Integrand size = 28, antiderivative size = 80 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\frac {\sqrt {-1+e^2 x^2} \text {arctanh}\left (\frac {\sqrt {c+a e^2} x}{\sqrt {a} \sqrt {-1+e^2 x^2}}\right )}{\sqrt {a} \sqrt {c+a e^2} \sqrt {-1+e x} \sqrt {1+e x}} \] Output:
(e^2*x^2-1)^(1/2)*arctanh((a*e^2+c)^(1/2)*x/a^(1/2)/(e^2*x^2-1)^(1/2))/a^( 1/2)/(a*e^2+c)^(1/2)/(e*x-1)^(1/2)/(e*x+1)^(1/2)
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c+a e^2} x}{\sqrt {a} \sqrt {-1+e x} \sqrt {1+e x}}\right )}{\sqrt {a} \sqrt {c+a e^2}} \] Input:
Integrate[1/(Sqrt[-1 + e*x]*Sqrt[1 + e*x]*(a + c*x^2)),x]
Output:
ArcTanh[(Sqrt[c + a*e^2]*x)/(Sqrt[a]*Sqrt[-1 + e*x]*Sqrt[1 + e*x])]/(Sqrt[ a]*Sqrt[c + a*e^2])
Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {648, 291, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt {e x-1} \sqrt {e x+1} \left (a+c x^2\right )} \, dx\) |
\(\Big \downarrow \) 648 |
\(\displaystyle \frac {\sqrt {e^2 x^2-1} \int \frac {1}{\left (c x^2+a\right ) \sqrt {e^2 x^2-1}}dx}{\sqrt {e x-1} \sqrt {e x+1}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\sqrt {e^2 x^2-1} \int \frac {1}{a-\frac {\left (a e^2+c\right ) x^2}{e^2 x^2-1}}d\frac {x}{\sqrt {e^2 x^2-1}}}{\sqrt {e x-1} \sqrt {e x+1}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {e^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {a e^2+c}}{\sqrt {a} \sqrt {e^2 x^2-1}}\right )}{\sqrt {a} \sqrt {e x-1} \sqrt {e x+1} \sqrt {a e^2+c}}\) |
Input:
Int[1/(Sqrt[-1 + e*x]*Sqrt[1 + e*x]*(a + c*x^2)),x]
Output:
(Sqrt[-1 + e^2*x^2]*ArcTanh[(Sqrt[c + a*e^2]*x)/(Sqrt[a]*Sqrt[-1 + e^2*x^2 ])])/(Sqrt[a]*Sqrt[c + a*e^2]*Sqrt[-1 + e*x]*Sqrt[1 + e*x])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c *e + d*f*x^2)^FracPart[m]) Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && !(EqQ[p, 2] && LtQ[m, -1])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 329, normalized size of antiderivative = 4.11
method | result | size |
default | \(-\frac {\sqrt {e x -1}\, \sqrt {e x +1}\, c \operatorname {csgn}\left (e \right )^{2} \left (\ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x -\sqrt {-a c}}\right ) a \,e^{2}-\ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x +\sqrt {-a c}}\right ) a \,e^{2}+\ln \left (\frac {2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x -\sqrt {-a c}}\right ) c -\ln \left (\frac {-2 \sqrt {-a c}\, e^{2} x +2 \sqrt {e^{2} x^{2}-1}\, \sqrt {-\frac {a \,e^{2}+c}{c}}\, c -2 c}{c x +\sqrt {-a c}}\right ) c \right )}{2 \sqrt {e^{2} x^{2}-1}\, \left (-e \sqrt {-a c}+c \right ) \left (e \sqrt {-a c}+c \right ) \sqrt {-a c}\, \sqrt {-\frac {a \,e^{2}+c}{c}}}\) | \(329\) |
Input:
int(1/(e*x-1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2*(e*x-1)^(1/2)*(e*x+1)^(1/2)*c*csgn(e)^2*(ln(2*((-a*c)^(1/2)*e^2*x+(e^ 2*x^2-1)^(1/2)*(-(a*e^2+c)/c)^(1/2)*c-c)/(c*x-(-a*c)^(1/2)))*a*e^2-ln(2*(- (-a*c)^(1/2)*e^2*x+(e^2*x^2-1)^(1/2)*(-(a*e^2+c)/c)^(1/2)*c-c)/(c*x+(-a*c) ^(1/2)))*a*e^2+ln(2*((-a*c)^(1/2)*e^2*x+(e^2*x^2-1)^(1/2)*(-(a*e^2+c)/c)^( 1/2)*c-c)/(c*x-(-a*c)^(1/2)))*c-ln(2*(-(-a*c)^(1/2)*e^2*x+(e^2*x^2-1)^(1/2 )*(-(a*e^2+c)/c)^(1/2)*c-c)/(c*x+(-a*c)^(1/2)))*c)/(e^2*x^2-1)^(1/2)/(-e*( -a*c)^(1/2)+c)/(e*(-a*c)^(1/2)+c)/(-a*c)^(1/2)/(-(a*e^2+c)/c)^(1/2)
