3.239 \(\int e^{n \tan ^{-1}(a+b x)} x \, dx\)

Optimal. Leaf size=147 \[ \frac{2^{-\frac{i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right )}{b^2 (-n+2 i)}+\frac{(-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{2 b^2} \]

[Out]

((1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/(2*b^2) + ((2*a + n)*(1 - I*a - I*b*x)^(1 +
(I/2)*n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (1 - I*a - I*b*x)/2])/(2^((I/2)*n)*b^2*(2*I - n)
)

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Rubi [A]  time = 0.0672028, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5095, 80, 69} \[ \frac{2^{-\frac{i n}{2}} (2 a+n) (-i a-i b x+1)^{1+\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;\frac{1}{2} (-i a-i b x+1)\right )}{b^2 (-n+2 i)}+\frac{(-i a-i b x+1)^{1+\frac{i n}{2}} (i a+i b x+1)^{1-\frac{i n}{2}}}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTan[a + b*x])*x,x]

[Out]

((1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/(2*b^2) + ((2*a + n)*(1 - I*a - I*b*x)^(1 +
(I/2)*n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (1 - I*a - I*b*x)/2])/(2^((I/2)*n)*b^2*(2*I - n)
)

Rule 5095

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1 -
I*a*c - I*b*c*x)^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 69

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

Rubi steps

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

Mathematica [A]  time = 0.114278, size = 128, normalized size = 0.87 \[ \frac{i (-i (a+b x+i))^{1+\frac{i n}{2}} \left (\frac{2^{1-\frac{i n}{2}} (2 a+n) \, _2F_1\left (\frac{i n}{2}+1,\frac{i n}{2};\frac{i n}{2}+2;-\frac{1}{2} i (a+b x+i)\right )}{-2-i n}+(a+b x-i) (i a+i b x+1)^{-\frac{i n}{2}}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTan[a + b*x])*x,x]

[Out]

((I/2)*((-I)*(I + a + b*x))^(1 + (I/2)*n)*((-I + a + b*x)/(1 + I*a + I*b*x)^((I/2)*n) + (2^(1 - (I/2)*n)*(2*a
+ n)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-I/2)*(I + a + b*x)])/(-2 - I*n)))/b^2

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( bx+a \right ) }}x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctan(b*x+a))*x,x)

[Out]

int(exp(n*arctan(b*x+a))*x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(b*x+a))*x,x, algorithm="maxima")

[Out]

integrate(x*e^(n*arctan(b*x + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(b*x+a))*x,x, algorithm="fricas")

[Out]

integral(x*e^(n*arctan(b*x + a)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x e^{n \operatorname{atan}{\left (a + b x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atan(b*x+a))*x,x)

[Out]

Integral(x*exp(n*atan(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(b*x+a))*x,x, algorithm="giac")

[Out]

integrate(x*e^(n*arctan(b*x + a)), x)