Integrand size = 43, antiderivative size = 133 \[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=-\frac {\sqrt {2} \sqrt {A} \sqrt {B} \arctan \left (\frac {A}{B}-\frac {\sqrt {A} \sqrt {e \left (\frac {A}{B}-\frac {B}{A}+2 x\right )}}{\sqrt {B} \sqrt {e}}\right )}{\sqrt {e}}+\frac {\sqrt {2} \sqrt {A} \sqrt {B} \arctan \left (\frac {A}{B}+\frac {\sqrt {A} \sqrt {e \left (\frac {A}{B}-\frac {B}{A}+2 x\right )}}{\sqrt {B} \sqrt {e}}\right )}{\sqrt {e}} \]
-arctan(A/B-A^(1/2)*(e*(A/B-B/A+2*x))^(1/2)/B^(1/2)/e^(1/2))*2^(1/2)*A^(1/ 2)*B^(1/2)/e^(1/2)+arctan(A/B+A^(1/2)*(e*(A/B-B/A+2*x))^(1/2)/B^(1/2)/e^(1 /2))*2^(1/2)*A^(1/2)*B^(1/2)/e^(1/2)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=-\frac {i \sqrt {2} \sqrt {A} \sqrt {B} \sqrt {\frac {A}{B}-\frac {B}{A}+2 x} \left (\text {arctanh}\left (\frac {\sqrt {A} \sqrt {B} \sqrt {\frac {A}{B}-\frac {B}{A}+2 x}}{A-i B}\right )-\text {arctanh}\left (\frac {\sqrt {A} \sqrt {B} \sqrt {\frac {A}{B}-\frac {B}{A}+2 x}}{A+i B}\right )\right )}{\sqrt {e \left (\frac {A}{B}-\frac {B}{A}+2 x\right )}} \]
((-I)*Sqrt[2]*Sqrt[A]*Sqrt[B]*Sqrt[A/B - B/A + 2*x]*(ArcTanh[(Sqrt[A]*Sqrt [B]*Sqrt[A/B - B/A + 2*x])/(A - I*B)] - ArcTanh[(Sqrt[A]*Sqrt[B]*Sqrt[A/B - B/A + 2*x])/(A + I*B)]))/Sqrt[e*(A/B - B/A + 2*x)]
Time = 0.40 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {654, 27, 1475, 1083, 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.
| \(\displaystyle \int \frac {A+B x}{\left (x^2+1\right ) \sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x}} \, dx\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle 2 \int \frac {2 \left (\left (A^2+B^2\right ) e+2 A B \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right )\right )}{A \left (\frac {\left (A^2+B^2\right )^2 e^2}{A^2 B^2}-\frac {4 \left (A^2-B^2\right ) \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right ) e}{A B}+4 \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right )^2\right )}d\sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 \int \frac {\left (A^2+B^2\right ) e+2 A B \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right )}{\frac {\left (A^2+B^2\right )^2 e^2}{A^2 B^2}-\frac {4 \left (A^2-B^2\right ) \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right ) e}{A B}+4 \left (\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e\right )^2}d\sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}}{A}\) |
| \(\Big \downarrow \) 1475 |
| \(\displaystyle \frac {4 \left (\frac {1}{4} A B \int \frac {1}{\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+\frac {\left (A^2+B^2\right ) e}{2 A B}+x e-\frac {\sqrt {2} \sqrt {A} \sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e} \sqrt {e}}{\sqrt {B}}}d\sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}+\frac {1}{4} A B \int \frac {1}{\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+\frac {\left (A^2+B^2\right ) e}{2 A B}+x e+\frac {\sqrt {2} \sqrt {A} \sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e} \sqrt {e}}{\sqrt {B}}}d\sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}\right )}{A}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {4 \left (-\frac {1}{2} A B \int \frac {1}{-\frac {2 B e}{A}-\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e-x e}d\left (2 \sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}-\frac {\sqrt {2} \sqrt {A} \sqrt {e}}{\sqrt {B}}\right )-\frac {1}{2} A B \int \frac {1}{-\frac {2 B e}{A}-\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e-x e}d\left (\frac {\sqrt {2} \sqrt {A} \sqrt {e}}{\sqrt {B}}+2 \sqrt {\frac {1}{2} \left (\frac {A}{B}-\frac {B}{A}\right ) e+x