3.4.63 \(\int \frac {1}{1+a+b x^3} \, dx\) [363]

Optimal. Leaf size=125 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )}{\sqrt {3} (1+a)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{1+a}+\sqrt [3]{b} x\right )}{3 (1+a)^{2/3} \sqrt [3]{b}}-\frac {\log \left ((1+a)^{2/3}-\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+1}}}{\sqrt {3}}\right )}{\sqrt {3} (a+1)^{2/3} \sqrt [3]{b}}-\frac {\log \left (-\sqrt [3]{a+1} \sqrt [3]{b} x+(a+1)^{2/3}+b^{2/3} x^2\right )}{6 (a+1)^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a+1}+\sqrt [3]{b} x\right )}{3 (a+1)^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + a + b*x^3)^(-1),x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 101, normalized size = 0.81 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {-1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{1+a}}}{\sqrt {3}}\right )+2 \log \left (\sqrt [3]{1+a}+\sqrt [3]{b} x\right )-\log \left ((1+a)^{2/3}-\sqrt [3]{1+a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 (1+a)^{2/3} \sqrt [3]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + a + b*x^3)^(-1),x]

[Out]

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

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Maple [A]
time = 0.13, size = 105, normalized size = 0.84

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a +1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b}\)	\(28\)
default	\(\frac {\ln \left (x +\left (\frac {1+a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1+a}{b}\right )^{\frac {1}{3}} x +\left (\frac {1+a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1+a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {1+a}{b}\right )^{\frac {2}{3}}}\)	\(105\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a+1),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 114, normalized size = 0.91 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a + 1}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a+1),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (90) = 180\).
time = 0.37, size = 446, normalized size = 3.57 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {-\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, {\left (a + 1\right )} b x^{3} - 3 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )} x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (a + 1\right )} b x^{2} + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {-\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} - 2 \, a - 1}{b x^{3} + a + 1}\right ) - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) + 2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a + 1\right )} b \sqrt {\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right )} \sqrt {\frac {\left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}}}{b}}}{a^{2} + 2 \, a + 1}\right ) - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x^{2} - \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {1}{3}} {\left (a + 1\right )}\right ) + 2 \, \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}} \log \left ({\left (a + 1\right )} b x + \left ({\left (a^{2} + 2 \, a + 1\right )} b\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} + 2 \, a + 1\right )} b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.11, size = 32, normalized size = 0.26 \begin {gather*} \operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{2} b + 54 a b + 27 b\right ) - 1, \left ( t \mapsto t \log {\left (3 t a + 3 t + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a+1),x)

[Out]

RootSum(_t**3*(27*a**2*b + 54*a*b + 27*b) - 1, Lambda(_t, _t*log(3*_t*a + 3*_t + x)))

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Giac [A]
time = 1.33, size = 143, normalized size = 1.14 \begin {gather*} \frac {{\left (-a b^{2} - b^{2}\right )}^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b + \sqrt {3} b} + \frac {{\left (-a b^{2} - b^{2}\right )}^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} + \left (-\frac {a + 1}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b + b\right )}} - \frac {\left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a + 1}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a+1),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.21, size = 137, normalized size = 1.10 \begin {gather*} \frac {\ln \left (a+b^{1/3}\,x\,{\left (a+1\right )}^{2/3}+1\right )}{3\,b^{1/3}\,{\left (a+1\right )}^{2/3}}+\frac {\ln \left (3\,b^2\,x+\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}-\frac {\ln \left (3\,b^2\,x-\frac {\left (9\,a\,b^2+9\,b^2\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{1/3}\,{\left (a+1\right )}^{2/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^3 + 1),x)

[Out]

log(a + b^(1/3)*x*(a + 1)^(2/3) + 1)/(3*b^(1/3)*(a + 1)^(2/3)) + (log(3*b^2*x + ((9*a*b^2 + 9*b^2)*(3^(1/2)*1i
- 1))/(6*b^(1/3)*(a + 1)^(2/3)))*(3^(1/2)*1i - 1))/(6*b^(1/3)*(a + 1)^(2/3)) - (log(3*b^2*x - ((9*a*b^2 + 9*b
^2)*(3^(1/2)*1i + 1))/(6*b^(1/3)*(a + 1)^(2/3)))*(3^(1/2)*1i + 1))/(6*b^(1/3)*(a + 1)^(2/3))

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