Optimal. Leaf size=141 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3938, 3934, 203, 3983} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} f \sqrt {c-d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 3934
Rule 3938
Rule 3983
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx &=\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c+d \sec (e+f x)}} \, dx}{a}-\int \frac {\sec (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}} \, dx\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+a c x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{2+(a c-a d) x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c} f}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {c-d} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{\sqrt {a} \sqrt {c-d} f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.35, size = 171, normalized size = 1.21 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \sec (e+f x) \sqrt {c \cos (e+f x)+d} \left (\sqrt {2} \sqrt {c-d} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c \cos (e+f x)+d}}\right )-\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c-d} \sin \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c \cos (e+f x)+d}}\right )\right )}{\sqrt {c} f \sqrt {c-d} \sqrt {a (\sec (e+f x)+1)} \sqrt {c+d \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.69, size = 913, normalized size = 6.48 \[ \left [\frac {\sqrt {2} a c \sqrt {-\frac {1}{a c - a d}} \log \left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a c - a d}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (3 \, c - d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c + d\right )} \cos \left (f x + e\right ) - c + 3 \, d}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 2 \, \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{2 \, a c f}, \frac {\sqrt {2} a c \sqrt {-\frac {1}{a c - a d}} \log \left (\frac {2 \, \sqrt {2} {\left (c - d\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \sqrt {-\frac {1}{a c - a d}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (3 \, c - d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (c + d\right )} \cos \left (f x + e\right ) - c + 3 \, d}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 4 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right )}{2 \, a c f}, \frac {\frac {\sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a c - a d} \sin \left (f x + e\right )}\right )}{\sqrt {a c - a d}} - \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a c f}, \frac {\frac {\sqrt {2} a c \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a c - a d} \sin \left (f x + e\right )}\right )}{\sqrt {a c - a d}} - 2 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right )}{a c f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 2.13, size = 424, normalized size = 3.01 \[ -\frac {2 \sqrt {\frac {a \left (1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )\right ) \left (\sqrt {2}\, \sqrt {-\left (c -d \right )^{4} c}\, \arctan \left (\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (c -d \right )^{2} c \sqrt {2}}{\sin \left (f x +e \right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, \sqrt {-\left (c -d \right )^{4} c}}\right ) \sqrt {c -d}+\ln \left (-\frac {\sqrt {c -d}\, \cos \left (f x +e \right )-\sin \left (f x +e \right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}-\sqrt {c -d}}{\sin \left (f x +e \right )}\right ) c^{3}-2 \ln \left (-\frac {\sqrt {c -d}\, \cos \left (f x +e \right )-\sin \left (f x +e \right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}-\sqrt {c -d}}{\sin \left (f x +e \right )}\right ) c^{2} d +\ln \left (-\frac {\sqrt {c -d}\, \cos \left (f x +e \right )-\sin \left (f x +e \right ) \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}-\sqrt {c -d}}{\sin \left (f x +e \right )}\right ) c \,d^{2}\right )}{f \sin \left (f x +e \right )^{2} \sqrt {-\frac {2 \left (d +c \cos \left (f x +e \right )\right )}{1+\cos \left (f x +e \right )}}\, a \sqrt {c -d}\, \left (c^{2}-2 c d +d^{2}\right ) c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sec {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________