Optimal. Leaf size=55 \[ \frac{8 a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.116714, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ \frac{8 a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{d}+(4 a) \int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{8 a^2 \sec (c+d x) \sqrt{a+a \sin (c+d x)}}{d}-\frac{2 a \sec (c+d x) (a+a \sin (c+d x))^{3/2}}{d}\\ \end{align*}
Mathematica [A] time = 4.05866, size = 36, normalized size = 0.65 \[ -\frac{2 a^2 (\sin (c+d x)-3) \sec (c+d x) \sqrt{a (\sin (c+d x)+1)}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.089, size = 45, normalized size = 0.8 \begin{align*} -2\,{\frac{{a}^{3} \left ( 1+\sin \left ( dx+c \right ) \right ) \left ( \sin \left ( dx+c \right ) -3 \right ) }{\cos \left ( dx+c \right ) \sqrt{a+a\sin \left ( dx+c \right ) }d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.64472, size = 258, normalized size = 4.69 \begin{align*} -\frac{2 \,{\left (3 \, a^{\frac{5}{2}} - \frac{2 \, a^{\frac{5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{9 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{4 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{9 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{2 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{3 \, a^{\frac{5}{2}} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )}}{d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60578, size = 99, normalized size = 1.8 \begin{align*} -\frac{2 \,{\left (a^{2} \sin \left (d x + c\right ) - 3 \, a^{2}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{d \cos \left (d x + c\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]