Optimal. Leaf size=122 \[ -\frac {3 \sqrt {3+2 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {2+3 \sec (c+d x)}}+\frac {\sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {2+3 \sec (c+d x)}}{d \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 122, 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 {\sqrt {5} \sqrt {3 \sec (c+d x)+2} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{d \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)}}-\frac {3 \sqrt {2 \cos (c+d x)+3} \sqrt {\sec (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )}{\sqrt {5} d \sqrt {3 \sec (c+d x)+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2740
Rule 3941
Rule 3943
Rule 3947
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {2+3 \sec (c+d x)}} \, dx &=\frac {1}{2} \int \frac {\sqrt {2+3 \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx-\frac {3}{2} \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {2+3 \sec (c+d x)}} \, dx\\ &=-\frac {\left (3 \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {3+2 \cos (c+d x)}} \, dx}{2 \sqrt {2+3 \sec (c+d x)}}+\frac {\sqrt {2+3 \sec (c+d x)} \int \sqrt {3+2 \cos (c+d x)} \, dx}{2 \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=-\frac {3 \sqrt {3+2 \cos (c+d x)} F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {2+3 \sec (c+d x)}}+\frac {\sqrt {5} E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right ) \sqrt {2+3 \sec (c+d x)}}{d \sqrt {3+2 \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 78, normalized size = 0.64 \begin {gather*} \frac {\sqrt {3+2 \cos (c+d x)} \left (5 E\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )-3 F\left (\frac {1}{2} (c+d x)|\frac {4}{5}\right )\right ) \sqrt {\sec (c+d x)}}{\sqrt {5} d \sqrt {2+3 \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.63, size = 405, normalized size = 3.32
method | result | size |
default | \(-\frac {\left (2 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {5}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}+i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {5}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}+2 i \sqrt {5}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+i \sqrt {5}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right ) \sqrt {5}}{5 \sin \left (d x +c \right )}, \sqrt {5}\right ) \sqrt {10}\, \sqrt {\frac {3+2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+20 \left (\cos ^{2}\left (d x +c \right )\right )+10 \cos \left (d x +c \right )-30\right ) \sqrt {\frac {3+2 \cos \left (d x +c \right )}{\cos \left (d x +c \right )}}}{10 d \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \left (3+2 \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}\) | \(405\) |
risch | \(-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\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 {{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{\sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right ) {\mathrm e}^{i \left (d x +c \right )}}}+\frac {2 \left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}\, \sqrt {-5 \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\sqrt {5}}{2}+\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {{\mathrm e}^{i \left (d x +c \right )}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}\, \left (-\sqrt {5}\, \EllipticE \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}\right )+\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) \EllipticF \left (\sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}+\frac {3}{2}+\frac {\sqrt {5}}{2}}{\frac {3}{2}+\frac {\sqrt {5}}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}\right )\right )}{5 \sqrt {{\mathrm e}^{3 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1\right ) {\mathrm e}^{i \left (d x +c \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 {{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}+1}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(531\) |
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 = 0.76, size = 95, normalized size = 0.78 \begin {gather*} \frac {i \, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) - i \, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) + i \, {\rm weierstrassZeta}\left (8, -4, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right )\right ) - i \, {\rm weierstrassZeta}\left (8, -4, {\rm weierstrassPInverse}\left (8, -4, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right )\right )}{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 {3 \sec {\left (c + d x \right )} + 2} \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 {3}{\cos \left (c+d\,x\right )}+2}\,\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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