Integrand size = 24, antiderivative size = 80 \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\frac {a \arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}-b \arctan \left (\sqrt [4]{-1+x^2}\right )+\frac {a \text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{-1+x^2}}\right )}{2 \sqrt {2}}+b \text {arctanh}\left (\sqrt [4]{-1+x^2}\right ) \]
-b*arctan((x^2-1)^(1/4))+b*arctanh((x^2-1)^(1/4))+1/4*a*arctan(1/2*x/(x^2- 1)^(1/4)*2^(1/2))*2^(1/2)+1/4*a*arctanh(1/2*x/(x^2-1)^(1/4)*2^(1/2))*2^(1/ 2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.30 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.96 \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\frac {x \left (b x \sqrt [4]{1-x^2} \left (-2+x^2\right ) \operatorname {AppellF1}\left (1,\frac {1}{4},1,2,x^2,\frac {x^2}{2}\right )-\frac {24 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )}{6 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},x^2,\frac {x^2}{2}\right )+x^2 \left (2 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},x^2,\frac {x^2}{2}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},x^2,\frac {x^2}{2}\right )\right )}\right )}{4 \left (-2+x^2\right ) \sqrt [4]{-1+x^2}} \]
(x*(b*x*(1 - x^2)^(1/4)*(-2 + x^2)*AppellF1[1, 1/4, 1, 2, x^2, x^2/2] - (2 4*a*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2])/(6*AppellF1[1/2, 1/4, 1, 3/2, x^2, x^2/2] + x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, x^2, x^2/2] + AppellF1[3/2 , 5/4, 1, 5/2, x^2, x^2/2]))))/(4*(-2 + x^2)*(-1 + x^2)^(1/4))
Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1343, 309, 353, 73, 827, 216, 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.
\(\displaystyle \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}} \, dx\) |
\(\Big \downarrow \) 1343 |
\(\displaystyle a \int \frac {1}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx+b \int \frac {x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx\) |
\(\Big \downarrow \) 309 |
\(\displaystyle b \int \frac {x}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 353 |
\(\displaystyle \frac {1}{2} b \int \frac {1}{\left (2-x^2\right ) \sqrt [4]{x^2-1}}dx^2+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle 2 b \int \frac {x^4}{1-x^8}d\sqrt [4]{x^2-1}+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 827 |
\(\displaystyle 2 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{x^2-1}-\frac {1}{2} \int \frac {1}{x^4+1}d\sqrt [4]{x^2-1}\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 2 b \left (\frac {1}{2} \int \frac {1}{1-x^4}d\sqrt [4]{x^2-1}-\frac {1}{2} \arctan \left (\sqrt [4]{x^2-1}\right )\right )+a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a \left (\frac {\arctan \left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {2} \sqrt [4]{x^2-1}}\right )}{2 \sqrt {2}}\right )+2 b \left (\frac {1}{2} \text {arctanh}\left (\sqrt [4]{x^2-1}\right )-\frac {1}{2} \arctan \left (\sqrt [4]{x^2-1}\right )\right )\) |
a*(ArcTan[x/(Sqrt[2]*(-1 + x^2)^(1/4))]/(2*Sqrt[2]) + ArcTanh[x/(Sqrt[2]*( -1 + x^2)^(1/4))]/(2*Sqrt[2])) + 2*b*(-1/2*ArcTan[(-1 + x^2)^(1/4)] + ArcT anh[(-1 + x^2)^(1/4)]/2)
3.1.69.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[-b^2/a, 4]}, Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTan[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x] + Simp[(b/(2*Sqrt[2]*a*d*q))*ArcTanh[q*(x/(Sqrt[2]*(a + b*x^2)^(1/4)))], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && NegQ[b^2/a]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q _), x_Symbol] :> Simp[g Int[(a + c*x^2)^p*(d + f*x^2)^q, x], x] + Simp[h Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h, p, q}, x]
\[\int \frac {b x +a}{\left (-x^{2}+2\right ) \left (x^{2}-1\right )^{\frac {1}{4}}}d x\]
Timed out. \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=- \int \frac {a}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx - \int \frac {b x}{x^{2} \sqrt [4]{x^{2} - 1} - 2 \sqrt [4]{x^{2} - 1}}\, dx \]
-Integral(a/(x**2*(x**2 - 1)**(1/4) - 2*(x**2 - 1)**(1/4)), x) - Integral( b*x/(x**2*(x**2 - 1)**(1/4) - 2*(x**2 - 1)**(1/4)), x)
\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}} \,d x } \]
\[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int { -\frac {b x + a}{{\left (x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 2\right )}} \,d x } \]
Timed out. \[ \int \frac {a+b x}{\left (2-x^2\right ) \sqrt [4]{-1+x^2}} \, dx=\int -\frac {a+b\,x}{{\left (x^2-1\right )}^{1/4}\,\left (x^2-2\right )} \,d x \]