3.94 \(\int e^{-\frac {1}{2} \tanh ^{-1}(a x)} \, dx\)

Optimal. Leaf size=221 \[ \frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2} a}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2} a} \]

[Out]

(-a*x+1)^(1/4)*(a*x+1)^(3/4)/a-1/2*arctan(-1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a*2^(1/2)-1/2*arctan(1+(-a*
x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4))/a*2^(1/2)+1/4*ln(1-(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+
1)^(1/2))/a*2^(1/2)-1/4*ln(1+(-a*x+1)^(1/4)*2^(1/2)/(a*x+1)^(1/4)+(-a*x+1)^(1/2)/(a*x+1)^(1/2))/a*2^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6125, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac {\sqrt [4]{1-a x} (a x+1)^{3/4}}{a}+\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}-\frac {\log \left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{2 \sqrt {2} a}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}\right )}{\sqrt {2} a}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}+1\right )}{\sqrt {2} a} \]

Antiderivative was successfully verified.

[In]

Int[E^(-ArcTanh[a*x]/2),x]

[Out]

((1 - a*x)^(1/4)*(1 + a*x)^(3/4))/a + ArcTan[1 - (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) - ArcT
an[1 + (Sqrt[2]*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(Sqrt[2]*a) + Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] - (Sqrt[2]
*(1 - a*x)^(1/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a) - Log[1 + Sqrt[1 - a*x]/Sqrt[1 + a*x] + (Sqrt[2]*(1 - a*x)^(1
/4))/(1 + a*x)^(1/4)]/(2*Sqrt[2]*a)

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 6125

Int[E^(ArcTanh[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(1 + a*x)^(n/2)/(1 - a*x)^(n/2), x] /; FreeQ[{a, n}, x] &&
!IntegerQ[(n - 1)/2]

Rubi steps

\begin {align*} \int e^{-\frac {1}{2} \tanh ^{-1}(a x)} \, dx &=\int \frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}} \, dx\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac {1}{2} \int \frac {1}{(1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-a x}\right )}{a}\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}-\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{a}\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}\\ &=\frac {\sqrt [4]{1-a x} (1+a x)^{3/4}}{a}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{\sqrt {2} a}+\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}-\frac {\log \left (1+\frac {\sqrt {1-a x}}{\sqrt {1+a x}}+\frac {\sqrt {2} \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}\right )}{2 \sqrt {2} a}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 35, normalized size = 0.16 \[ \frac {8 e^{\frac {3}{2} \tanh ^{-1}(a x)} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};-e^{2 \tanh ^{-1}(a x)}\right )}{3 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(-1/2*ArcTanh[a*x]),x]

[Out]

(8*E^((3*ArcTanh[a*x])/2)*Hypergeometric2F1[3/4, 2, 7/4, -E^(2*ArcTanh[a*x])])/(3*a)

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fricas [B]  time = 1.10, size = 518, normalized size = 2.34 \[ -\frac {4 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - 1\right ) + 4 \, \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \arctan \left (\sqrt {2} a \sqrt {-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} - \sqrt {2} a \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {1}{4}} + 1\right ) + \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} + {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} - \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - \sqrt {2} a \frac {1}{a^{4}}^{\frac {1}{4}} \log \left (-\frac {\sqrt {2} {\left (a^{4} x - a^{3}\right )} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}} \frac {1}{a^{4}}^{\frac {3}{4}} - {\left (a^{3} x - a^{2}\right )} \sqrt {\frac {1}{a^{4}}} + \sqrt {-a^{2} x^{2} + 1}}{a x - 1}\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-\frac {\sqrt {-a^{2} x^{2} + 1}}{a x - 1}}}{4 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/4*(4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x -
1))*(a^(-4))^(3/4) + (a^3*x - a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sq
rt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) - 1) + 4*sqrt(2)*a*(a^(-4))^(1/4)*arctan(sqrt(2)*a*sqrt(-(sqr
t(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2
*x^2 + 1))/(a*x - 1))*(a^(-4))^(1/4) - sqrt(2)*a*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(1/4) + 1) + sqr
t(2)*a*(a^(-4))^(1/4)*log((sqrt(2)*(a^4*x - a^3)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) + (a^3*x -
a^2)*sqrt(a^(-4)) - sqrt(-a^2*x^2 + 1))/(a*x - 1)) - sqrt(2)*a*(a^(-4))^(1/4)*log(-(sqrt(2)*(a^4*x - a^3)*sqr
t(-sqrt(-a^2*x^2 + 1)/(a*x - 1))*(a^(-4))^(3/4) - (a^3*x - a^2)*sqrt(a^(-4)) + sqrt(-a^2*x^2 + 1))/(a*x - 1))
- 4*sqrt(-a^2*x^2 + 1)*sqrt(-sqrt(-a^2*x^2 + 1)/(a*x - 1)))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x)

[Out]

int(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x+1)/(-a^2*x^2+1)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt((a*x + 1)/sqrt(-a^2*x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2),x)

[Out]

int(1/((a*x + 1)/(1 - a^2*x^2)^(1/2))^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a x + 1}{\sqrt {- a^{2} x^{2} + 1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x+1)/(-a**2*x**2+1)**(1/2))**(1/2),x)

[Out]

Integral(1/sqrt((a*x + 1)/sqrt(-a**2*x**2 + 1)), x)

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