Optimal. Leaf size=405 \[ -\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left (c^2-7 c d-10 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.833648, antiderivative size = 405, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2766, 2978, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{c+d \sin (e+f x)}}-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 f (c-d)^3 (c+d) (c+d \sin (e+f x))^{3/2}}+\frac{\left (c^2-7 c d-10 d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left (c^2-10 c d-7 d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3 a^2 f (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 f (c-d)^2 (\sin (e+f x)+1) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 f (c-d) (a \sin (e+f x)+a)^2 (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2766
Rule 2978
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{\int \frac{-\frac{1}{2} a (2 c-9 d)-\frac{5}{2} a d \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^{5/2}} \, dx}{3 a^2 (c-d)}\\ &=-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}+\frac{\int \frac{15 a^2 d^2+\frac{3}{2} a^2 (c-7 d) d \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{3 a^4 (c-d)^2}\\ &=-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 \int \frac{-\frac{9}{4} a^2 d^2 (9 c+7 d)-\frac{3}{4} a^2 d \left (c^2-7 c d-10 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{9 a^4 (c-d)^3 (c+d)}\\ &=-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{4 \int \frac{\frac{3}{4} a^2 d^2 \left (13 c^2+14 c d+5 d^2\right )-\frac{3}{8} a^2 d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{9 a^4 (c-d)^4 (c+d)^2}\\ &=-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (c^2-7 c d-10 d^2\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{6 a^2 (c-d)^3 (c+d)}-\frac{\left ((c+3 d) \left (c^2-10 c d-7 d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{6 a^2 (c-d)^4 (c+d)^2}\\ &=-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{\left ((c+3 d) \left (c^2-10 c d-7 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}{6 a^2 (c-d)^4 (c+d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (c^2-7 c d-10 d^2\right ) \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}{6 a^2 (c-d)^3 (c+d) \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{d \left (c^2-7 c d-10 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sin (e+f x))^{3/2}}-\frac{(c-7 d) \cos (e+f x)}{3 a^2 (c-d)^2 f (1+\sin (e+f x)) (c+d \sin (e+f x))^{3/2}}-\frac{\cos (e+f x)}{3 (c-d) f (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{3/2}}-\frac{d (c+3 d) \left (c^2-10 c d-7 d^2\right ) \cos (e+f x)}{3 a^2 (c-d)^4 (c+d)^2 f \sqrt{c+d \sin (e+f x)}}-\frac{(c+3 d) \left (c^2-10 c d-7 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 a^2 (c-d)^4 (c+d)^2 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (c^2-7 c d-10 d^2\right ) 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}}}{3 a^2 (c-d)^3 (c+d) f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.62043, size = 674, normalized size = 1.66 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \sqrt{c+d \sin (e+f x)} \left (-\frac{2 \left (-7 c^2 d+c^3-27 c d^2-15 d^3\right )}{3 (c-d)^4 (c+d)^2}+\frac{2 d^3 \cos (e+f x)}{3 (c-d)^3 (c+d) (c+d \sin (e+f x))^2}+\frac{4 \left (5 c d^3 \cos (e+f x)+3 d^4 \cos (e+f x)\right )}{3 (c-d)^4 (c+d)^2 (c+d \sin (e+f x))}+\frac{2 \left (c \sin \left (\frac{1}{2} (e+f x)\right )-9 d \sin \left (\frac{1}{2} (e+f x)\right )\right )}{3 (c-d)^4 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{1}{3 (c-d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{2 \sin \left (\frac{1}{2} (e+f x)\right )}{3 (c-d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}\right )}{f (a \sin (e+f x)+a)^2}+\frac{d \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4 \left (\frac{2 \left (-7 c^2 d+c^3-37 c d^2-21 d^3\right ) \cos ^2(e+f x) \sqrt{c+d \sin (e+f x)}}{d \left (1-\sin ^2(e+f x)\right )}-\frac{2 \left (26 c^2 d+28 c d^2+10 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{\sqrt{c+d \sin (e+f x)}}-\frac{\left (7 c^2 d-c^3+37 c d^2+21 d^3\right ) \left (\frac{2 (c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{\sqrt{c+d \sin (e+f x)}}\right )}{d}\right )}{6 f (c-d)^4 (c+d)^2 (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 7.269, size = 1758, normalized size = 4.3 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{2 \, a^{2} c^{3} + 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 2 \, a^{2} d^{3} +{\left (3 \, a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{4} -{\left (a^{2} c^{3} + 6 \, a^{2} c^{2} d + 9 \, a^{2} c d^{2} + 4 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} +{\left (a^{2} d^{3} \cos \left (f x + e\right )^{4} + 2 \, a^{2} c^{3} + 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 2 \, a^{2} d^{3} - 3 \,{\left (a^{2} c^{2} d + 2 \, a^{2} c d^{2} + a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \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 (a \sin \left (f x + e\right ) + a\right )}^{2}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]