Optimal. Leaf size=115 \[ -\frac {2 E\left (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {-3-2 \sec (c+d x)}}{3 d \sqrt {-2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {4 \sqrt {-2-3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3-2 \sec (c+d x)}} \]
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Rubi [A]
time = 0.13, antiderivative size = 115, 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,
2733, 3943, 2741} \begin {gather*} -\frac {4 \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-2 \sec (c+d x)-3}}-\frac {2 \sqrt {-2 \sec (c+d x)-3} E\left (\left .\frac {1}{2} (c+d x+\pi )\right |6\right )}{3 d \sqrt {-3 \cos (c+d x)-2} \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2733
Rule 2741
Rule 3941
Rule 3943
Rule 3947
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-2 \sec (c+d x)} \sqrt {\sec (c+d x)}} \, dx &=-\left (\frac {1}{3} \int \frac {\sqrt {-3-2 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx\right )-\frac {2}{3} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {-3-2 \sec (c+d x)}} \, dx\\ &=-\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 {\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 {2 E\left (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {-3-2 \sec (c+d x)}}{3 d \sqrt {-2-3 \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {4 \sqrt {-2-3 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+\pi +d x)\right |6\right ) \sqrt {\sec (c+d x)}}{3 d \sqrt {-3-2 \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 81, normalized size = 0.70 \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.
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Maple [C] Result contains complex when optimal does not.
time = 0.31, size = 394, normalized size = 3.43
method | result | size |
default | \(-\frac {\left (3 i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}-5 i \sin \left (d x +c \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}+3 i \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-5 i \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, \frac {i \sqrt {5}}{5}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {10}\, \sqrt {\frac {2+3 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-30 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right )+20\right ) \sqrt {-\frac {2+3 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{15 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (2+3 \cos \left (d x +c \right )\right )}\) | \(394\) |
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 \,{\mathrm e}^{2 i \left (d x +c \right )}-\frac {8 \,{\mathrm e}^{i \left (d x +c \right )}}{3}-2}{\sqrt {\left (-3 \,{\mathrm e}^{2 i \left (d x +c \right )}-4 \,{\mathrm e}^{i \left (d x +c \right )}-3\right ) {\mathrm e}^{i \left (d x +c \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}}}\) | \(576\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.53, size = 108, normalized size = 0.94 \begin {gather*} \frac {4 \, \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 \, \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 \, \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 \, \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.
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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.
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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.
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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.
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