Integrand size = 15, antiderivative size = 329 \[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\frac {(1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \arctan \left (\frac {\sqrt {\frac {13}{2}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{\sqrt {26} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {2 \sqrt {26+15 \sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {4 \sqrt [4]{3} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
1/26*(1+x)*arctan(1/2*26^(1/2)*((1+x)/(1+x+3^(1/2))^2)^(1/2)/((x^2-x+1)/(1 +x+3^(1/2))^2)^(1/2))*((x^2-x+1)/(1+x+3^(1/2))^2)^(1/2)*26^(1/2)/(x^3+1)^( 1/2)/((1+x)/(1+x+3^(1/2))^2)^(1/2)+4*3^(1/4)*(1+x)*EllipticPi((-1-x+3^(1/2 ))/(1+x+3^(1/2)),97-56*3^(1/2),I*3^(1/2)+2*I)*((x^2-x+1)/(1+x+3^(1/2))^2)^ (1/2)/(x^3+1)^(1/2)/(1/2*6^(1/2)-1/2*2^(1/2))/((1+x)/(1+x+3^(1/2))^2)^(1/2 )+2/3*(1+x)*EllipticF((1+x-3^(1/2))/(1+x+3^(1/2)),I*3^(1/2)+2*I)*((x^2-x+1 )/(1+x+3^(1/2))^2)^(1/2)*(3/2*6^(1/2)+5/2*2^(1/2))*3^(3/4)/(x^3+1)^(1/2)/( (1+x)/(1+x+3^(1/2))^2)^(1/2)
Result contains complex when optimal does not.
Time = 20.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=-\frac {4 \sqrt {2} \sqrt {\frac {i (1+x)}{3 i+\sqrt {3}}} \sqrt {1-x+x^2} \operatorname {EllipticPi}\left (\frac {2 \sqrt {3}}{7 i+\sqrt {3}},\arcsin \left (\frac {\sqrt {i+\sqrt {3}-2 i x}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )}{\left (7 i+\sqrt {3}\right ) \sqrt {1+x^3}} \]
(-4*Sqrt[2]*Sqrt[(I*(1 + x))/(3*I + Sqrt[3])]*Sqrt[1 - x + x^2]*EllipticPi [(2*Sqrt[3])/(7*I + Sqrt[3]), ArcSin[Sqrt[I + Sqrt[3] - (2*I)*x]/(Sqrt[2]* 3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/((7*I + Sqrt[3])*Sqrt[1 + x^3])
Time = 0.94 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2561, 759, 2567, 25, 2538, 412, 435, 104, 217}
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.
| \(\displaystyle \int \frac {1}{(x+3) \sqrt {x^3+1}} \, dx\) |
| \(\Big \downarrow \) 2561 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {x^3+1}}dx}{2-\sqrt {3}}-\frac {\int \frac {x+\sqrt {3}+1}{(x+3) \sqrt {x^3+1}}dx}{2-\sqrt {3}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\int \frac {x+\sqrt {3}+1}{(x+3) \sqrt {x^3+1}}dx}{2-\sqrt {3}}\) |
| \(\Big \downarrow \) 2567 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \int -\frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (2-\sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+\sqrt {3}+2\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \int \frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (2-\sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+\sqrt {3}+2\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int -\frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )-\left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int -\frac {x-\sqrt {3}+1}{\left (x+\sqrt {3}+1\right ) \sqrt {1-\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \left (-\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+4 \sqrt {3}+7\right )}d\left (-\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\frac {1}{2} \left (2-\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7} \sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1} \left (\frac {\left (7-4 \sqrt {3}\right ) \left (x-\sqrt {3}+1\right )}{x+\sqrt {3}+1}+4 \sqrt {3}+7\right )}d\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\left (2-\sqrt {3}\right ) \int \frac {1}{-\frac {52 \left (2-\sqrt {3}\right ) \sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}-8 \sqrt {3}}d\frac {\sqrt {\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}+1}}{\sqrt {\frac {\left (x-\sqrt {3}+1\right )^2}{\left (x+\sqrt {3}+1\right )^2}-4 \sqrt {3}+7}}+\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \left (2-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {4 \sqrt [4]{3} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (\sqrt {7-4 \sqrt {3}} \left (2+\sqrt {3}\right ) \operatorname {EllipticPi}\left (97-56 \sqrt {3},\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )+\frac {\sqrt {\frac {1}{26} \left (2-\sqrt {3}\right )} \arctan \left (\frac {\sqrt {\frac {13}{2} \left (2-\sqrt {3}\right )} \left (x-\sqrt {3}+1\right )}{\sqrt [4]{3} \left (x+\sqrt {3}+1\right )}\right )}{4 \sqrt [4]{3}}\right )}{\sqrt {2-\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}\) |
(2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ellip ticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4 )*(2 - Sqrt[3])*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (4*3^(1 /4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*((Sqrt[(2 - Sqrt[3])/2 6]*ArcTan[(Sqrt[(13*(2 - Sqrt[3]))/2]*(1 - Sqrt[3] + x))/(3^(1/4)*(1 + Sqr t[3] + x))])/(4*3^(1/4)) + Sqrt[7 - 4*Sqrt[3]]*(2 + Sqrt[3])*EllipticPi[97 - 56*Sqrt[3], ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3] ]))/(Sqrt[2 - Sqrt[3]]*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3])
3.1.14.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* ((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & & PosQ[a]
Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]}, Simp[-q/((1 + Sqrt[3])*d - c*q) Int[1/Sqrt[a + b*x^3], x] , x] + Simp[d/((1 + Sqrt[3])*d - c*q) Int[(1 + Sqrt[3] + q*x)/((c + d*x)* Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b *c^3*d^3 - 8*a^2*d^6, 0]
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> With[{q = Simplify[(1 + Sqrt[3])*(f/e)]}, Simp[4*3^(1/4)*Sqrt[2 - Sqrt[3]]*f*(1 + q*x)*(Sqrt[(1 - q*x + q^2*x^2)/(1 + Sqrt[3] + q*x)^2]/(q* Sqrt[a + b*x^3]*Sqrt[(1 + q*x)/(1 + Sqrt[3] + q*x)^2])) Subst[Int[1/(((1 - Sqrt[3])*d - c*q + ((1 + Sqrt[3])*d - c*q)*x)*Sqrt[1 - x^2]*Sqrt[7 - 4*Sq rt[3] + x^2]), x], x, (-1 + Sqrt[3] - q*x)/(1 + Sqrt[3] + q*x)], x]] /; Fre eQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b*e^3 - 2*(5 + 3*Sqrt [3])*a*f^3, 0] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
Time = 1.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.37
| method | result | size |
| default | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3}{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(123\) |
| elliptic | \(\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x +1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {3}{4}+\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(123\) |
(3/2-1/2*I*3^(1/2))*((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2 ))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)) )^(1/2)/(x^3+1)^(1/2)*EllipticPi(((x+1)/(3/2-1/2*I*3^(1/2)))^(1/2),-3/4+1/ 4*I*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))
\[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
\[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\int \frac {1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 3\right )}\, dx \]
\[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
\[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} {\left (x + 3\right )}} \,d x } \]
Time = 9.15 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.50 \[ \int \frac {1}{(3+x) \sqrt {1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {3}{4}-\frac {\sqrt {3}\,1{}\mathrm {i}}{4};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
((3^(1/2)*1i + 3)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2 )*((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)*ellipticPi(- (3^(1/2)*1i)/4 - 3/4, 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)))/(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))