Optimal. Leaf size=169 \[ \frac{8 a^2 (19 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{6 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.544518, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4265, 4087, 4013, 3809, 3804} \[ \frac{8 a^2 (19 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{6 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4265
Rule 4087
Rule 4013
Rule 3809
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3 a A}{2}+\frac{1}{2} a (2 A+7 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{6 A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{1}{35} \left ((19 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a (19 A+35 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{105 d}+\frac{6 A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}+\frac{1}{105} \left (4 a (19 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{8 a^2 (19 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a (19 A+35 C) \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{105 d}+\frac{6 A \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac{2 A \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.887585, size = 85, normalized size = 0.5 \[ \frac{a \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} ((253 A+140 C) \cos (c+d x)+78 A \cos (2 (c+d x))+15 A \cos (3 (c+d x))+494 A+700 C)}{210 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.293, size = 98, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 15\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+39\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+52\,A\cos \left ( dx+c \right ) +35\,C\cos \left ( dx+c \right ) +104\,A+175\,C \right ) }{105\,d\sin \left ( dx+c \right ) }\sqrt{\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.12708, size = 497, normalized size = 2.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.491414, size = 269, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (15 \, A a \cos \left (d x + c\right )^{3} + 39 \, A a \cos \left (d x + c\right )^{2} +{\left (52 \, A + 35 \, C\right )} a \cos \left (d x + c\right ) +{\left (104 \, A + 175 \, C\right )} a\right )} \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 )}{105 \,{\left (d \cos \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]