Optimal. Leaf size=284 \[ \frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.01406, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4265, 4086, 4017, 4015, 3805, 3804} \[ \frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (5 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4265
Rule 4086
Rule 4017
Rule 4015
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+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))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac{11}{2}}(c+d x)} \, dx\\ &=\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (5 A+11 B)+\frac{1}{2} a (4 A+11 C) \sec (c+d x)\right )}{\sec ^{\frac{9}{2}}(c+d x)} \, dx}{11 a}\\ &=\frac{2 a (5 A+11 B) \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{3}{4} a^2 (32 A+44 B+33 C)+\frac{1}{4} a^2 (56 A+44 B+99 C) \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx}{99 a}\\ &=\frac{2 a^2 (32 A+44 B+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (5 A+11 B) \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (8 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{1}{8} a^3 (1160 A+1364 B+1485 C)+\frac{1}{8} a^3 (776 A+836 B+1089 C) \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{693 a}\\ &=\frac{2 a^3 (1160 A+1364 B+1485 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+44 B+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (5 A+11 B) \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (a^2 (2840 A+3212 B+3795 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{1155}\\ &=\frac{2 a^3 (2840 A+3212 B+3795 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (1160 A+1364 B+1485 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+44 B+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (5 A+11 B) \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}+\frac{\left (2 a^2 (2840 A+3212 B+3795 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}{3465}\\ &=\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^3 (1160 A+1364 B+1485 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (32 A+44 B+33 C) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{231 d}+\frac{2 a (5 A+11 B) \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac{2 A \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 2.32137, size = 157, normalized size = 0.55 \[ \frac{a^2 \sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} ((69890 A+68552 B+66660 C) \cos (c+d x)+16 (1625 A+1397 B+990 C) \cos (2 (c+d x))+8675 A \cos (3 (c+d x))+2240 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+114640 A+5720 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+124366 B+1980 C \cos (3 (c+d x))+137280 C)}{27720 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.28, size = 189, normalized size = 0.7 \begin{align*} -{\frac{2\,{a}^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 315\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1120\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+385\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1775\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1430\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+495\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2130\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2409\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+1980\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2840\,A\cos \left ( dx+c \right ) +3212\,B\cos \left ( dx+c \right ) +3795\,C\cos \left ( dx+c \right ) +5680\,A+6424\,B+7590\,C \right ) }{3465\,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 = 2.50638, size = 1249, normalized size = 4.4 \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 = 0.508834, size = 459, normalized size = 1.62 \begin{align*} \frac{2 \,{\left (315 \, A a^{2} \cos \left (d x + c\right )^{5} + 35 \,{\left (32 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \,{\left (355 \, A + 286 \, B + 99 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \,{\left (710 \, A + 803 \, B + 660 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (2840 \, A + 3212 \, B + 3795 \, C\right )} a^{2}\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 )}{3465 \,{\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} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{\frac{11}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]