Optimal. Leaf size=185 \[ -\frac{9}{32 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27491, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2681, 2687, 2667, 51, 63, 206} \[ -\frac{9}{32 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2681
Rule 2687
Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}+\frac{9 \int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{14 a}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac{9 \int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{20 a^2}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{16 a}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{32 a d}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac{9}{32 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{64 a^2 d}\\ &=-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac{9}{32 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{32 a^2 d}\\ &=\frac{9 \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}-\frac{\sec ^2(c+d x)}{7 d (a+a \sin (c+d x))^{5/2}}-\frac{3}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a+a \sin (c+d x))^{3/2}}-\frac{9}{32 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0895356, size = 42, normalized size = 0.23 \[ -\frac{a \, _2F_1\left (-\frac{7}{2},2;-\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{14 d (a \sin (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.194, size = 141, normalized size = 0.8 \begin{align*} 2\,{\frac{{a}^{3}}{d} \left ( -1/16\,{\frac{1}{{a}^{5}} \left ( 1/4\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a\sin \left ( dx+c \right ) -a}}-{\frac{9\,\sqrt{2}}{8\,\sqrt{a}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) }-1/8\,{\frac{1}{{a}^{5}\sqrt{a+a\sin \left ( dx+c \right ) }}}-1/16\,{\frac{1}{{a}^{4} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3/2}}}-1/20\,{\frac{1}{{a}^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{5/2}}}-1/28\,{\frac{1}{{a}^{2} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{7/2}}} \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 = 2.53116, size = 601, normalized size = 3.25 \begin{align*} \frac{315 \, \sqrt{2}{\left (3 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\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 (315 \, \cos \left (d x + c\right )^{4} - 1092 \, \cos \left (d x + c\right )^{2} - 120 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 200\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{4480 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} +{\left (a^{3} d \cos \left (d x + c\right )^{4} - 4 \, a^{3} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\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 [A] time = 1.14311, size = 201, normalized size = 1.09 \begin{align*} -\frac{1}{2240} \, a^{3}{\left (\frac{315 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{5} d} + \frac{70 \, \sqrt{a \sin \left (d x + c\right ) + a}}{{\left (a \sin \left (d x + c\right ) - a\right )} a^{5} d} + \frac{8 \,{\left (70 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3} + 35 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a + 28 \,{\left (a \sin \left (d x + c\right ) + a\right )} a^{2} + 20 \, a^{3}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{5} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]