Optimal. Leaf size=33 \[ \frac {(c-i c \tan (e+f x))^n}{f (-\tan (e+f x)+i)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3588, 74} \[ \frac {(c-i c \tan (e+f x))^n}{f (-\tan (e+f x)+i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 74
Rule 3588
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^n (-i (2+n)+(-2+n) \tan (e+f x))}{(-i+\tan (e+f x))^2} \, dx &=-\frac {(i c) \operatorname {Subst}\left (\int \frac {(c-i c x)^{-1+n} (-i (2+n)+(-2+n) x)}{(-i+x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(c-i c \tan (e+f x))^n}{f (i-\tan (e+f x))^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.18, size = 56, normalized size = 1.70 \[ \frac {(c \sec (e+f x))^n \exp (n (-\log (c \sec (e+f x))+\log (c-i c \tan (e+f x))))}{f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.54, size = 54, normalized size = 1.64 \[ -\frac {\left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} {\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left ({\left (n - 2\right )} \tan \left (f x + e\right ) - i \, n - 2 i\right )} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}}{{\left (\tan \left (f x + e\right ) - i\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 4.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (c -i c \tan \left (f x +e \right )\right )^{n} \left (-i \left (2+n \right )+\left (-2+n \right ) \tan \left (f x +e \right )\right )}{\left (\tan \left (f x +e \right )-i\right )^{2}}\, dx \]
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 [B] time = 9.38, size = 90, normalized size = 2.73 \[ \frac {{\left (-\frac {c\,\left (-2\,{\cos \left (e+f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (e+f\,x\right )}^2}\right )}^n\,\left (-4\,{\cos \left (e+f\,x\right )}^2-2\,{\cos \left (2\,e+2\,f\,x\right )}^2+\sin \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}+\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+2\right )}{4\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.47, size = 66, normalized size = 2.00 \[ \begin {cases} \frac {\left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f \tan ^{2}{\left (e + f x \right )} - 2 i f \tan {\left (e + f x \right )} - f} & \text {for}\: f \neq 0 \\\frac {x \left (\left (n - 2\right ) \tan {\relax (e )} - i \left (n + 2\right )\right ) \left (- i c \tan {\relax (e )} + c\right )^{n}}{\left (\tan {\relax (e )} - i\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________