Optimal. Leaf size=127 \[ -\frac {4 \sqrt {2+3 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {3+2 \sec (c+d x)}}+\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {3+2 \sec (c+d x)}}{3 d \sqrt {2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3947, 3941,
2732, 3943, 2740} \begin {gather*} \frac {2 \sqrt {5} \sqrt {2 \sec (c+d x)+3} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 d \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)}}-\frac {4 \sqrt {3 \cos (c+d x)+2} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )}{3 \sqrt {5} d \sqrt {2 \sec (c+d x)+3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2740
Rule 3941
Rule 3943
Rule 3947
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {3+2 \sec (c+d x)}} \, dx &=\frac {1}{3} \int \frac {\sqrt {3+2 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx-\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {3+2 \sec (c+d x)}} \, dx\\ &=-\frac {\left (2 \sqrt {2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {2+3 \cos (c+d x)}} \, dx}{3 \sqrt {3+2 \sec (c+d x)}}+\frac {\sqrt {3+2 \sec (c+d x)} \int \sqrt {2+3 \cos (c+d x)} \, dx}{3 \sqrt {2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=-\frac {4 \sqrt {2+3 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {3+2 \sec (c+d x)}}+\frac {2 \sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right ) \sqrt {3+2 \sec (c+d x)}}{3 d \sqrt {2+3 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 81, normalized size = 0.64 \begin {gather*} \frac {2 \sqrt {2+3 \cos (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )-2 F\left (\frac {1}{2} (c+d x)|\frac {6}{5}\right )\right ) \sqrt {\sec (c+d x)}}{3 \sqrt {5} d \sqrt {3+2 \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 1.10, size = 409, normalized size = 3.22
method | result | size |
default | \(\frac {\left (3 \EllipticF \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {5}-\EllipticE \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {5}+3 \EllipticF \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {5}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\EllipticE \left (\frac {\left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, i \sqrt {5}\right ) \sqrt {5}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-30 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right )+20\right ) \sqrt {\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{15 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (2+3 \cos \left (d x +c \right )\right )}\) | \(409\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right ) \sqrt {2}}{3 d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}{3 \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}}+\frac {\left (\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}\, \sqrt {30}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}\, \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{-\frac {2}{3}-\frac {i \sqrt {5}}{3}}}\, \left (-\frac {2 i \sqrt {5}\, \EllipticE \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )}{3}+\left (-\frac {2}{3}+\frac {i \sqrt {5}}{3}\right ) \EllipticF \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {2}{3}+\frac {i \sqrt {5}}{3}}{\frac {2}{3}+\frac {i \sqrt {5}}{3}}}, \frac {\sqrt {30}\, \sqrt {i \left (-\frac {2}{3}-\frac {i \sqrt {5}}{3}\right ) \sqrt {5}}}{10}\right )\right )}{5 \sqrt {3 \,{\mathrm e}^{3 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )}}{d \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+3}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(573\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.65, size = 108, normalized size = 0.85 \begin {gather*} \frac {4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) - 4 i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right ) - 9 i \, \sqrt {6} {\rm weierstrassZeta}\left (-\frac {44}{27}, \frac {784}{729}, {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )\right )}{27 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 \sec {\left (c + d x \right )} + 3} \sqrt {\sec {\left (c + d x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {\frac {2}{\cos \left (c+d\,x\right )}+3}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________