Optimal. Leaf size=295 \[ -\frac {g \sin (e+f x) \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b)}{f (c \cos (e+f x)+c) \sqrt {a+b \sec (e+f x)}}+\frac {g (a-b) \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} F\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \sqrt {a+b \sec (e+f x)}}+\frac {2 b g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.92, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.282, Rules used = {3971, 3859, 2807, 2805, 3975, 2768, 2752, 2663, 2661, 2655, 2653} \[ -\frac {g \sin (e+f x) \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b)}{f (c \cos (e+f x)+c) \sqrt {a+b \sec (e+f x)}}+\frac {g (a-b) \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} F\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}}+\frac {g \sqrt {g \sec (e+f x)} (a \cos (e+f x)+b) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \sqrt {a+b \sec (e+f x)}}+\frac {2 b g \sqrt {g \sec (e+f x)} \sqrt {\frac {a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right )}{c f \sqrt {a+b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2768
Rule 2805
Rule 2807
Rule 3859
Rule 3971
Rule 3975
Rubi steps
\begin {align*} \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx &=-\left ((-a+b) \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx\right )+\frac {b \int \frac {(g \sec (e+f x))^{3/2}}{\sqrt {a+b \sec (e+f x)}} \, dx}{c}\\ &=-\frac {\left ((-a+b) g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} (c+c \cos (e+f x))} \, dx}{\sqrt {a+b \sec (e+f x)}}+\frac {\left (b g \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}{c \sqrt {a+b \sec (e+f x)}}\\ &=-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}}+\frac {\left (a (-a+b) g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {-\frac {c}{2}-\frac {1}{2} c \cos (e+f x)}{\sqrt {b+a \cos (e+f x)}} \, dx}{(a-b) c^2 \sqrt {a+b \sec (e+f x)}}+\frac {\left (b g \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}{c \sqrt {a+b \sec (e+f x)}}\\ &=\frac {2 b g \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)}}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}}-\frac {\left ((-a+b) g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)}} \, dx}{2 c \sqrt {a+b \sec (e+f x)}}-\frac {\left ((-a+b) g \sqrt {b+a \cos (e+f x)} \sqrt {g \sec (e+f x)}\right ) \int \sqrt {b+a \cos (e+f x)} \, dx}{2 (a-b) c \sqrt {a+b \sec (e+f x)}}\\ &=\frac {2 b g \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)}}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}}-\frac {\left ((-a+b) g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (e+f x)}{a+b}} \, dx}{2 (a-b) c \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {a+b \sec (e+f x)}}-\frac {\left ((-a+b) g \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}}} \, dx}{2 c \sqrt {a+b \sec (e+f x)}}\\ &=\frac {g (b+a \cos (e+f x)) E\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {\frac {b+a \cos (e+f x)}{a+b}} \sqrt {a+b \sec (e+f x)}}+\frac {(a-b) g \sqrt {\frac {b+a \cos (e+f x)}{a+b}} F\left (\frac {1}{2} (e+f x)|\frac {2 a}{a+b}\right ) \sqrt {g \sec (e+f x)}}{c f \sqrt {a+b \sec (e+f x)}}+\frac {2 b g \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)}}{c f \sqrt {a+b \sec (e+f x)}}-\frac {g (b+a \cos (e+f x)) \sqrt {g \sec (e+f x)} \sin (e+f x)}{f (c+c \cos (e+f x)) \sqrt {a+b \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 20.05, size = 0, normalized size = 0.00 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+b \sec (e+f x)}}{c+c \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{c \sec \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.22, size = 292, normalized size = 0.99 \[ \frac {i \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \left (2 a \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )-2 b \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )-a \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )-b \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, \sqrt {-\frac {a -b}{a +b}}\right )+4 \EllipticPi \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, -1, i \sqrt {\frac {a -b}{a +b}}\right ) b \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (\frac {g}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \left (\cos ^{2}\left (f x +e \right )\right )}{c f \left (b +a \cos \left (f x +e \right )\right ) \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{c \sec \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{c+\frac {c}{\cos \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}} \sqrt {a + b \sec {\left (e + f x \right )}}}{\sec {\left (e + f x \right )} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________