3.359 \(\int \frac{e^{n \tan ^{-1}(a x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=281 \[ \frac{a^2 \left (1-n^2\right ) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \, _2F_1\left (1,\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1-i a x}{i a x+1}\right )}{(1+i n) \sqrt{a^2 c x^2+c}}-\frac{a n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x^2 \sqrt{a^2 c x^2+c}} \]

[Out]

-((1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/2)*Sqrt[1 + a^2*x^2])/(2*x^2*Sqrt[c + a^2*c*x^2]) - (a*n*(1
- I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/2)*Sqrt[1 + a^2*x^2])/(2*x*Sqrt[c + a^2*c*x^2]) + (a^2*(1 - n^2
)*(1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((-1 - I*n)/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1 + I*n)/2, (3
+ I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/((1 + I*n)*Sqrt[c + a^2*c*x^2])

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Rubi [A]  time = 0.25253, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {5085, 5082, 129, 151, 12, 131} \[ \frac{a^2 \left (1-n^2\right ) \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \, _2F_1\left (1,\frac{1}{2} (i n+1);\frac{1}{2} (i n+3);\frac{1-i a x}{i a x+1}\right )}{(1+i n) \sqrt{a^2 c x^2+c}}-\frac{a n \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)}}{2 x^2 \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcTan[a*x])/(x^3*Sqrt[c + a^2*c*x^2]),x]

[Out]

-((1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/2)*Sqrt[1 + a^2*x^2])/(2*x^2*Sqrt[c + a^2*c*x^2]) - (a*n*(1
- I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/2)*Sqrt[1 + a^2*x^2])/(2*x*Sqrt[c + a^2*c*x^2]) + (a^2*(1 - n^2
)*(1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((-1 - I*n)/2)*Sqrt[1 + a^2*x^2]*Hypergeometric2F1[1, (1 + I*n)/2, (3
+ I*n)/2, (1 - I*a*x)/(1 + I*a*x)])/((1 + I*n)*Sqrt[c + a^2*c*x^2])

Rule 5085

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(c^IntPart[p]*(c + d
*x^2)^FracPart[p])/(1 + a^2*x^2)^FracPart[p], Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c
, d, m, n, p}, x] && EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])

Rule 5082

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + (I*n)/2)*(1 + I*a*x)^(p - (I*n)/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rule 129

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 x)}}{x^3 \sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{n \tan ^{-1}(a x)}}{x^3 \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int \frac{(1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}}}{x^3} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{(1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \left (-a n+a^2 x\right )}{x^2} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x^2 \sqrt{c+a^2 c x^2}}-\frac{a n (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \int \frac{a^2 \left (1-n^2\right ) (1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}}}{x} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x^2 \sqrt{c+a^2 c x^2}}-\frac{a n (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x \sqrt{c+a^2 c x^2}}-\frac{\left (a^2 \left (1-n^2\right ) \sqrt{1+a^2 x^2}\right ) \int \frac{(1-i a x)^{-\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}}}{x} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{(1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x^2 \sqrt{c+a^2 c x^2}}-\frac{a n (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (1-i n)} \sqrt{1+a^2 x^2}}{2 x \sqrt{c+a^2 c x^2}}+\frac{a^2 \left (1-n^2\right ) (1-i a x)^{\frac{1}{2} (1+i n)} (1+i a x)^{\frac{1}{2} (-1-i n)} \sqrt{1+a^2 x^2} \, _2F_1\left (1,\frac{1}{2} (1+i n);\frac{1}{2} (3+i n);\frac{1-i a x}{1+i a x}\right )}{(1+i n) \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0766287, size = 159, normalized size = 0.57 \[ \frac{i \sqrt{a^2 x^2+1} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{1}{2}-\frac{i n}{2}} \left (2 a^2 \left (n^2-1\right ) x^2 \, _2F_1\left (1,\frac{i n}{2}+\frac{1}{2};\frac{i n}{2}+\frac{3}{2};\frac{a x+i}{i-a x}\right )-(n-i) (a x-i) (a n x+1)\right )}{2 (n-i) x^2 \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcTan[a*x])/(x^3*Sqrt[c + a^2*c*x^2]),x]

[Out]

((I/2)*(1 - I*a*x)^(1/2 + (I/2)*n)*(1 + I*a*x)^(-1/2 - (I/2)*n)*Sqrt[1 + a^2*x^2]*(-((-I + n)*(-I + a*x)*(1 +
a*n*x)) + 2*a^2*(-1 + n^2)*x^2*Hypergeometric2F1[1, 1/2 + (I/2)*n, 3/2 + (I/2)*n, (I + a*x)/(I - a*x)]))/((-I
+ n)*x^2*Sqrt[c + a^2*c*x^2])

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Maple [F]  time = 0.293, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}}{{x}^{3}}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c)^(1/2),x)

[Out]

int(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate(e^(n*arctan(a*x))/(sqrt(a^2*c*x^2 + c)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{5} + c x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*e^(n*arctan(a*x))/(a^2*c*x^5 + c*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{n \operatorname{atan}{\left (a x \right )}}}{x^{3} \sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*atan(a*x))/x**3/(a**2*c*x**2+c)**(1/2),x)

[Out]

Integral(exp(n*atan(a*x))/(x**3*sqrt(c*(a**2*x**2 + 1))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt{a^{2} c x^{2} + c} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arctan(a*x))/x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate(e^(n*arctan(a*x))/(sqrt(a^2*c*x^2 + c)*x^3), x)