) \operatorname {EllipticPi}\left (\frac {2 b}{3+b},\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(3+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (12 b^2 c^3 d+24 b c^2 d^2-72 c d^3-4 b^2 c d^3-24 b d^4\right ) \cos (e+f x) \left ((b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )+3 d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b (b c-3 d) d^2 \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-7 b^2 c^2 d^2+12 b c d^3+3 b^2 d^4\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (b c-3 d) (c-d) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (3+b) (b c-3 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )-\left (-18+b^2\right ) d \operatorname {EllipticPi}\left (\frac {b (c+d)}{b c-3 d},i \text {arcsinh}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right ),\frac {c+d}{c-d}\right )\right )\right ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+3 d+b (c+d \sin (e+f x)))}{b^2 (b c-3 d) d \sqrt {-\frac {1}{c+d}} (3+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{6 (b c-3 d)^2 (c-d)^2 (c+d)^2 f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1071\) vs. \(2(474)=948\).
Time = 7.98 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.73
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sin \left (f x + e\right ) + a\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a+b\,\sin \left (e+f\,x\right )\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
[In]
[Out]