Optimal. Leaf size=121 \[ \frac{16 i \sqrt{e \sec (c+d x)}}{15 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac{2 i}{5 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.220198, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3502, 3497, 3488} \[ \frac{16 i \sqrt{e \sec (c+d x)}}{15 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac{2 i}{5 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 3497
Rule 3488
Rubi steps
\begin{align*} \int \frac{1}{(e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{4 \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{5 a}\\ &=\frac{2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac{8 \int \frac{\sqrt{e \sec (c+d x)}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{15 e^2}\\ &=\frac{2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)}}+\frac{16 i \sqrt{e \sec (c+d x)}}{15 d e^2 \sqrt{a+i a \tan (c+d x)}}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.250152, size = 68, normalized size = 0.56 \[ -\frac{i \sec ^2(c+d x) (4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d \sqrt{a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.319, size = 105, normalized size = 0.9 \begin{align*}{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( 3\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +4\,i\cos \left ( dx+c \right ) +8\,\sin \left ( dx+c \right ) \right ) }{15\,ad{e}^{3}} \left ({\frac{e}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.93319, size = 176, normalized size = 1.45 \begin{align*} \frac{3 i \, \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) - 5 i \, \cos \left (\frac{3}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 30 i \, \cos \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 3 \, \sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ) + 5 \, \sin \left (\frac{3}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right ) + 30 \, \sin \left (\frac{1}{5} \, \arctan \left (\sin \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right ), \cos \left (\frac{5}{2} \, d x + \frac{5}{2} \, c\right )\right )\right )}{30 \, \sqrt{a} d e^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.06385, size = 265, normalized size = 2.19 \begin{align*} \frac{\sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-5 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 25 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 33 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-\frac{5}{2} i \, d x - \frac{5}{2} i \, c\right )}}{30 \, a d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \left (e \sec{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \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{1}{\left (e \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]