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

Optimal. Leaf size=191 \[ \frac{2 i (-i a-i b x+1)^{\frac{i n}{2}} (i a+i b x+1)^{-\frac{i n}{2}} \, _2F_1\left (1,\frac{i n}{2};\frac{i n}{2}+1;\frac{(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{n}-\frac{i 2^{1-\frac{i n}{2}} (-i a-i b x+1)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (-i a-i b x+1)\right )}{n} \]

[Out]

((2*I)*(1 - I*a - I*b*x)^((I/2)*n)*Hypergeometric2F1[1, (I/2)*n, 1 + (I/2)*n, ((I - a)*(1 - I*a - I*b*x))/((I
+ a)*(1 + I*a + I*b*x))])/(n*(1 + I*a + I*b*x)^((I/2)*n)) - (I*2^(1 - (I/2)*n)*(1 - I*a - I*b*x)^((I/2)*n)*Hyp
ergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a - I*b*x)/2])/n

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Rubi [A]  time = 0.0742903, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5095, 105, 69, 131} \[ \frac{2 i (-i a-i b x+1)^{\frac{i n}{2}} (i a+i b x+1)^{-\frac{i n}{2}} \, _2F_1\left (1,\frac{i n}{2};\frac{i n}{2}+1;\frac{(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{n}-\frac{i 2^{1-\frac{i n}{2}} (-i a-i b x+1)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;\frac{1}{2} (-i a-i b x+1)\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTan[a + b*x])/x,x]

[Out]

((2*I)*(1 - I*a - I*b*x)^((I/2)*n)*Hypergeometric2F1[1, (I/2)*n, 1 + (I/2)*n, ((I - a)*(1 - I*a - I*b*x))/((I
+ a)*(1 + I*a + I*b*x))])/(n*(1 + I*a + I*b*x)^((I/2)*n)) - (I*2^(1 - (I/2)*n)*(1 - I*a - I*b*x)^((I/2)*n)*Hyp
ergeometric2F1[(I/2)*n, (I/2)*n, 1 + (I/2)*n, (1 - I*a - I*b*x)/2])/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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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]))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

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

Mathematica [A]  time = 0.0344468, size = 170, normalized size = 0.89 \[ \frac{2 i (i a+i b x+1)^{-\frac{i n}{2}} (-i (a+b x+i))^{\frac{i n}{2}} \left (\, _2F_1\left (1,\frac{i n}{2};\frac{i n}{2}+1;\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )-2^{-\frac{i n}{2}} (i a+i b x+1)^{\frac{i n}{2}} \, _2F_1\left (\frac{i n}{2},\frac{i n}{2};\frac{i n}{2}+1;-\frac{1}{2} i (a+b x+i)\right )\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTan[a + b*x])/x,x]

[Out]

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

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Maple [F]  time = 0.198, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\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 \frac{e^{\left (n \arctan \left (b x + a\right )\right )}}{x}\,{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(e^(n*arctan(b*x + a))/x, x)

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

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

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