Optimal. Leaf size=95 \[ -\frac{4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.192069, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2772, 2771} \[ -\frac{4 a \cos (e+f x)}{3 f (c+d)^2 \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}-\frac{2 a \cos (e+f x)}{3 f (c+d) \sqrt{a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2772
Rule 2771
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx &=-\frac{2 a \cos (e+f x)}{3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac{2 \int \frac{\sqrt{a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{3 (c+d)}\\ &=-\frac{2 a \cos (e+f x)}{3 (c+d) f \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac{4 a \cos (e+f x)}{3 (c+d)^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.273571, size = 100, normalized size = 1.05 \[ -\frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) (3 c+2 d \sin (e+f x)+d)}{3 f (c+d)^2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.215, size = 222, normalized size = 2.3 \begin{align*}{\frac{4\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}{d}^{3}+2\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}c{d}^{2}+2\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{3}+8\,{c}^{2} \left ( \cos \left ( fx+e \right ) \right ) ^{2}d+2\,c \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{2}-6\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{3}+6\,{c}^{3}\sin \left ( fx+e \right ) +10\,{c}^{2}d\sin \left ( fx+e \right ) +2\,\sin \left ( fx+e \right ){d}^{2}c-2\,{d}^{3}\sin \left ( fx+e \right ) -6\,{c}^{3}-10\,{c}^{2}d-2\,c{d}^{2}+2\,{d}^{3}}{3\,f \left ( c+d \right ) ^{2}\cos \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{2}+{c}^{2}-{d}^{2} \right ) ^{2}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{c+d\sin \left ( fx+e \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.95327, size = 459, normalized size = 4.83 \begin{align*} -\frac{2 \,{\left ({\left (3 \, c^{2} + c d\right )} \sqrt{a} - \frac{{\left (3 \, c^{2} - 9 \, c d - 2 \, d^{2}\right )} \sqrt{a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{2 \,{\left (3 \, c^{2} - 4 \, c d + 3 \, d^{2}\right )} \sqrt{a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{2 \,{\left (3 \, c^{2} - 4 \, c d + 3 \, d^{2}\right )} \sqrt{a} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac{{\left (3 \, c^{2} - 9 \, c d - 2 \, d^{2}\right )} \sqrt{a} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac{{\left (3 \, c^{2} + c d\right )} \sqrt{a} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}{\left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \,{\left (c^{2} + 2 \, c d + d^{2} + \frac{2 \,{\left (c^{2} + 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{{\left (c^{2} + 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}{\left (c + \frac{2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac{5}{2}} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.36704, size = 703, normalized size = 7.4 \begin{align*} \frac{2 \,{\left (2 \, d \cos \left (f x + e\right )^{2} +{\left (3 \, c + d\right )} \cos \left (f x + e\right ) +{\left (2 \, d \cos \left (f x + e\right ) - 3 \, c + d\right )} \sin \left (f x + e\right ) + 3 \, c - d\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right ) + c}}{3 \,{\left ({\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{3} +{\left (2 \, c^{3} d + 5 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} -{\left (c^{4} + 2 \, c^{3} d + 2 \, c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right ) -{\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f +{\left ({\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} - 2 \,{\left (c^{3} d + 2 \, c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) -{\left (c^{4} + 4 \, c^{3} d + 6 \, c^{2} d^{2} + 4 \, c d^{3} + d^{4}\right )} f\right )} \sin \left (f x + e\right )\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{\sqrt{a \sin \left (f x + e\right ) + a}}{{\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]