Optimal. Leaf size=80 \[ \frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3515, 3502, 3488} \[ \frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3488
Rule 3502
Rule 3515
Rubi steps
\begin {align*} \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\left (\sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}+\frac {\left (2 \sqrt {e \cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{3 a}\\ &=\frac {2 i \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {4 i \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}}{3 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 48, normalized size = 0.60 \[ \frac {2 (2 \tan (c+d x)-i) \sqrt {e \cos (c+d x)}}{3 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 68, normalized size = 0.85 \[ \frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-\frac {3}{2} i \, d x - \frac {3}{2} i \, c\right )}}{3 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \cos \left (d x + c\right )}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.38, size = 74, normalized size = 0.92 \[ \frac {2 \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (i \left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right ) \sin \left (d x +c \right )-2 i\right )}{3 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 80, normalized size = 1.00 \[ \frac {\sqrt {e} {\left (i \, \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 3 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right ) + \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right )\right )}}{3 \, \sqrt {a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.73, size = 82, normalized size = 1.02 \[ \frac {\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+\sin \left (2\,c+2\,d\,x\right )-3{}\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}}{3\,a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \cos {\left (c + d x \right )}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________