Optimal. Leaf size=242 \[ -\frac{2 \left (a^2+3 b^2\right ) (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{3 a^4 d}+\frac{2 \left (3 a^2 A-5 a b B+5 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^3 d}+\frac{2 b^3 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.755326, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4034, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{2 \left (a^2+3 b^2\right ) (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^4 d}+\frac{2 \left (3 a^2 A-5 a b B+5 A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 a^3 d}+\frac{2 b^3 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^4 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4034
Rule 4104
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx &=\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \int \frac{\frac{5}{2} (A b-a B)-\frac{3}{2} a A \sec (c+d x)-\frac{3}{2} A b \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx}{5 a}\\ &=\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{4 \int \frac{\frac{3}{4} \left (3 a^2 A+5 A b^2-5 a b B\right )+\frac{1}{4} a (4 A b+5 a B) \sec (c+d x)-\frac{5}{4} b (A b-a B) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx}{15 a^2}\\ &=\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{4 \int \frac{\frac{3}{4} a \left (3 a^2 A+5 A b^2-5 a b B\right )-\left (-\frac{1}{4} a^2 (4 A b+5 a B)+\frac{3}{4} b \left (3 a^2 A+5 A b^2-5 a b B\right )\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{15 a^4}+\frac{\left (b^3 (A b-a B)\right ) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx}{a^4}\\ &=\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (a^2+3 b^2\right ) (A b-a B)\right ) \int \sqrt{\sec (c+d x)} \, dx}{3 a^4}+\frac{\left (3 a^2 A+5 A b^2-5 a b B\right ) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{5 a^3}+\frac{\left (b^3 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a^4}\\ &=\frac{2 b^3 (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^4 (a+b) d}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{\left (\left (a^2+3 b^2\right ) (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a^4}+\frac{\left (\left (3 a^2 A+5 A b^2-5 a b B\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 a^3}\\ &=\frac{2 \left (3 a^2 A+5 A b^2-5 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)}}{5 a^3 d}-\frac{2 \left (a^2+3 b^2\right ) (A b-a B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 a^4 d}+\frac{2 b^3 (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 a}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^4 (a+b) d}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 6.93873, size = 617, normalized size = 2.55 \[ \frac{\frac{2 \left (9 a^2 A-5 a b B+5 A b^2\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left (\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+\Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{b \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (9 a^2 A-15 a b B+15 A b^2\right ) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left (a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right ),-1\right )+a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 a b \sec ^2(c+d x)+2 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+2 a b\right )}{a^2 b \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a \cos (c+d x)+b)}-\frac{2 \left (10 a^2 B+8 a A b\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left (-\frac{b}{a};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{a \left (1-\cos ^2(c+d x)\right ) (a \cos (c+d x)+b)}}{30 a^2 d}+\frac{\sqrt{\sec (c+d x)} \left (\frac{(a B-A b) \sin (2 (c+d x))}{3 a^2}+\frac{A \sin (c+d x)}{10 a}+\frac{A \sin (3 (c+d x))}{10 a}\right )}{d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.22, size = 1074, 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]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
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{B \sec \left (d x + c\right ) + A}{{\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]