Optimal. Leaf size=43 \[ \frac {a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{d f (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3537, 64} \[ \frac {a (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{d f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 64
Rule 3537
Rubi steps
\begin {align*} \int (d \tan (e+f x))^n (a-i a \tan (e+f x)) \, dx &=-\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,-i a \tan (e+f x)\right )}{f}\\ &=\frac {a \, _2F_1(1,1+n;2+n;-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 44, normalized size = 1.02 \[ \frac {a \tan (e+f x) (d \tan (e+f x))^n \, _2F_1(1,n+1;n+2;-i \tan (e+f x))}{f (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a \left (\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.82, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (f x +e \right )\right )^{n} \left (a -i a \tan \left (f x +e \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a-a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i a \left (\int i \left (d \tan {\left (e + f x \right )}\right )^{n}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________