Optimal. Leaf size=249 \[ \frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{2 a^3 d}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}-\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.533404, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4085, 4020, 3787, 3769, 3771, 2641, 2639} \[ \frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{2 a^3 d}-\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4085
Rule 4020
Rule 3787
Rule 3769
Rule 3771
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx &=-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{\int \frac{-\frac{1}{2} a (13 A+3 C)+\frac{1}{2} a (7 A-3 C) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{\int \frac{-\frac{3}{2} a^2 (23 A+3 C)+25 a^2 A \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx}{15 a^4}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \frac{-\frac{45}{4} a^3 (11 A+C)+\frac{3}{4} a^3 (119 A+9 C) \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(3 (11 A+C)) \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{4 a^3}-\frac{(119 A+9 C) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}\\ &=\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(11 A+C) \int \sqrt{\sec (c+d x)} \, dx}{4 a^3}-\frac{\left ((119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{(119 A+9 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\left ((11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{4 a^3}\\ &=-\frac{(119 A+9 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(11 A+C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{2 a^3 d}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 7.11061, size = 1008, normalized size = 4.05 \[ \frac{238 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left (\frac{c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) \sec \left (\frac{c}{2}\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+A\right ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{6 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left (\frac{c}{2}\right ) \left (e^{2 i d x} \left (-1+e^{2 i c}\right ) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-e^{2 i (c+d x)}\right )-3 \sqrt{1+e^{2 i (c+d x)}}\right ) \sec \left (\frac{c}{2}\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+A\right ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{44 A \sqrt{\cos (c+d x)} \csc \left (\frac{c}{2}\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sec \left (\frac{c}{2}\right ) \sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \sin (c) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left (\frac{c}{2}\right ) \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right ) \sec \left (\frac{c}{2}\right ) \sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \sin (c) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left (C \sec ^2(c+d x)+A\right ) \left (-\frac{4 \sec \left (\frac{c}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+C \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}-\frac{4 (A+C) \tan \left (\frac{c}{2}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}+\frac{16 \sec \left (\frac{c}{2}\right ) \left (11 A \sin \left (\frac{d x}{2}\right )+6 C \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}+\frac{16 (11 A+6 C) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}-\frac{8 \sec \left (\frac{c}{2}\right ) \left (43 A \sin \left (\frac{d x}{2}\right )+9 C \sin \left (\frac{d x}{2}\right )\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{4 (30 \cos (2 c) A+89 A+9 C) \cos (d x) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right )}{5 d}+\frac{16 A \cos (2 d x) \sin (2 c)}{3 d}-\frac{96 A \cos (c) \sin (d x)}{d}+\frac{16 A \cos (2 c) \sin (2 d x)}{3 d}-\frac{8 (43 A+9 C) \tan \left (\frac{c}{2}\right )}{3 d}\right ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 2.18, size = 465, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt{\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{5} + 3 \, a^{3} \sec \left (d x + c\right )^{4} + 3 \, a^{3} \sec \left (d x + c\right )^{3} + a^{3} \sec \left (d x + c\right )^{2}}, x\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{C \sec \left (d x + c\right )^{2} + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3} \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]