3.523 \(\int \frac{e^x}{b+a e^{3 x}} \, dx\)

Optimal. Leaf size=100 \[ \frac{\log \left (\sqrt [3]{a} e^x+\sqrt [3]{b}\right )}{2 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (a e^{3 x}+b\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} e^x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]

[Out]

-(ArcTan[(b^(1/3) - 2*a^(1/3)*E^x)/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*a^(1/3)*b^(2/3))) + Log[b^(1/3) + a^(1/3)*E^x]/
(2*a^(1/3)*b^(2/3)) - Log[b + a*E^(3*x)]/(6*a^(1/3)*b^(2/3))

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Rubi [A]  time = 0.0943053, antiderivative size = 123, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2249, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3} e^{2 x}-\sqrt [3]{a} \sqrt [3]{b} e^x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\log \left (\sqrt [3]{a} e^x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} e^x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[E^x/(b + a*E^(3*x)),x]

[Out]

-(ArcTan[(b^(1/3) - 2*a^(1/3)*E^x)/(Sqrt[3]*b^(1/3))]/(Sqrt[3]*a^(1/3)*b^(2/3))) + Log[b^(1/3) + a^(1/3)*E^x]/
(3*a^(1/3)*b^(2/3)) - Log[b^(2/3) - a^(1/3)*b^(1/3)*E^x + a^(2/3)*E^(2*x)]/(6*a^(1/3)*b^(2/3))

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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 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 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]

Rubi steps

\begin{align*} \int \frac{e^x}{b+a e^{3 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{b+a x^3} \, dx,x,e^x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,e^x\right )}{3 b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{3 b^{2/3}}\\ &=\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{2 \sqrt [3]{b}}\\ &=\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} e^x+a^{2/3} e^{2 x}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} b^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}}+\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} e^x+a^{2/3} e^{2 x}\right )}{6 \sqrt [3]{a} b^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0551443, size = 97, normalized size = 0.97 \[ -\frac{\log \left (a^{2/3} e^{2 x}-\sqrt [3]{a} \sqrt [3]{b} e^x+b^{2/3}\right )-2 \log \left (\sqrt [3]{a} e^x+\sqrt [3]{b}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{6 \sqrt [3]{a} b^{2/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^x/(b + a*E^(3*x)),x]

[Out]

-(2*Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*E^x)/b^(1/3))/Sqrt[3]] - 2*Log[b^(1/3) + a^(1/3)*E^x] + Log[b^(2/3) - a^(1/
3)*b^(1/3)*E^x + a^(2/3)*E^(2*x)])/(6*a^(1/3)*b^(2/3))

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Maple [A]  time = 0.007, size = 95, normalized size = 1. \begin{align*}{\frac{1}{3\,a}\ln \left ({{\rm e}^{x}}+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,a}\ln \left ( \left ({{\rm e}^{x}} \right ) ^{2}-\sqrt [3]{{\frac{b}{a}}}{{\rm e}^{x}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{{{\rm e}^{x}}{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(b+a*exp(3*x)),x)

[Out]

1/3/a/(b/a)^(2/3)*ln(exp(x)+(b/a)^(1/3))-1/6/a/(b/a)^(2/3)*ln(exp(x)^2-(b/a)^(1/3)*exp(x)+(b/a)^(2/3))+1/3/a/(
b/a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*exp(x)-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(b+a*exp(3*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.87355, size = 795, normalized size = 7.95 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, a b e^{\left (3 \, x\right )} - 3 \, \left (a b^{2}\right )^{\frac{1}{3}} b e^{x} - b^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b e^{\left (2 \, x\right )} + \left (a b^{2}\right )^{\frac{2}{3}} e^{x} - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{-\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{a e^{\left (3 \, x\right )} + b}\right ) - \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b e^{\left (2 \, x\right )} - \left (a b^{2}\right )^{\frac{2}{3}} e^{x} + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b e^{x} + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a b^{2}\right )^{\frac{2}{3}} e^{x} - \left (a b^{2}\right )^{\frac{1}{3}} b\right )} \sqrt{\frac{\left (a b^{2}\right )^{\frac{1}{3}}}{a}}}{b^{2}}\right ) - \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b e^{\left (2 \, x\right )} - \left (a b^{2}\right )^{\frac{2}{3}} e^{x} + \left (a b^{2}\right )^{\frac{1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac{2}{3}} \log \left (a b e^{x} + \left (a b^{2}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(b+a*exp(3*x)),x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a*b^2)^(1/3)/a)*log((2*a*b*e^(3*x) - 3*(a*b^2)^(1/3)*b*e^x - b^2 + 3*sqrt(1/3)*(2
*a*b*e^(2*x) + (a*b^2)^(2/3)*e^x - (a*b^2)^(1/3)*b)*sqrt(-(a*b^2)^(1/3)/a))/(a*e^(3*x) + b)) - (a*b^2)^(2/3)*l
og(a*b*e^(2*x) - (a*b^2)^(2/3)*e^x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log(a*b*e^x + (a*b^2)^(2/3)))/(a*b^2),
1/6*(6*sqrt(1/3)*a*b*sqrt((a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*(a*b^2)^(2/3)*e^x - (a*b^2)^(1/3)*b)*sqrt((a*b
^2)^(1/3)/a)/b^2) - (a*b^2)^(2/3)*log(a*b*e^(2*x) - (a*b^2)^(2/3)*e^x + (a*b^2)^(1/3)*b) + 2*(a*b^2)^(2/3)*log
(a*b*e^x + (a*b^2)^(2/3)))/(a*b^2)]

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Sympy [A]  time = 0.160282, size = 22, normalized size = 0.22 \begin{align*} \operatorname{RootSum}{\left (27 z^{3} a b^{2} - 1, \left ( i \mapsto i \log{\left (3 i b + e^{x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(b+a*exp(3*x)),x)

[Out]

RootSum(27*_z**3*a*b**2 - 1, Lambda(_i, _i*log(3*_i*b + exp(x))))

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Giac [A]  time = 1.12688, size = 157, normalized size = 1.57 \begin{align*} -\frac{\left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | -\left (-\frac{b}{a}\right )^{\frac{1}{3}} + e^{x} \right |}\right )}{3 \, b} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{b}{a}\right )^{\frac{1}{3}} + 2 \, e^{x}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a b} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} \log \left (\left (-\frac{b}{a}\right )^{\frac{1}{3}} e^{x} + \left (-\frac{b}{a}\right )^{\frac{2}{3}} + e^{\left (2 \, x\right )}\right )}{6 \, a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(b+a*exp(3*x)),x, algorithm="giac")

[Out]

-1/3*(-b/a)^(1/3)*log(abs(-(-b/a)^(1/3) + e^x))/b + 1/3*sqrt(3)*(-a^2*b)^(1/3)*arctan(1/3*sqrt(3)*((-b/a)^(1/3
) + 2*e^x)/(-b/a)^(1/3))/(a*b) + 1/6*(-a^2*b)^(1/3)*log((-b/a)^(1/3)*e^x + (-b/a)^(2/3) + e^(2*x))/(a*b)