3.44 \(\int e^{\sec ^{-1}(a x)} x \, dx\)

Optimal. Leaf size=91 \[ \frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},3;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2} \]

[Out]

(-8/5-4/5*I)*exp((1+2*I)*arcsec(a*x))*hypergeom([2, 1-1/2*I],[2-1/2*I],-(1/a/x+I*(1-1/a^2/x^2)^(1/2))^2)/a^2+(
16/5+8/5*I)*exp((1+2*I)*arcsec(a*x))*hypergeom([3, 1-1/2*I],[2-1/2*I],-(1/a/x+I*(1-1/a^2/x^2)^(1/2))^2)/a^2

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Rubi [A]  time = 0.11, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5266, 12, 4471, 2251} \[ \frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},3;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSec[a*x]*x,x]

[Out]

((-8/5 - (4*I)/5)*E^((1 + 2*I)*ArcSec[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcSec[a*x])])/a^
2 + ((16/5 + (8*I)/5)*E^((1 + 2*I)*ArcSec[a*x])*Hypergeometric2F1[1 - I/2, 3, 2 - I/2, -E^((2*I)*ArcSec[a*x])]
)/a^2

Rule 12

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

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rule 4471

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol]
:> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
IGtQ[m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]

Rule 5266

Int[(u_.)*(f_)^(ArcSec[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[1/b, Subst[Int[(u /. x -> -(a/b) +
Sec[x]/b)*f^(c*x^n)*Sec[x]*Tan[x], x], x, ArcSec[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int e^{\sec ^{-1}(a x)} x \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^x \sec ^2(x) \tan (x)}{a} \, dx,x,\sec ^{-1}(a x)\right )}{a}\\ &=\frac {\operatorname {Subst}\left (\int e^x \sec ^2(x) \tan (x) \, dx,x,\sec ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {8 i e^{(1+2 i) x}}{\left (1+e^{2 i x}\right )^3}-\frac {4 i e^{(1+2 i) x}}{\left (1+e^{2 i x}\right )^2}\right ) \, dx,x,\sec ^{-1}(a x)\right )}{a^2}\\ &=-\frac {(4 i) \operatorname {Subst}\left (\int \frac {e^{(1+2 i) x}}{\left (1+e^{2 i x}\right )^2} \, dx,x,\sec ^{-1}(a x)\right )}{a^2}+\frac {(8 i) \operatorname {Subst}\left (\int \frac {e^{(1+2 i) x}}{\left (1+e^{2 i x}\right )^3} \, dx,x,\sec ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2}+\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},3;2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )}{a^2}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 107, normalized size = 1.18 \[ \frac {\left (\frac {1}{5}+\frac {i}{10}\right ) e^{\sec ^{-1}(a x)} \left ((-2+i) a x \left (\sqrt {1-\frac {1}{a^2 x^2}}-a x\right )+(1+2 i) \, _2F_1\left (-\frac {i}{2},1;1-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )-e^{2 i \sec ^{-1}(a x)} \, _2F_1\left (1,1-\frac {i}{2};2-\frac {i}{2};-e^{2 i \sec ^{-1}(a x)}\right )\right )}{a^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSec[a*x]*x,x]

[Out]

((1/5 + I/10)*E^ArcSec[a*x]*((-2 + I)*a*x*(Sqrt[1 - 1/(a^2*x^2)] - a*x) + (1 + 2*I)*Hypergeometric2F1[-1/2*I,
1, 1 - I/2, -E^((2*I)*ArcSec[a*x])] - E^((2*I)*ArcSec[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, -E^((2*I)*A
rcSec[a*x])]))/a^2

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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x e^{\left (\operatorname {arcsec}\left (a x\right )\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsec(a*x))*x,x, algorithm="fricas")

[Out]

integral(x*e^(arcsec(a*x)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (\operatorname {arcsec}\left (a x\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsec(a*x))*x,x, algorithm="giac")

[Out]

integrate(x*e^(arcsec(a*x)), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\mathrm {arcsec}\left (a x \right )} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsec(a*x))*x,x)

[Out]

int(exp(arcsec(a*x))*x,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\left (\operatorname {arcsec}\left (a x\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsec(a*x))*x,x, algorithm="maxima")

[Out]

integrate(x*e^(arcsec(a*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(acos(1/(a*x))),x)

[Out]

int(x*exp(acos(1/(a*x))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x e^{\operatorname {asec}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asec(a*x))*x,x)

[Out]

Integral(x*exp(asec(a*x)), x)

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