e}\right )\right )}{A}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {4 \left (\frac {A^{3/2} \sqrt {B} \arctan \left (\frac {\sqrt {A} \left (2 \sqrt {\frac {1}{2} e \left (\frac {A}{B}-\frac {B}{A}\right )+e x}-\frac {\sqrt {2} \sqrt {A} \sqrt {e}}{\sqrt {B}}\right )}{\sqrt {2} \sqrt {B} \sqrt {e}}\right )}{2 \sqrt {2} \sqrt {e}}+\frac {A^{3/2} \sqrt {B} \arctan \left (\frac {\sqrt {A} \left (2 \sqrt {\frac {1}{2} e \left (\frac {A}{B}-\frac {B}{A}\right )+e x}+\frac {\sqrt {2} \sqrt {A} \sqrt {e}}{\sqrt {B}}\right )}{\sqrt {2} \sqrt {B} \sqrt {e}}\right )}{2 \sqrt {2} \sqrt {e}}\right )}{A}\) |
(4*((A^(3/2)*Sqrt[B]*ArcTan[(Sqrt[A]*(-((Sqrt[2]*Sqrt[A]*Sqrt[e])/Sqrt[B]) + 2*Sqrt[((A/B - B/A)*e)/2 + e*x]))/(Sqrt[2]*Sqrt[B]*Sqrt[e])])/(2*Sqrt[2 ]*Sqrt[e]) + (A^(3/2)*Sqrt[B]*ArcTan[(Sqrt[A]*((Sqrt[2]*Sqrt[A]*Sqrt[e])/S qrt[B] + 2*Sqrt[((A/B - B/A)*e)/2 + e*x]))/(Sqrt[2]*Sqrt[B]*Sqrt[e])])/(2* Sqrt[2]*Sqrt[e])))/A
3.15.64.3.1 Defintions of rubi rules used
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[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[2*(d/e) - b/c, 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]] /; F reeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[2*(d/e) - b/c, 0] || ( !LtQ[2*(d/e) - b/c, 0] && EqQ[d - e*Rt[a/c, 2] , 0]))
Time = 1.95 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83
| method | result | size |
| pseudoelliptic | \(\frac {A B \sqrt {2}\, \left (\arctan \left (\frac {\sqrt {\frac {e \left (2 x A B +A^{2}-B^{2}\right )}{A B}}\, A B -\sqrt {A^{3} e B}}{\sqrt {A e B}\, B}\right )+\arctan \left (\frac {\sqrt {\frac {e \left (2 x A B +A^{2}-B^{2}\right )}{A B}}\, A B +\sqrt {A^{3} e B}}{\sqrt {A e B}\, B}\right )\right )}{\sqrt {A e B}}\) | \(110\) |
| default | \(2 \sqrt {2}\, B^{2} A \left (\frac {\arctan \left (\frac {2 A B \sqrt {2 e x +\frac {e \left (A^{2}-B^{2}\right )}{A B}}-2 \sqrt {A^{3} e B}}{2 B \sqrt {A e B}}\right )}{2 B \sqrt {A e B}}+\frac {\arctan \left (\frac {2 A B \sqrt {2 e x +\frac {e \left (A^{2}-B^{2}\right )}{A B}}+2 \sqrt {A^{3} e B}}{2 B \sqrt {A e B}}\right )}{2 B \sqrt {A e B}}\right )\) | \(135\) |
| derivativedivides | \(4 B^{2} A \left (\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4 e x +\frac {2 A^{2} e -2 B^{2} e}{A B}}\, A B -2 \sqrt {2}\, \sqrt {A^{3} e B}\right ) \sqrt {2}}{4 B \sqrt {A e B}}\right )}{4 B \sqrt {A e B}}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (2 \sqrt {4 e x +\frac {2 A^{2} e -2 B^{2} e}{A B}}\, A B +2 \sqrt {2}\, \sqrt {A^{3} e B}\right ) \sqrt {2}}{4 B \sqrt {A e B}}\right )}{4 B \sqrt {A e B}}\right )\) | \(156\) |
A*B*2^(1/2)*(arctan(((e*(2*A*B*x+A^2-B^2)/A/B)^(1/2)*A*B-(A^3*e*B)^(1/2))/ (A*e*B)^(1/2)/B)+arctan(((e*(2*A*B*x+A^2-B^2)/A/B)^(1/2)*A*B+(A^3*e*B)^(1/ 2))/(A*e*B)^(1/2)/B))/(A*e*B)^(1/2)
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.23 \[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=\left [\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {A B}{e}} \log \left (\frac {A^{2} x^{2} - 4 \, A B x - A^{2} + 2 \, B^{2} + 2 \, {\left (A x - B\right )} \sqrt {-\frac {A B}{e}} \sqrt {\frac {2 \, A B e x + {\left (A^{2} - B^{2}\right )} e}{A B}}}{x^{2} + 1}\right ), \sqrt {2} \sqrt {\frac {A B}{e}} \arctan \left (\frac {{\left (A x - B\right )} \sqrt {\frac {A B}{e}} \sqrt {\frac {2 \, A B e x + {\left (A^{2} - B^{2}\right )} e}{A B}}}{2 \, A B x + A^{2} - B^{2}}\right )\right ] \]
[1/2*sqrt(2)*sqrt(-A*B/e)*log((A^2*x^2 - 4*A*B*x - A^2 + 2*B^2 + 2*(A*x - B)*sqrt(-A*B/e)*sqrt((2*A*B*e*x + (A^2 - B^2)*e)/(A*B)))/(x^2 + 1)), sqrt( 2)*sqrt(A*B/e)*arctan((A*x - B)*sqrt(A*B/e)*sqrt((2*A*B*e*x + (A^2 - B^2)* e)/(A*B))/(2*A*B*x + A^2 - B^2))]
