Optimal. Leaf size=325 \[ \frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{b \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.968121, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2802, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{b \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f \left (a^2-b^2\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (-3 a^2 d+2 a b c+b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{f (a-b) (a+b)^2 (b c-a d) \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2802
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sin (e+f x))^2 \sqrt{c+d \sin (e+f x)}} \, dx &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac{\int \frac{\frac{1}{2} \left (-2 a b c+2 a^2 d-b^2 d\right )-a b d \sin (e+f x)-\frac{1}{2} b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right ) (b c-a d)}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac{b \int \sqrt{c+d \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac{\int \frac{\frac{1}{2} b d \left (a b c-2 a^2 d+b^2 d\right )-\frac{1}{2} b^2 d (b c-a d) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right ) d (b c-a d)}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}-\frac{\int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right )}+\frac{\left (2 a b c-3 a^2 d+b^2 d\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}+\frac{\left (b \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{2 \left (a^2-b^2\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac{b 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)}}{\left (a^2-b^2\right ) (b c-a d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{2 \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left (\left (2 a b c-3 a^2 d+b^2 d\right ) \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}{2 \left (a^2-b^2\right ) (b c-a d) \sqrt{c+d \sin (e+f x)}}\\ &=\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))}+\frac{b 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)}}{\left (a^2-b^2\right ) (b c-a d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{F\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}}}{\left (a^2-b^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{\left (2 a b c-3 a^2 d+b^2 d\right ) \Pi \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) (a+b)^2 (b c-a d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.25027, size = 871, normalized size = 2.68 \[ \frac{-\frac{2 \left (4 d a^2-4 b c a-3 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}+\frac{8 i a \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{\sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) (a+b) (a d-b c) f}-\frac{b^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{\left (a^2-b^2\right ) (a d-b c) f (a+b \sin (e+f x))} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 3.542, size = 690, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]