Optimal. Leaf size=73 \[ \frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11219, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2675, 2667, 63, 206} \[ \frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2675
Rule 2667
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{1}{4} a \int \sec (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac{\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{2 d}\\ &=\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.249573, size = 72, normalized size = 0.99 \[ \frac{a \left (\sqrt{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right )-\frac{2 \sqrt{a (\sin (c+d x)+1)}}{\sin (c+d x)-1}\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.102, size = 70, normalized size = 1. \begin{align*} 2\,{\frac{{a}^{3}}{d} \left ( -1/4\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a \left ( a\sin \left ( dx+c \right ) -a \right ) }}+1/8\,{\frac{\sqrt{2}}{{a}^{3/2}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69733, size = 266, normalized size = 3.64 \begin{align*} \frac{{\left (\sqrt{2} a \sin \left (d x + c\right ) - \sqrt{2} a\right )} \sqrt{a} \log \left (-\frac{a \sin \left (d x + c\right ) + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, \sqrt{a \sin \left (d x + c\right ) + a} a}{8 \,{\left (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(-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]