Optimal. Leaf size=292 \[ -\frac{(13 A-33 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \text{EllipticF}\left (\frac{1}{2} (c+d x),2\right )}{6 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{(13 A-33 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{7 (7 A-17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{(A-B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.559542, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4019, 3787, 3768, 3771, 2639, 2641} \[ \frac{7 (7 A-17 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{(13 A-33 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-33 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}-\frac{7 (7 A-17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac{(A-B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4019
Rule 3787
Rule 3768
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sec ^{\frac{9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\sec ^{\frac{7}{2}}(c+d x) \left (\frac{7}{2} a (A-B)-\frac{1}{2} a (3 A-13 B) \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left (\frac{25}{2} a^2 (A-2 B)-\frac{3}{2} a^2 (8 A-23 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{7 (7 A-17 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \sec ^{\frac{3}{2}}(c+d x) \left (\frac{21}{4} a^3 (7 A-17 B)-\frac{15}{4} a^3 (13 A-33 B) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{7 (7 A-17 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(13 A-33 B) \int \sec ^{\frac{5}{2}}(c+d x) \, dx}{4 a^3}+\frac{(7 (7 A-17 B)) \int \sec ^{\frac{3}{2}}(c+d x) \, dx}{20 a^3}\\ &=\frac{7 (7 A-17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac{(13 A-33 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{7 (7 A-17 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(13 A-33 B) \int \sqrt{\sec (c+d x)} \, dx}{12 a^3}-\frac{(7 (7 A-17 B)) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{20 a^3}\\ &=\frac{7 (7 A-17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac{(13 A-33 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{7 (7 A-17 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\left ((13 A-33 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}-\frac{\left (7 (7 A-17 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=-\frac{7 (7 A-17 B) \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{(13 A-33 B) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{6 a^3 d}+\frac{7 (7 A-17 B) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac{(13 A-33 B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac{(A-B) \sec ^{\frac{9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(A-2 B) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac{7 (7 A-17 B) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 7.96317, size = 953, normalized size = 3.26 \[ \frac{49 \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 ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{119 \sqrt{2} B 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 ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{26 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{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{22 B \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{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left (\frac{2 \sec \left (\frac{c}{2}\right ) \left (B \sin \left (\frac{d x}{2}\right )-A \sin \left (\frac{d x}{2}\right )\right ) \sec ^5\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}+\frac{2 (B-A) \tan \left (\frac{c}{2}\right ) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{5 d}+\frac{4 \sec \left (\frac{c}{2}\right ) \left (13 B \sin \left (\frac{d x}{2}\right )-8 A \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}+\frac{4 (13 B-8 A) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{15 d}+\frac{4 \sec \left (\frac{c}{2}\right ) \left (29 B \sin \left (\frac{d x}{2}\right )-13 A \sin \left (\frac{d x}{2}\right )\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}-\frac{14 (17 B-7 A) \cos (d x) \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right )}{5 d}+\frac{16 B \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{4 (33 \cos (c) B+4 B-13 A \cos (c)) \sec (c) \tan \left (\frac{c}{2}\right )}{3 d}\right ) \cos ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.949, size = 876, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B \sec \left (d x + c\right )^{5} + A \sec \left (d x + c\right )^{4}\right )} \sqrt{\sec \left (d x + c\right )}}{a^{3} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \left (d x + c\right ) + a^{3}}, 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{{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{9}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]