Optimal. Leaf size=80 \[ \frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d (a+i a \tan (c+d x)) \sqrt {e \sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3502, 3771, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d (a+i a \tan (c+d x)) \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 3502
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))} \, dx &=\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a}\\ &=\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}+\frac {3 \int \sqrt {\cos (c+d x)} \, dx}{5 a \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 i}{5 d \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.88, size = 109, normalized size = 1.36 \[ \frac {(\tan (c+d x)+i) \left (-2 e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )+3 i \sin (2 (c+d x))+4 \cos (2 (c+d x))+4\right )}{5 a d \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} - 7 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 4 i \, e^{\left (3 i \, d x + 3 i \, c\right )} - 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e^{\left (i \, d x + i \, c\right )} - i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 10 \, {\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (3 i \, d x + 3 i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 6 i \, e^{\left (i \, d x + i \, c\right )} - 3 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (a d e e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a d e e^{\left (2 i \, d x + 2 i \, c\right )} + a d e e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{10 \, {\left (a d e e^{\left (4 i \, d x + 4 i \, c\right )} - a d e e^{\left (3 i \, d x + 3 i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.63, size = 358, normalized size = 4.48 \[ \frac {2 \left (3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )+i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-\left (\cos ^{4}\left (d x +c \right )\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \sqrt {\frac {e}{\cos \left (d x +c \right )}}}{5 a d \sin \left (d x +c \right )^{5} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {1}{\sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )} - i \sqrt {e \sec {\left (c + d x \right )}}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________