Optimal. Leaf size=111 \[ -\frac {(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n}+\frac {A (a+i a \tan (c+d x))^n}{d n}-\frac {i B (a+i a \tan (c+d x))^{n+1}}{a d (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3592, 3527, 3481, 68} \[ -\frac {(A-i B) (a+i a \tan (c+d x))^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d n}+\frac {A (a+i a \tan (c+d x))^n}{d n}-\frac {i B (a+i a \tan (c+d x))^{n+1}}{a d (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 68
Rule 3481
Rule 3527
Rule 3592
Rubi steps
\begin {align*} \int \tan (c+d x) (a+i a \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=-\frac {i B (a+i a \tan (c+d x))^{1+n}}{a d (1+n)}+\int (a+i a \tan (c+d x))^n (-B+A \tan (c+d x)) \, dx\\ &=\frac {A (a+i a \tan (c+d x))^n}{d n}-\frac {i B (a+i a \tan (c+d x))^{1+n}}{a d (1+n)}-(i A+B) \int (a+i a \tan (c+d x))^n \, dx\\ &=\frac {A (a+i a \tan (c+d x))^n}{d n}-\frac {i B (a+i a \tan (c+d x))^{1+n}}{a d (1+n)}-\frac {(a (A-i B)) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+n}}{a-x} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {A (a+i a \tan (c+d x))^n}{d n}-\frac {(A-i B) \, _2F_1\left (1,n;1+n;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^n}{2 d n}-\frac {i B (a+i a \tan (c+d x))^{1+n}}{a d (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 32.52, size = 270, normalized size = 2.43 \[ 2^{n-1} e^{-2 i d n x} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n \left (\frac {(A+i B) e^{2 i d n x} \left (1+e^{2 i (c+d x)}\right )^n \, _2F_1\left (n,n+2;n+1;-e^{2 i (c+d x)}\right )}{d n}+\frac {e^{2 i c} \left (-\frac {(A-i B) e^{2 i (c+d (n+2) x)} \left (1+e^{2 i (c+d x)}\right )^n \, _2F_1\left (n+2,n+2;n+3;-e^{2 i (c+d x)}\right )}{n+2}-\frac {2 i B e^{2 i d (n+1) x}}{(n+1) \left (1+e^{2 i (c+d x)}\right )}\right )}{d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left ({\left (-i \, A - B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, B e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A - B\right )} \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}}{e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 4.97, size = 0, normalized size = 0.00 \[ \int \tan \left (d x +c \right ) \left (a +i a \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {tan}\left (c+d\,x\right )\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \left (A + B \tan {\left (c + d x \right )}\right ) \tan {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________