Optimal. Leaf size=82 \[ \frac {i \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {i (a+i a \tan (c+d x))^{1+m}}{a d (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3624, 3562, 70}
\begin {gather*} \frac {i (a+i a \tan (c+d x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d m}-\frac {i (a+i a \tan (c+d x))^{m+1}}{a d (m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 70
Rule 3562
Rule 3624
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx &=-\frac {i (a+i a \tan (c+d x))^{1+m}}{a d (1+m)}-\int (a+i a \tan (c+d x))^m \, dx\\ &=-\frac {i (a+i a \tan (c+d x))^{1+m}}{a d (1+m)}+\frac {(i a) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {i \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {i (a+i a \tan (c+d x))^{1+m}}{a d (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 16.30, size = 0, normalized size = 0.00 \begin {gather*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.99, size = 0, normalized size = 0.00 \[\int \left (\tan ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{m}\, dx\]
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*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{m} \tan ^{2}{\left (c + d x \right )}\, dx \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 = -0.01 \begin {gather*} \int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________