Integrand size = 11, antiderivative size = 148 \[ \int \frac {1}{a+b x^{4/3}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {a}+\sqrt {b} x^{2/3}}\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}} \] Output:
-3/4*arctan(1-2^(1/2)*b^(1/4)*x^(1/3)/a^(1/4))*2^(1/2)/a^(1/4)/b^(3/4)+3/4 *arctan(1+2^(1/2)*b^(1/4)*x^(1/3)/a^(1/4))*2^(1/2)/a^(1/4)/b^(3/4)-3/4*arc tanh(2^(1/2)*a^(1/4)*b^(1/4)*x^(1/3)/(a^(1/2)+b^(1/2)*x^(2/3)))*2^(1/2)/a^ (1/4)/b^(3/4)
Time = 0.14 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.68 \[ \int \frac {1}{a+b x^{4/3}} \, dx=-\frac {3 \left (\arctan \left (\frac {\sqrt {a}-\sqrt {b} x^{2/3}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt {a}+\sqrt {b} x^{2/3}}\right )\right )}{2 \sqrt {2} \sqrt [4]{a} b^{3/4}} \] Input:
Integrate[(a + b*x^(4/3))^(-1),x]
Output:
(-3*(ArcTan[(Sqrt[a] - Sqrt[b]*x^(2/3))/(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))] + ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3))/(Sqrt[a] + Sqrt[b]*x^(2/3))]) )/(2*Sqrt[2]*a^(1/4)*b^(3/4))
Time = 0.60 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.50, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {774, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a+b x^{4/3}} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 3 \int \frac {x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle 3 \left (\frac {\int \frac {\sqrt {b} x^{2/3}+\sqrt {a}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle 3 \left (\frac {\frac {\int \frac {1}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 3 \left (\frac {\frac {\int \frac {1}{-x^{2/3}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{2/3}-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^{2/3}}{b x^{4/3}+a}d\sqrt [3]{x}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{b} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{b} \left (x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt [3]{x}}{x^{2/3}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}+\sqrt [4]{a}}{x^{2/3}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt [3]{x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt [3]{x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt [3]{x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a}+\sqrt {b} x^{2/3}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt [3]{x}+\sqrt {a}+\sqrt {b} x^{2/3}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
Input:
Int[(a + b*x^(4/3))^(-1),x]
Output:
3*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x^(1/3))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/ 4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x^(1/3))/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1 /4)))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^(1/3) + Sqrt[b]*x^(2/3)]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4) *b^(1/4)*x^(1/3) + Sqrt[b]*x^(2/3)]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b ]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{\frac {2}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(110\) |
| default | \(\frac {\left (\frac {a^{3}}{b^{3}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a^{3}}{b^{3}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a^{3}}{b^{3}}}}{x^{2}-\left (\frac {a^{3}}{b^{3}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a^{3}}{b^{3}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a^{3}}{b^{3}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a^{3}}{b^{3}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+b^{2} \left (\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{\frac {2}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{8}-a b \,\textit {\_Z}^{4}+a^{2}\right )}{\sum }\frac {\left (2 \textit {\_R}^{6} b -\textit {\_R}^{2} a \right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 b \,\textit {\_R}^{7}-\textit {\_R}^{3} a}}{4 b^{3}}\right )-a b \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{\frac {2}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{\frac {1}{3}} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{2} \textit {\_Z}^{8}-a b \,\textit {\_Z}^{4}+a^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{6} b +\textit {\_R}^{2} a \right ) \ln \left (x^{\frac {1}{3}}-\textit {\_R} \right )}{2 b \,\textit {\_R}^{7}-\textit {\_R}^{3} a}}{4 a \,b^{2}}\right )\) | \(477\) |
Input:
int(1/(a+b*x^(4/3)),x,method=_RETURNVERBOSE)
Output:
3/8/b/(a/b)^(1/4)*2^(1/2)*(ln((x^(2/3)-(a/b)^(1/4)*x^(1/3)*2^(1/2)+(a/b)^( 1/2))/(x^(2/3)+(a/b)^(1/4)*x^(1/3)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/ (a/b)^(1/4)*x^(1/3)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/3)-1))
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\frac {3}{4} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) - \frac {3}{4} i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (i \, a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) + \frac {3}{4} i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) - \frac {3}{4} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{\frac {1}{3}}\right ) \] Input:
