Optimal. Leaf size=211 \[ \frac{99 \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{99}{256 a d \sqrt{a \sin (c+d x)+a}}-\frac{33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a \sin (c+d x)+a}}-\frac{99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.342764, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{99 \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{99}{256 a d \sqrt{a \sin (c+d x)+a}}-\frac{33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a \sin (c+d x)+a}}-\frac{99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2681
Rule 2687
Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{11 \int \frac{\sec ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{14 a}\\ &=-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99}{112} \int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{160 a}\\ &=-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99}{128} \int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{(99 a) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{128 d}\\ &=-\frac{33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{256 d}\\ &=-\frac{33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac{99}{256 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{512 a d}\\ &=-\frac{33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac{99}{256 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{256 a d}\\ &=\frac{99 \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac{99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac{99}{256 a d \sqrt{a+a \sin (c+d x)}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0905095, size = 44, normalized size = 0.21 \[ -\frac{a^2 \, _2F_1\left (-\frac{7}{2},3;-\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right )}{28 d (a \sin (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 152, normalized size = 0.7 \begin{align*} -2\,{\frac{{a}^{5}}{d} \left ( 1/32\,{\frac{1}{{a}^{6}} \left ( 1/16\,{\frac{a\sqrt{a+a\sin \left ( dx+c \right ) } \left ( 19\,\sin \left ( dx+c \right ) -23 \right ) }{ \left ( a\sin \left ( dx+c \right ) -a \right ) ^{2}}}-{\frac{99\,\sqrt{2}}{32\,\sqrt{a}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) }+{\frac{5}{32\,{a}^{6}\sqrt{a+a\sin \left ( dx+c \right ) }}}+1/16\,{\frac{1}{{a}^{5} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3/2}}}+{\frac{3}{80\,{a}^{4} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{5/2}}}+{\frac{1}{56\,{a}^{3} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{7/2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.57181, size = 572, normalized size = 2.71 \begin{align*} \frac{3465 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{4}\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 (5775 \, \cos \left (d x + c\right )^{4} - 1188 \, \cos \left (d x + c\right )^{2} + 11 \,{\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 480\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{35840 \,{\left (a^{2} d \cos \left (d x + c\right )^{6} - 2 \, a^{2} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.13994, size = 225, normalized size = 1.07 \begin{align*} -\frac{1}{17920} \, a^{5}{\left (\frac{3465 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{6} d} + \frac{70 \,{\left (19 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 42 \, \sqrt{a \sin \left (d x + c\right ) + a} a\right )}}{{\left (a \sin \left (d x + c\right ) - a\right )}^{2} a^{6} d} + \frac{32 \,{\left (175 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{3} + 70 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{2} a + 42 \,{\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^{6} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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