Optimal. Leaf size=652 \[ \frac {2 (b c-a d) \cot (e+f x) \sqrt {a+b \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \sqrt {\frac {(b c-a d) (\sec (e+f x)-1)}{(c+d) (a+b \sec (e+f x))}} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} F\left (\sin ^{-1}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{a b f \sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}}-\frac {2 c (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt {\frac {(a+b) (b c-a d) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left (\frac {a (c+d)}{(a+b) c};\sin ^{-1}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{a f (a+b) \sqrt {c+d \sec (e+f x)}}+\frac {2 d (c+d) \cot (e+f x) (a+b \sec (e+f x))^{3/2} \sqrt {\frac {(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \sqrt {-\frac {(a+b) (a d-b c) (\sec (e+f x)-1) (c+d \sec (e+f x))}{(c+d)^2 (a+b \sec (e+f x))^2}} \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\sqrt {\frac {(a+b) (c+d \sec (e+f x))}{(c+d) (a+b \sec (e+f x))}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f (a+b) \sqrt {c+d \sec (e+f x)}} \]
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Rubi [F] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx &=\int \frac {(c+d \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx\\ \end {align*}
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Mathematica [C] time = 32.78, size = 49385, normalized size = 75.74 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.10, size = 491, normalized size = 0.75 \[ \frac {2 \left (2 \EllipticPi \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) d^{2}+2 \EllipticPi \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{a -b}, \frac {\sqrt {\frac {c -d}{c +d}}}{\sqrt {\frac {a -b}{a +b}}}\right ) c^{2}-\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c^{2}+2 \EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) c d -\EllipticF \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {\frac {\left (a +b \right ) \left (c -d \right )}{\left (a -b \right ) \left (c +d \right )}}\right ) d^{2}\right ) \cos \left (f x +e \right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}}{f \left (-1+\cos \left (f x +e \right )\right ) \left (d +c \cos \left (f x +e \right )\right ) \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {a -b}{a +b}}} \]
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 \[ \int \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{\sqrt {b \sec \left (f x + e\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \]
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 \[ \int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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