Optimal. Leaf size=166 \[ \frac {2 g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}} \]
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Rubi [A]
time = 0.60, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {4064, 3944,
2886, 2884, 4060} \begin {gather*} \frac {2 g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{d f (c+d) \sqrt {a+b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2884
Rule 2886
Rule 3944
Rule 4060
Rule 4064
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{5/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx &=\frac {g \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx}{d}-\frac {(c g) \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))} \, dx}{d}\\ &=\frac {\left (g^2 \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {b+a \cos (e+f x)}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left (c g^2 \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))} \, dx}{d \sqrt {a+b \sec (e+f x)}}\\ &=\frac {\left (g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}}} \, dx}{d \sqrt {a+b \sec (e+f x)}}-\frac {\left (c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}} (d+c \cos (e+f x))} \, dx}{d \sqrt {a+b \sec (e+f x)}}\\ &=\frac {2 g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d f \sqrt {a+b \sec (e+f x)}}-\frac {2 c g^2 \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \Pi \left (\frac {2 c}{c+d};\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{d (c+d) f \sqrt {a+b \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 13.68, size = 246, normalized size = 1.48 \begin {gather*} -\frac {2 i g \sqrt {-\frac {a (-1+\cos (e+f x))}{a+b}} \sqrt {\frac {a (1+\cos (e+f x))}{a-b}} \sqrt {b+a \cos (e+f x)} \cot (e+f x) \left ((-b c+a d) \Pi \left (1-\frac {a}{b};i \sinh ^{-1}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right )|\frac {-a+b}{a+b}\right )+b c \Pi \left (\frac {(a-b) c}{-b c+a d};i \sinh ^{-1}\left (\sqrt {\frac {1}{a-b}} \sqrt {b+a \cos (e+f x)}\right )|\frac {-a+b}{a+b}\right )\right ) (g \sec (e+f x))^{3/2}}{\sqrt {\frac {1}{a-b}} b d (-b c+a d) f \sqrt {a+b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 6.28, size = 346, normalized size = 2.08
method | result | size |
default | \(\frac {2 i \left (\EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right ) d c +\EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right ) d^{2}+2 c^{2} \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -1, i \sqrt {\frac {a -b}{a +b}}\right )-2 \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -1, i \sqrt {\frac {a -b}{a +b}}\right ) d^{2}-2 c^{2} \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -\frac {c -d}{c +d}, i \sqrt {\frac {a -b}{a +b}}\right )\right ) \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\cos ^{3}\left (f x +e \right )\right )}{f \left (a \cos \left (f x +e \right )+b \right ) \left (\frac {1}{\cos \left (f x +e \right )+1}\right )^{\frac {5}{2}} \sin \left (f x +e \right )^{4} d \left (c +d \right ) \left (c -d \right )}\) | \(346\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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