\[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=\sqrt {2} \left (\int \frac {A}{x^{2} \sqrt {\frac {A e}{B} + 2 e x - \frac {B e}{A}} + \sqrt {\frac {A e}{B} + 2 e x - \frac {B e}{A}}}\, dx + \int \frac {B x}{x^{2} \sqrt {\frac {A e}{B} + 2 e x - \frac {B e}{A}} + \sqrt {\frac {A e}{B} + 2 e x - \frac {B e}{A}}}\, dx\right ) \]
sqrt(2)*(Integral(A/(x**2*sqrt(A*e/B + 2*e*x - B*e/A) + sqrt(A*e/B + 2*e*x - B*e/A)), x) + Integral(B*x/(x**2*sqrt(A*e/B + 2*e*x - B*e/A) + sqrt(A*e /B + 2*e*x - B*e/A)), x))
\[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=\int { \frac {2 \, {\left (B x + A\right )}}{\sqrt {4 \, e x + \frac {2 \, {\left (A^{2} e - B^{2} e\right )}}{A B}} {\left (x^{2} + 1\right )}} \,d x } \]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 14.05 (sec) , antiderivative size = 18107, normalized size of antiderivative = 136.14 \[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=\text {Too large to display} \]
1/2*sqrt(2)*(6*A*B*e^2*abs(A)*abs(B)*cos(1/2*real_part(arccos(A^3*B*e/abs( A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))^2*cosh(1/2*imag_par t(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))) )^3*sin(1/2*real_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs( A^3*B*e + A*B^3*e)))) - 2*A*B*e^2*abs(A)*abs(B)*cosh(1/2*imag_part(arccos( A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))^3*sin(1 /2*real_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))^3 - 18*A*B*e^2*abs(A)*abs(B)*cos(1/2*real_part(arccos(A^3*B*e /abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))^2*cosh(1/2*ima g_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3 *e))))^2*sin(1/2*real_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e /abs(A^3*B*e + A*B^3*e))))*sinh(1/2*imag_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e)))) + 6*A*B*e^2*abs(A)*abs(B)*co sh(1/2*imag_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B *e + A*B^3*e))))^2*sin(1/2*real_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))^3*sinh(1/2*imag_part(arccos(A^3*B*e/a bs(A^3*B*e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e)))) + 18*A*B*e^2*abs (A)*abs(B)*cos(1/2*real_part(arccos(A^3*B*e/abs(A^3*B*e + A*B^3*e) - A*B^3 *e/abs(A^3*B*e + A*B^3*e))))^2*cosh(1/2*imag_part(arccos(A^3*B*e/abs(A^3*B *e + A*B^3*e) - A*B^3*e/abs(A^3*B*e + A*B^3*e))))*sin(1/2*real_part(arc...
Time = 0.22 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.55 \[ \int \frac {A+B x}{\sqrt {\frac {A^2 e-B^2 e}{2 A B}+e x} \left (1+x^2\right )} \, dx=\frac {\sqrt {2}\,\sqrt {A}\,\sqrt {B}\,\left (\mathrm {atan}\left (\frac {A^{3/2}\,{\left (4\,e\,x+\frac {2\,A^2\,e-2\,B^2\,e}{A\,B}\right )}^{3/2}\,\sqrt {2\,B}}{8\,e^{3/2}\,\left (A^2+B^2\right )}-\frac {A^{5/2}\,\sqrt {8\,e\,x+\frac {2\,\left (2\,A^2\,e-2\,B^2\,e\right )}{A\,B}}}{4\,\sqrt {B}\,\sqrt {e}\,\left (A^2+B^2\right )}+\frac {3\,\sqrt {2}\,\sqrt {A}\,B^{3/2}\,\sqrt {4\,e\,x+\frac {2\,A^2\,e-2\,B^2\,e}{A\,B}}}{4\,\sqrt {e}\,\left (A^2+B^2\right )}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {A}\,\sqrt {4\,e\,x+\frac {2\,A^2\,e-2\,B^2\,e}{A\,B}}}{4\,\sqrt {B}\,\sqrt {e}}\right )\right )}{\sqrt {e}} \]
(2^(1/2)*A^(1/2)*B^(1/2)*(atan((A^(3/2)*(4*e*x + (2*A^2*e - 2*B^2*e)/(A*B) )^(3/2)*(2*B)^(1/2))/(8*e^(3/2)*(A^2 + B^2)) - (A^(5/2)*(8*e*x + (2*(2*A^2 *e - 2*B^2*e))/(A*B))^(1/2))/(4*B^(1/2)*e^(1/2)*(A^2 + B^2)) + (3*2^(1/2)* A^(1/2)*B^(3/2)*(4*e*x + (2*A^2*e - 2*B^2*e)/(A*B))^(1/2))/(4*e^(1/2)*(A^2 + B^2))) + atan((2^(1/2)*A^(1/2)*(4*e*x + (2*A^2*e - 2*B^2*e)/(A*B))^(1/2 ))/(4*B^(1/2)*e^(1/2)))))/e^(1/2)