Optimal. Leaf size=166 \[ \frac{2 g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{d f \sqrt{a+b \sec (e+f x)}}-\frac{2 c g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{d f (c+d) \sqrt{a+b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.855632, antiderivative size = 166, normalized size of antiderivative = 1., 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 = {3979, 3859, 2807, 2805, 3975} \[ \frac{2 g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (2;\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{d f \sqrt{a+b \sec (e+f x)}}-\frac{2 c g^2 \sqrt{g \sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{a+b}} \Pi \left (\frac{2 c}{c+d};\frac{1}{2} (e+f x)|\frac{2 a}{a+b}\right )}{d f (c+d) \sqrt{a+b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3979
Rule 3859
Rule 2807
Rule 2805
Rule 3975
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*}
Mathematica [C] time = 4.04738, size = 246, normalized size = 1.48 \[ -\frac{2 i g \cot (e+f x) (g \sec (e+f x))^{3/2} \sqrt{-\frac{a (\cos (e+f x)-1)}{a+b}} \sqrt{\frac{a (\cos (e+f x)+1)}{a-b}} \sqrt{a \cos (e+f x)+b} \left ((a d-b c) \Pi \left (1-\frac{a}{b};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right )|\frac{b-a}{a+b}\right )+b c \Pi \left (\frac{(a-b) c}{a d-b c};i \sinh ^{-1}\left (\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (e+f x)}\right )|\frac{b-a}{a+b}\right )\right )}{b d f \sqrt{\frac{1}{a-b}} (a d-b c) \sqrt{a+b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.354, size = 344, normalized size = 2.1 \begin{align*}{\frac{-2\,i \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{fd \left ( c+d \right ) \left ( c-d \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \left ({\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ) dc+{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},\sqrt{-{\frac{a-b}{a+b}}} \right ){d}^{2}+2\,{c}^{2}{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-1,i\sqrt{{\frac{a-b}{a+b}}} \right ) -2\,{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-1,i\sqrt{{\frac{a-b}{a+b}}} \right ){d}^{2}-2\,{c}^{2}{\it EllipticPi} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},-{\frac{c-d}{c+d}},i\sqrt{{\frac{a-b}{a+b}}} \right ) \right ) \sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}} \left ({\frac{g}{\cos \left ( fx+e \right ) }} \right ) ^{{\frac{5}{2}}} \left ( \left ( 1+\cos \left ( fx+e \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \sec \left (f x + e\right )\right )^{\frac{5}{2}}}{\sqrt{b \sec \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right ) + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]