Optimal. Leaf size=231 \[ -\frac {2 c \cot (e+f x) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left (\frac {c}{c+d};\sin ^{-1}\left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {c+d}}-\frac {(c-d) \sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
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Rubi [A] time = 0.26, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3927, 3780, 3968} \[ -\frac {2 c \cot (e+f x) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left (\frac {c}{c+d};\sin ^{-1}\left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {c+d}}-\frac {(c-d) \sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
Antiderivative was successfully verified.
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Rule 3780
Rule 3927
Rule 3968
Rubi steps
\begin {align*} \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx &=\frac {c \int \sqrt {c+d \sec (e+f x)} \, dx}{a}+(-c+d) \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx\\ &=-\frac {2 c \cot (e+f x) \Pi \left (\frac {c}{c+d};\sin ^{-1}\left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right )|\frac {c-d}{c+d}\right ) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (1+\sec (e+f x))}{c+d \sec (e+f x)}} (c+d \sec (e+f x))}{a \sqrt {c+d} f}-\frac {(c-d) E\left (\sin ^{-1}\left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}}\\ \end {align*}
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Mathematica [B] time = 18.23, size = 810, normalized size = 3.51 \[ \frac {(c+d \sec (e+f x))^{3/2} \left (2 \sec \left (\frac {1}{2} (e+f x)\right ) \left (d \sin \left (\frac {1}{2} (e+f x)\right )-c \sin \left (\frac {1}{2} (e+f x)\right )\right )-2 (d-c) \sin (e+f x)\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f (d+c \cos (e+f x)) (\sec (e+f x) a+a)}+\frac {2 (c+d \sec (e+f x))^{3/2} \left (c^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )+d^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )-2 c d \tan ^5\left (\frac {1}{2} (e+f x)\right )-2 c^2 \tan ^3\left (\frac {1}{2} (e+f x)\right )+2 c d \tan ^3\left (\frac {1}{2} (e+f x)\right )-4 c^2 \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )+c+d}{c+d}} \tan ^2\left (\frac {1}{2} (e+f x)\right )+c^2 \tan \left (\frac {1}{2} (e+f x)\right )-d^2 \tan \left (\frac {1}{2} (e+f x)\right )+\left (c^2-d^2\right ) E\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )+c+d}{c+d}}+2 c (c-d) F\left (\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right ) \sqrt {\frac {-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )+c+d}{c+d}}-4 c^2 \Pi \left (-1;\sin ^{-1}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )+c+d}{c+d}}\right ) \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{f (d+c \cos (e+f x))^{3/2} \sqrt {\sec (e+f x)} (\sec (e+f x) a+a) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac {1}{2} (e+f x)\right )+1\right )^{3/2} \sqrt {\frac {-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )+c+d}{\tan ^2\left (\frac {1}{2} (e+f x)\right )+1}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.92, size = 295, normalized size = 1.28 \[ -\frac {\sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \left (-1+\cos \left (f x +e \right )\right ) \left (2 \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-2 c \EllipticF \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) d +\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-4 c^{2} \EllipticPi \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, -1, \sqrt {\frac {c -d}{c +d}}\right )\right )}{a f \left (d +c \cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{2}} \]
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 \[ \int \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{a+\frac {a}{\cos \left (e+f\,x\right )}} \,d x \]
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 \[ \frac {\int \frac {c \sqrt {c + d \sec {\left (e + f x \right )}}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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