Optimal. Leaf size=107 \[ \frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {4089}
\begin {gather*} \frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right )}{b^2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4089
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (A-A \sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx &=\frac {2 A \sqrt {a-b} (a+b) \cot (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a-b}}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{b^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 7.40, size = 211, normalized size = 1.97 \begin {gather*} \frac {A (a+b) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \left (\sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1+\sec (e+f x)}-\sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \sqrt {\sec (e+f x)} \sin (e+f x)\right )}{b f \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} \sqrt {a+b \sec (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(456\) vs.
\(2(98)=196\).
time = 11.47, size = 457, normalized size = 4.27
method | result | size |
default | \(-\frac {2 A \sqrt {\frac {a \cos \left (f x +e \right )+b}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, a +\cos \left (f x +e \right ) \EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, b +\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, a +\EllipticE \left (\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}, \sqrt {\frac {a -b}{a +b}}\right ) \sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {a \cos \left (f x +e \right )+b}{\left (\cos \left (f x +e \right )+1\right ) \left (a +b \right )}}\, b -\left (\cos ^{2}\left (f x +e \right )\right ) a +a \cos \left (f x +e \right )-\cos \left (f x +e \right ) b +b \right )}{f \sin \left (f x +e \right )^{5} \left (a \cos \left (f x +e \right )+b \right ) b}\) | \(457\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - A \left (\int \left (- \frac {\sec {\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\right )\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sec {\left (e + f x \right )}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A-\frac {A}{\cos \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________