Optimal. Leaf size=217 \[ -\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.50717, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {3817, 4020, 4022, 4013, 3808, 206} \[ -\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3817
Rule 4020
Rule 4022
Rule 4013
Rule 3808
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{\int \frac{-\frac{11 a}{2}+3 a \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}-\frac{\int \frac{-\frac{95 a^2}{4}+17 a^2 \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{\int \frac{\frac{299 a^3}{8}-\frac{95}{4} a^3 \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx}{12 a^5}\\ &=-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{163 \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{163 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{16 a^2 d}\\ &=\frac{163 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{299 \sqrt{\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.2171, size = 165, normalized size = 0.76 \[ -\frac{\sec (c+d x) \left (\tan (c+d x) \sqrt{1-\sec (c+d x)} (379 \sec (c+d x)+16 \cos (2 (c+d x)) (5 \sec (c+d x)-1)+487)+978 \sqrt{2} \sin (c+d x) \cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right )\right )}{48 d \sqrt{-(\sec (c+d x)-1) \sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.208, size = 254, normalized size = 1.2 \begin{align*} -{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{96\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 489\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+978\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+64\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+489\,\arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) -384\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-686\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+408\,\cos \left ( dx+c \right ) +598 \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [A] time = 2.21058, size = 1256, normalized size = 5.79 \begin{align*} \left [\frac{489 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + \frac{4 \,{\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 503 \, \cos \left (d x + c\right )^{2} - 299 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{192 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, -\frac{489 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )}}{a \sin \left (d x + c\right )}\right ) - \frac{2 \,{\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 503 \, \cos \left (d x + c\right )^{2} - 299 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{96 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]