Optimal. Leaf size=177 \[ \frac {2 \left (3 a^2 B+6 a A b+b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 b (5 a B+3 A b) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 b B \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4026, 4047, 3771, 2641, 4046, 2639} \[ \frac {2 \left (3 a^2 B+6 a A b+b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 \left (a^2 A-2 a b B-A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 b (5 a B+3 A b) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 b B \sin (c+d x) \sqrt {\sec (c+d x)} (a+b \sec (c+d x))}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3771
Rule 4026
Rule 4046
Rule 4047
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt {\sec (c+d x)}} \, dx &=\frac {2 b B \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac {2}{3} \int \frac {\frac {1}{2} a (3 a A-b B)+\frac {1}{2} \left (6 a A b+3 a^2 B+b^2 B\right ) \sec (c+d x)+\frac {1}{2} b (3 A b+5 a B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 b B \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\frac {2}{3} \int \frac {\frac {1}{2} a (3 a A-b B)+\frac {1}{2} b (3 A b+5 a B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (6 a A b+3 a^2 B+b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx\\ &=\frac {2 b (3 A b+5 a B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 b B \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\left (a^2 A-A b^2-2 a b B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{3} \left (\left (6 a A b+3 a^2 B+b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (6 a A b+3 a^2 B+b^2 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 b (3 A b+5 a B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 b B \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}+\left (\left (a^2 A-A b^2-2 a b B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (a^2 A-A b^2-2 a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (6 a A b+3 a^2 B+b^2 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 d}+\frac {2 b (3 A b+5 a B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}+\frac {2 b B \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \sin (c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.28, size = 125, normalized size = 0.71 \[ \frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\left (3 a^2 B+6 a A b+b^2 B\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+3 \left (a^2 A-2 a b B-A b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\frac {b \sin (c+d x) (3 (2 a B+A b) \cos (c+d x)+b B)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B b^{2} \sec \left (d x + c\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \sec \left (d x + c\right )}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 10.25, size = 677, normalized size = 3.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{2}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^2}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________