\(\int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx\) [327]

Optimal result

Integrand size = 24, antiderivative size = 82 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {i \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^m}{2 d m}-\frac {i (a+i a \tan (c+d x))^{1+m}}{a d (1+m)} \]

[Out]

1/2*I*hypergeom([1, m],[1+m],1/2+1/2*I*tan(d*x+c))*(a+I*a*tan(d*x+c))^m/d/m-I*(a+I*a*tan(d*x+c))^(1+m)/a/d/(1+
m)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3624, 3562, 70} \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {i (a+i a \tan (c+d x))^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {1}{2} (i \tan (c+d x)+1)\right )}{2 d m}-\frac {i (a+i a \tan (c+d x))^{m+1}}{a d (m+1)} \]

[In]

Int[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m,x]

[Out]

((I/2)*Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[c + d*x])/2]*(a + I*a*Tan[c + d*x])^m)/(d*m) - (I*(a + I*a*Ta
n[c + d*x])^(1 + m))/(a*d*(1 + m))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 3562

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[-b/d, Subst[Int[(a + x)^(n - 1)/(a - x), x]
, x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 + b^2, 0]

Rule 3624

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rubi steps \begin{align*} \text {integral}& = -\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 \operatorname {Hypergeometric2F1}\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 [A] (verified)

Time = 0.45 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.89 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\frac {(a+i a \tan (c+d x))^m \left (i (1+m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {1}{2} (1+i \tan (c+d x))\right )+2 m (-i+\tan (c+d x))\right )}{2 d m (1+m)} \]

[In]

Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^m,x]

[Out]

((a + I*a*Tan[c + d*x])^m*(I*(1 + m)*Hypergeometric2F1[1, m, 1 + m, (1 + I*Tan[c + d*x])/2] + 2*m*(-I + Tan[c
+ d*x])))/(2*d*m*(1 + m))

Maple [F]

\[\int \left (\tan ^{2}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{m}d x\]

[In]

int(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m,x)

[Out]

int(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m,x)

Fricas [F]

\[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m,x, algorithm="fricas")

[Out]

integral(-(2*a*e^(2*I*d*x + 2*I*c)/(e^(2*I*d*x + 2*I*c) + 1))^m*(e^(4*I*d*x + 4*I*c) - 2*e^(2*I*d*x + 2*I*c) +
1)/(e^(4*I*d*x + 4*I*c) + 2*e^(2*I*d*x + 2*I*c) + 1), x)

Sympy [F]

\[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{m} \tan ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(tan(d*x+c)**2*(a+I*a*tan(d*x+c))**m,x)

[Out]

Integral((I*a*(tan(c + d*x) - I))**m*tan(c + d*x)**2, x)

Maxima [F]

\[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m,x, algorithm="maxima")

[Out]

integrate((I*a*tan(d*x + c) + a)^m*tan(d*x + c)^2, x)

Giac [F]

\[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^m,x, algorithm="giac")

[Out]

integrate((I*a*tan(d*x + c) + a)^m*tan(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^m \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m \,d x \]

[In]

int(tan(c + d*x)^2*(a + a*tan(c + d*x)*1i)^m,x)

[Out]

int(tan(c + d*x)^2*(a + a*tan(c + d*x)*1i)^m, x)