Optimal. Leaf size=92 \[ \frac{i A \sqrt{c-i c \tan (e+f x)}}{c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.199729, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3588, 78, 37} \[ \frac{i A \sqrt{c-i c \tan (e+f x)}}{c f \sqrt{a+i a \tan (e+f x)}}-\frac{B+i A}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3588
Rule 78
Rule 37
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{(a A) \operatorname{Subst}\left (\int \frac{1}{(a+i a x)^{3/2} \sqrt{c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i A+B}{f \sqrt{a+i a \tan (e+f x)} \sqrt{c-i c \tan (e+f x)}}+\frac{i A \sqrt{c-i c \tan (e+f x)}}{c f \sqrt{a+i a \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.77169, size = 77, normalized size = 0.84 \[ -\frac{\sqrt{c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x)) (B \cos (e+f x)-A \sin (e+f x))}{c f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.193, size = 99, normalized size = 1.1 \begin{align*}{\frac{A \left ( \tan \left ( fx+e \right ) \right ) ^{3}-B \left ( \tan \left ( fx+e \right ) \right ) ^{2}+A\tan \left ( fx+e \right ) -B}{afc \left ( -\tan \left ( fx+e \right ) +i \right ) ^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}\sqrt{a \left ( 1+i\tan \left ( fx+e \right ) \right ) }\sqrt{-c \left ( -1+i\tan \left ( fx+e \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.52534, size = 170, normalized size = 1.85 \begin{align*} -\frac{{\left ({\left (2 \, A - 2 i \, B\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 i \, B \cos \left (2 \, f x + 2 \, e\right ) - 2 \,{\left (-i \, A - B\right )} \sin \left (4 \, f x + 4 \, e\right ) + 4 \, B \sin \left (2 \, f x + 2 \, e\right ) - 2 \, A - 2 i \, B\right )} \sqrt{a} \sqrt{c}}{{\left (-4 i \, a c \cos \left (3 \, f x + 3 \, e\right ) - 4 i \, a c \cos \left (f x + e\right ) + 4 \, a c \sin \left (3 \, f x + 3 \, e\right ) + 4 \, a c \sin \left (f x + e\right )\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.31045, size = 290, normalized size = 3.15 \begin{align*} \frac{{\left ({\left (-i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, B e^{\left (3 i \, f x + 3 i \, e\right )} - 2 \, B e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, B e^{\left (i \, f x + i \, e\right )} + i \, A - B\right )} \sqrt{\frac{a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{2 \, a c f} \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{A + B \tan{\left (e + f x \right )}}{\sqrt{a \left (i \tan{\left (e + f x \right )} + 1\right )} \sqrt{- c \left (i \tan{\left (e + f x \right )} - 1\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 \frac{B \tan \left (f x + e\right ) + A}{\sqrt{i \, a \tan \left (f x + e\right ) + a} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]