Integrand size = 18, antiderivative size = 31 \[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {1+x^3}}\right ) \] Output:
-2/3*arctanh((1/3+2/3*x+1/3*x^2)/(x^3+1)^(1/2))
Time = 0.94 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}+\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {1+x^3}}\right ) \] Input:
Integrate[(1 + x)/((-2 + x)*Sqrt[1 + x^3]),x]
Output:
(-2*ArcTanh[(1/3 + (2*x)/3 + x^2/3)/Sqrt[1 + x^3]])/3
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2563, 219}
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.
\(\displaystyle \int \frac {x+1}{(x-2) \sqrt {x^3+1}} \, dx\) |
\(\Big \downarrow \) 2563 |
\(\displaystyle -2 \int \frac {1}{9-\frac {(x+1)^4}{x^3+1}}d\frac {(x+1)^2}{\sqrt {x^3+1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2}{3} \text {arctanh}\left (\frac {(x+1)^2}{3 \sqrt {x^3+1}}\right )\) |
Input:
Int[(1 + x)/((-2 + x)*Sqrt[1 + x^3]),x]
Output:
(-2*ArcTanh[(1 + x)^2/(3*Sqrt[1 + x^3])])/3
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-2*(e/d) Subst[Int[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/ Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] & & EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).
Time = 0.31 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
method | result | size |
trager | \(\frac {\ln \left (\frac {-x^{3}+6 x \sqrt {x^{3}+1}-12 x^{2}+6 \sqrt {x^{3}+1}+6 x -10}{\left (x -2\right )^{3}}\right )}{3}\) | \(44\) |
default | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-\frac {i \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(240\) |
elliptic | \(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{2}-\frac {i \sqrt {3}}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(240\) |
Input:
int((1+x)/(x-2)/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*ln((-x^3+6*x*(x^3+1)^(1/2)-12*x^2+6*(x^3+1)^(1/2)+6*x-10)/(x-2)^3)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\frac {1}{3} \, \log \left (\frac {x^{3} + 12 \, x^{2} - 6 \, \sqrt {x^{3} + 1} {\left (x + 1\right )} - 6 \, x + 10}{x^{3} - 6 \, x^{2} + 12 \, x - 8}\right ) \] Input:
integrate((1+x)/(-2+x)/(x^3+1)^(1/2),x, algorithm="fricas")
Output:
1/3*log((x^3 + 12*x^2 - 6*sqrt(x^3 + 1)*(x + 1) - 6*x + 10)/(x^3 - 6*x^2 + 12*x - 8))
\[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\int \frac {x + 1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x - 2\right )}\, dx \] Input:
integrate((1+x)/(-2+x)/(x**3+1)**(1/2),x)
Output:
Integral((x + 1)/(sqrt((x + 1)*(x**2 - x + 1))*(x - 2)), x)
\[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\int { \frac {x + 1}{\sqrt {x^{3} + 1} {\left (x - 2\right )}} \,d x } \] Input:
integrate((1+x)/(-2+x)/(x^3+1)^(1/2),x, algorithm="maxima")
Output:
integrate((x + 1)/(sqrt(x^3 + 1)*(x - 2)), x)
\[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\int { \frac {x + 1}{\sqrt {x^{3} + 1} {\left (x - 2\right )}} \,d x } \] Input:
integrate((1+x)/(-2+x)/(x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate((x + 1)/(sqrt(x^3 + 1)*(x - 2)), x)
Time = 7.07 (sec) , antiderivative size = 204, normalized size of antiderivative = 6.58 \[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int((x + 1)/((x^3 + 1)^(1/2)*(x - 2)),x)
Output:
((3^(1/2)*1i + 3)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2 )*(ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/ 2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)) - ellipticPi((3^(1/2)*1i)/6 + 1/2, asin(( (x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)* 1i)/2 - 3/2)))*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2))/(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*(( 3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2)) ^(1/2)
\[ \int \frac {1+x}{(-2+x) \sqrt {1+x^3}} \, dx=\int \frac {\sqrt {x^{3}+1}}{x^{3}-3 x^{2}+3 x -2}d x \] Input:
int((1+x)/(-2+x)/(x^3+1)^(1/2),x)
Output:
int(sqrt(x**3 + 1)/(x**3 - 3*x**2 + 3*x - 2),x)