Optimal. Leaf size=43 \[ \frac {a \, _2F_1(1,1+n;2+n;-i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, 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 = {3618, 66}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 66
Rule 3618
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 ) \text {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 = 0.08, size = 44, normalized size = 1.02 \begin {gather*} \frac {a \, _2F_1(1,1+n;2+n;-i \tan (e+f x)) \tan (e+f x) (d \tan (e+f x))^n}{f (1+n)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.37, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________