3.6.23 \(\int \frac {e^x}{b+a e^{3 x}} \, dx\) [523]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, 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 = {2281, 206, 31, 648, 631, 210, 642} \begin {gather*} -\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 {\text {ArcTan}\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}}+\frac {\log \left (\sqrt [3]{a} e^x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}} \end {gather*}

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 31

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

Rule 206

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 210

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

Rule 631

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 642

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 648

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 2281

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]

Rubi steps

\begin {align*} \int \frac {e^x}{b+a e^{3 x}} \, dx &=\text {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,e^x\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,e^x\right )}{3 b^{2/3}}+\frac {\text {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 {\text {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 {\text {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 {\text {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 = 0.09, size = 97, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )+\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x/(b + a*E^(3*x)),x]

[Out]

-1/6*(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)])/(a^(1/3)*b^(2/3))

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 95, normalized size = 0.95

method	result	size
risch	\(\munderset {\textit {\_R} =\RootOf \left (27 a \,b^{2} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (3 b \textit {\_R} +{\mathrm e}^{x}\right )\)	\(26\)
default	\(\frac {\ln \left ({\mathrm e}^{x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left ({\mathrm e}^{2 x}-\left (\frac {b}{a}\right )^{\frac {1}{3}} {\mathrm e}^{x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,{\mathrm e}^{x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

________________________________________________________________________________________

Maxima [A]
time = 2.75, size = 100, normalized size = 1.00 \begin {gather*} \frac {\sqrt {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 \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\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 \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (\left (\frac {b}{a}\right )^{\frac {1}{3}} + e^{x}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.53, size = 311, normalized size = 3.11 \begin {gather*} \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 {gather*}

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)]

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 22, normalized size = 0.22 \begin {gather*} \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 {gather*}

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))))

________________________________________________________________________________________

Giac [A]
time = 1.14, size = 116, normalized size = 1.16 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Mupad [B]
time = 1.51, size = 104, normalized size = 1.04 \begin {gather*} \frac {\ln \left (\frac {b^{1/3}}{a^{7/3}}+\frac {{\mathrm {e}}^x}{a^2}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (\frac {{\mathrm {e}}^x}{a^2}+\frac {b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{7/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{a^2}-\frac {b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{7/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(b + a*exp(3*x)),x)

[Out]

log(b^(1/3)/a^(7/3) + exp(x)/a^2)/(3*a^(1/3)*b^(2/3)) + (log(exp(x)/a^2 + (b^(1/3)*(3^(1/2)*1i - 1))/(2*a^(7/3
)))*(3^(1/2)*1i - 1))/(6*a^(1/3)*b^(2/3)) - (log(exp(x)/a^2 - (b^(1/3)*(3^(1/2)*1i + 1))/(2*a^(7/3)))*(3^(1/2)
*1i + 1))/(6*a^(1/3)*b^(2/3))

________________________________________________________________________________________