Optimal. Leaf size=91 \[ \frac{3 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.127245, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2676, 2667, 50, 63, 206} \[ \frac{3 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2676
Rule 2667
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx &=\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac{1}{2} \left (3 a^2\right ) \int \sec (c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+x}}{a-x} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac{3 a^3 \sqrt{a+a \sin (c+d x)}}{d}+\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{3 a^3 \sqrt{a+a \sin (c+d x)}}{d}+\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}-\frac{\left (6 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{d}\\ &=-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{d}+\frac{3 a^3 \sqrt{a+a \sin (c+d x)}}{d}+\frac{a \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2}}{d}\\ \end{align*}
Mathematica [C] time = 0.089461, size = 42, normalized size = 0.46 \[ \frac{a (a \sin (c+d x)+a)^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{10 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.158, size = 83, normalized size = 0.9 \begin{align*} 2\,{\frac{{a}^{3}}{d} \left ( \sqrt{a+a\sin \left ( dx+c \right ) }+4\,a \left ( -1/4\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a\sin \left ( dx+c \right ) -a}}-3/8\,{\frac{\sqrt{2}}{\sqrt{a}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \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 = 1.68494, size = 297, normalized size = 3.26 \begin{align*} \frac{3 \, \sqrt{2}{\left (a^{3} \sin \left (d x + c\right ) - a^{3}\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 \,{\left (a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{2 \,{\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]