Optimal. Leaf size=157 \[ \frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.251288, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ \frac{-B+i A}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a c f \sqrt{c-i c \tan (e+f x)}}-\frac{(-B+2 i A) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3588
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)}{\sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac{(a c) \operatorname{Subst}\left (\int \frac{A+B x}{(a+i a x)^{3/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}+\frac{((2 A+i B) c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac{(2 i A-B) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac{(2 A+i B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac{i A-B}{f \sqrt{a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac{(2 i A-B) \sqrt{a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac{(2 i A-B) \sqrt{a+i a \tan (e+f x)}}{3 a c f \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 7.03764, size = 103, normalized size = 0.66 \[ \frac{i \sqrt{c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) ((B-2 i A) \sin (2 (e+f x))+(A+2 i B) \cos (2 (e+f x))-3 A)}{6 c^2 f \sqrt{a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.19, size = 151, normalized size = 1. \begin{align*}{\frac{-{\frac{i}{3}} \left ( 2\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{4}-iB \left ( \tan \left ( fx+e \right ) \right ) ^{3}-B \left ( \tan \left ( fx+e \right ) \right ) ^{4}+3\,iA \left ( \tan \left ( fx+e \right ) \right ) ^{2}-2\,A \left ( \tan \left ( fx+e \right ) \right ) ^{3}-iB\tan \left ( fx+e \right ) +iA-2\,A\tan \left ( fx+e \right ) +B \right ) }{af{c}^{2} \left ( -\tan \left ( fx+e \right ) +i \right ) ^{2} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44176, size = 389, normalized size = 2.48 \begin{align*} \frac{{\left ({\left (-i \, A - B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-7 i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (4 i \, A + 4 \, B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} +{\left (-3 i \, A - 3 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left (4 i \, A + 4 \, B\right )} e^{\left (i \, f x + i \, e\right )} + 3 i \, A - 3 \, 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 )}}{12 \, a c^{2} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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}{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]