3.309 \(\int (e \tan (c+d x))^m (a-i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=43 \[ \frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;-i \tan (c+d x))}{d e (m+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.041055, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ \frac{a (e \tan (c+d x))^{m+1} \, _2F_1(1,m+1;m+2;-i \tan (c+d x))}{d e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3537

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

Rule 64

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

Rubi steps

\begin{align*} \int (e \tan (c+d x))^m (a-i a \tan (c+d x)) \, dx &=-\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{i e x}{a}\right )^m}{-a^2+a x} \, dx,x,-i a \tan (c+d x)\right )}{d}\\ &=\frac{a \, _2F_1(1,1+m;2+m;-i \tan (c+d x)) (e \tan (c+d x))^{1+m}}{d e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0631259, size = 44, normalized size = 1.02 \[ \frac{a \tan (c+d x) (e \tan (c+d x))^m \, _2F_1(1,m+1;m+2;-i \tan (c+d x))}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.188, size = 0, normalized size = 0. \begin{align*} \int \left ( e\tan \left ( dx+c \right ) \right ) ^{m} \left ( a-ia\tan \left ( dx+c \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-i \, a \tan \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{2 \, a \left (\frac{-i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + i \, e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - a \left (\int - \left (e \tan{\left (c + d x \right )}\right )^{m}\, dx + \int i \left (e \tan{\left (c + d x \right )}\right )^{m} \tan{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-i \, a \tan \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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