Optimal. Leaf size=29 \[ -\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0556067, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 32} \[ -\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int \sqrt{a+x} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.122623, size = 34, normalized size = 1.17 \[ \frac{2 (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 24, normalized size = 0.8 \begin{align*}{\frac{-{\frac{2\,i}{3}}}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.971031, size = 28, normalized size = 0.97 \begin{align*} -\frac{2 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{3 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.27043, size = 132, normalized size = 4.55 \begin{align*} -\frac{4 i \, \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (3 i \, d x + 3 i \, c\right )}}{3 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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 \sqrt{a \left (i \tan{\left (c + d x \right )} + 1\right )} \sec ^{2}{\left (c + d x \right )}\, 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 \sqrt{i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]