integrate(1/(a+b*x^(4/3)),x, algorithm="fricas")
Output:
3/4*(-1/(a*b^3))^(1/4)*log(a*b^2*(-1/(a*b^3))^(3/4) + x^(1/3)) - 3/4*I*(-1 /(a*b^3))^(1/4)*log(I*a*b^2*(-1/(a*b^3))^(3/4) + x^(1/3)) + 3/4*I*(-1/(a*b ^3))^(1/4)*log(-I*a*b^2*(-1/(a*b^3))^(3/4) + x^(1/3)) - 3/4*(-1/(a*b^3))^( 1/4)*log(-a*b^2*(-1/(a*b^3))^(3/4) + x^(1/3))
Time = 1.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.70 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\begin {cases} \frac {\tilde {\infty }}{\sqrt [3]{x}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x}{a} & \text {for}\: b = 0 \\- \frac {3}{b \sqrt [3]{x}} & \text {for}\: a = 0 \\\frac {3 \log {\left (\sqrt [3]{x} - \sqrt [4]{- \frac {a}{b}} \right )}}{4 b \sqrt [4]{- \frac {a}{b}}} - \frac {3 \log {\left (\sqrt [3]{x} + \sqrt [4]{- \frac {a}{b}} \right )}}{4 b \sqrt [4]{- \frac {a}{b}}} + \frac {3 \operatorname {atan}{\left (\frac {\sqrt [3]{x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{2 b \sqrt [4]{- \frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(a+b*x**(4/3)),x)
Output:
Piecewise((zoo/x**(1/3), Eq(a, 0) & Eq(b, 0)), (x/a, Eq(b, 0)), (-3/(b*x** (1/3)), Eq(a, 0)), (3*log(x**(1/3) - (-a/b)**(1/4))/(4*b*(-a/b)**(1/4)) - 3*log(x**(1/3) + (-a/b)**(1/4))/(4*b*(-a/b)**(1/4)) + 3*atan(x**(1/3)/(-a/ b)**(1/4))/(2*b*(-a/b)**(1/4)), True))
Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.19 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} x^{\frac {1}{3}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {3 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} x^{\frac {1}{3}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {3 \, \sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{\frac {1}{3}} + \sqrt {b} x^{\frac {2}{3}} + \sqrt {a}\right )}{8 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {3 \, \sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{\frac {1}{3}} + \sqrt {b} x^{\frac {2}{3}} + \sqrt {a}\right )}{8 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} \] Input:
integrate(1/(a+b*x^(4/3)),x, algorithm="maxima")
Output:
3/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*x^(1/3 ))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 3/4*sqrt(2)*ar ctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*x^(1/3))/sqrt(sqrt( a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - 3/8*sqrt(2)*log(sqrt(2)*a^( 1/4)*b^(1/4)*x^(1/3) + sqrt(b)*x^(2/3) + sqrt(a))/(a^(1/4)*b^(3/4)) + 3/8* sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*x^(1/3) + sqrt(b)*x^(2/3) + sqrt(a))/ (a^(1/4)*b^(3/4))
Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.26 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, x^{\frac {1}{3}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x^{\frac {2}{3}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} \] Input:
integrate(1/(a+b*x^(4/3)),x, algorithm="giac")
Output:
3/4*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*x^(1 /3))/(a/b)^(1/4))/(a*b^3) + 3/4*sqrt(2)*(a*b^3)^(3/4)*arctan(-1/2*sqrt(2)* (sqrt(2)*(a/b)^(1/4) - 2*x^(1/3))/(a/b)^(1/4))/(a*b^3) - 3/8*sqrt(2)*(a*b^ 3)^(3/4)*log(sqrt(2)*x^(1/3)*(a/b)^(1/4) + x^(2/3) + sqrt(a/b))/(a*b^3) + 3/8*sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*x^(1/3)*(a/b)^(1/4) + x^(2/3) + sqr t(a/b))/(a*b^3)
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.28 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,x^{1/3}}{{\left (-a\right )}^{1/4}}\right )-3\,\mathrm {atanh}\left (\frac {b^{1/4}\,x^{1/3}}{{\left (-a\right )}^{1/4}}\right )}{2\,{\left (-a\right )}^{1/4}\,b^{3/4}} \] Input:
int(1/(a + b*x^(4/3)),x)
Output:
(3*atan((b^(1/4)*x^(1/3))/(-a)^(1/4)) - 3*atanh((b^(1/4)*x^(1/3))/(-a)^(1/ 4)))/(2*(-a)^(1/4)*b^(3/4))
Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+b x^{4/3}} \, dx=\frac {3 \sqrt {2}\, \left (-2 \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 x^{\frac {1}{3}} \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 x^{\frac {1}{3}} \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+\mathrm {log}\left (-x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+x^{\frac {2}{3}} \sqrt {b}\right )-\mathrm {log}\left (x^{\frac {1}{3}} b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+x^{\frac {2}{3}} \sqrt {b}\right )\right )}{8 b^{\frac {3}{4}} a^{\frac {1}{4}}} \] Input:
int(1/(a+b*x^(4/3)),x)
Output:
(3*b**(1/4)*a**(3/4)*sqrt(2)*( - 2*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*x** (1/3)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2))) + 2*atan((b**(1/4)*a**(1/4)*sq rt(2) + 2*x**(1/3)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2))) + log( - x**(1/3) *b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)*sqrt(b)) - log(x**(1/3)*b* *(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + x**(2/3)*sqrt(b))))/(8*a*b)