Optimal. Leaf size=165 \[ -\frac{8 a^2 (35 A c+21 A d+21 B c+19 B d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+7 B c-2 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 a (35 A c+21 A d+21 B c+19 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.315633, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2968, 3023, 2751, 2647, 2646} \[ -\frac{8 a^2 (35 A c+21 A d+21 B c+19 B d) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{2 (7 A d+7 B c-2 B d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 a (35 A c+21 A d+21 B c+19 B d) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 B d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2968
Rule 3023
Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\int (a+a \sin (e+f x))^{3/2} \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac{2 \int (a+a \sin (e+f x))^{3/2} \left (\frac{1}{2} a (7 A c+5 B d)+\frac{1}{2} a (7 B c+7 A d-2 B d) \sin (e+f x)\right ) \, dx}{7 a}\\ &=-\frac{2 (7 B c+7 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac{1}{35} (35 A c+21 B c+21 A d+19 B d) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac{2 a (35 A c+21 B c+21 A d+19 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 (7 B c+7 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac{1}{105} (4 a (35 A c+21 B c+21 A d+19 B d)) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{8 a^2 (35 A c+21 B c+21 A d+19 B d) \cos (e+f x)}{105 f \sqrt{a+a \sin (e+f x)}}-\frac{2 a (35 A c+21 B c+21 A d+19 B d) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 (7 B c+7 A d-2 B d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}\\ \end{align*}
Mathematica [A] time = 1.07593, size = 144, normalized size = 0.87 \[ -\frac{a \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) ((140 A c+252 A d+252 B c+253 B d) \sin (e+f x)-6 (7 A d+7 B c+13 B d) \cos (2 (e+f x))+700 A c+546 A d+546 B c-15 B d \sin (3 (e+f x))+494 B d)}{210 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.996, size = 150, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{2} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 15\,Bd \left ( \sin \left ( fx+e \right ) \right ) ^{3}+21\,Ad \left ( \sin \left ( fx+e \right ) \right ) ^{2}+21\,Bc \left ( \sin \left ( fx+e \right ) \right ) ^{2}+39\,B \left ( \sin \left ( fx+e \right ) \right ) ^{2}d+35\,A\sin \left ( fx+e \right ) c+63\,A\sin \left ( fx+e \right ) d+63\,B\sin \left ( fx+e \right ) c+52\,B\sin \left ( fx+e \right ) d+175\,Ac+126\,Ad+126\,Bc+104\,Bd \right ) }{105\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7824, size = 660, normalized size = 4. \begin{align*} \frac{2 \,{\left (15 \, B a d \cos \left (f x + e\right )^{4} + 3 \,{\left (7 \, B a c +{\left (7 \, A + 13 \, B\right )} a d\right )} \cos \left (f x + e\right )^{3} - 28 \,{\left (5 \, A + 3 \, B\right )} a c - 4 \,{\left (21 \, A + 19 \, B\right )} a d -{\left (7 \,{\left (5 \, A + 6 \, B\right )} a c +{\left (42 \, A + 43 \, B\right )} a d\right )} \cos \left (f x + e\right )^{2} -{\left (7 \,{\left (25 \, A + 21 \, B\right )} a c +{\left (147 \, A + 143 \, B\right )} a d\right )} \cos \left (f x + e\right ) +{\left (15 \, B a d \cos \left (f x + e\right )^{3} + 28 \,{\left (5 \, A + 3 \, B\right )} a c + 4 \,{\left (21 \, A + 19 \, B\right )} a d - 3 \,{\left (7 \, B a c +{\left (7 \, A + 8 \, B\right )} a d\right )} \cos \left (f x + e\right )^{2} -{\left (7 \,{\left (5 \, A + 9 \, B\right )} a c +{\left (63 \, A + 67 \, B\right )} a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{105 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\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(-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]