Optimal. Leaf size=89 \[ -\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0470163, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2647, 2646} \[ -\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a \sin (c+d x)+a}}-\frac{16 a^2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}-\frac{2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{5} (8 a) \int (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac{16 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 d}-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{64 a^3 \cos (c+d x)}{15 d \sqrt{a+a \sin (c+d x)}}-\frac{16 a^2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{15 d}-\frac{2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.31971, size = 117, normalized size = 1.31 \[ -\frac{(a (\sin (c+d x)+1))^{5/2} \left (-150 \sin \left (\frac{1}{2} (c+d x)\right )+25 \sin \left (\frac{3}{2} (c+d x)\right )+3 \sin \left (\frac{5}{2} (c+d x)\right )+150 \cos \left (\frac{1}{2} (c+d x)\right )+25 \cos \left (\frac{3}{2} (c+d x)\right )-3 \cos \left (\frac{5}{2} (c+d x)\right )\right )}{30 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.074, size = 65, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) \left ( 3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+14\,\sin \left ( dx+c \right ) +43 \right ) }{15\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \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 (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60321, size = 292, normalized size = 3.28 \begin{align*} \frac{2 \,{\left (3 \, a^{2} \cos \left (d x + c\right )^{3} - 11 \, a^{2} \cos \left (d x + c\right )^{2} - 46 \, a^{2} \cos \left (d x + c\right ) - 32 \, a^{2} -{\left (3 \, a^{2} \cos \left (d x + c\right )^{2} + 14 \, a^{2} \cos \left (d x + c\right ) - 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\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{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]