Time = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.21 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\left [\frac {\log \left (-\frac {2 \, a^{2} e^{2} - {\left (4 \, a^{2} e^{4} + 4 \, a c e^{2} + c^{2}\right )} x^{2} - 2 \, {\left (\sqrt {a^{2} e^{2} + a c} {\left (2 \, a e^{2} + c\right )} x + 2 \, {\left (a^{2} e^{3} + a c e\right )} x\right )} \sqrt {e x + 1} \sqrt {e x - 1} + a c - 2 \, \sqrt {a^{2} e^{2} + a c} {\left ({\left (2 \, a e^{3} + c e\right )} x^{2} - a e\right )}}{c x^{2} + a}\right )}{2 \, \sqrt {a^{2} e^{2} + a c}}, \frac {\sqrt {-a^{2} e^{2} - a c} \arctan \left (\frac {\sqrt {-a^{2} e^{2} - a c} \sqrt {e x + 1} \sqrt {e x - 1} c x - \sqrt {-a^{2} e^{2} - a c} {\left (c e x^{2} + a e\right )}}{a^{2} e^{2} + a c}\right )}{a^{2} e^{2} + a c}\right ] \] Input:
integrate(1/(e*x-1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a),x, algorithm="fricas")
Output:
[1/2*log(-(2*a^2*e^2 - (4*a^2*e^4 + 4*a*c*e^2 + c^2)*x^2 - 2*(sqrt(a^2*e^2 + a*c)*(2*a*e^2 + c)*x + 2*(a^2*e^3 + a*c*e)*x)*sqrt(e*x + 1)*sqrt(e*x - 1) + a*c - 2*sqrt(a^2*e^2 + a*c)*((2*a*e^3 + c*e)*x^2 - a*e))/(c*x^2 + a)) /sqrt(a^2*e^2 + a*c), sqrt(-a^2*e^2 - a*c)*arctan((sqrt(-a^2*e^2 - a*c)*sq rt(e*x + 1)*sqrt(e*x - 1)*c*x - sqrt(-a^2*e^2 - a*c)*(c*e*x^2 + a*e))/(a^2 *e^2 + a*c))/(a^2*e^2 + a*c)]
\[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\int \frac {1}{\left (a + c x^{2}\right ) \sqrt {e x - 1} \sqrt {e x + 1}}\, dx \] Input:
integrate(1/(e*x-1)**(1/2)/(e*x+1)**(1/2)/(c*x**2+a),x)
Output:
Integral(1/((a + c*x**2)*sqrt(e*x - 1)*sqrt(e*x + 1)), x)
\[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )} \sqrt {e x + 1} \sqrt {e x - 1}} \,d x } \] Input:
integrate(1/(e*x-1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a),x, algorithm="maxima")
Output:
integrate(1/((c*x^2 + a)*sqrt(e*x + 1)*sqrt(e*x - 1)), x)
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=-\frac {\arctan \left (\frac {c {\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{4} + 8 \, a e^{2} + 4 \, c}{8 \, \sqrt {-a^{2} e^{2} - a c} e}\right )}{\sqrt {-a^{2} e^{2} - a c}} \] Input:
integrate(1/(e*x-1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a),x, algorithm="giac")
Output:
-arctan(1/8*(c*(sqrt(e*x + 1) - sqrt(e*x - 1))^4 + 8*a*e^2 + 4*c)/(sqrt(-a ^2*e^2 - a*c)*e))/sqrt(-a^2*e^2 - a*c)
Time = 15.19 (sec) , antiderivative size = 2280, normalized size of antiderivative = 28.50 \[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((a + c*x^2)*(e*x - 1)^(1/2)*(e*x + 1)^(1/2)),x)
Output:
-(atan((((((13631488*a*c^3*e^5 + 1310720*a^2*c^2*e^7)*((e*x - 1)^(1/2) - 1 i)^3)/((e*x + 1)^(1/2) - 1)^3 - (((((e*x - 1)^(1/2) - 1i)*(4194304*a^2*c^4 *e^5 + 4718592*a^3*c^3*e^7))/((e*x + 1)^(1/2) - 1) + (((e*x - 1)^(1/2) - 1 i)^3*(54525952*a^2*c^4*e^5 + 60293120*a^3*c^3*e^7 + 5242880*a^4*c^2*e^9))/ ((e*x + 1)^(1/2) - 1)^3 + (9437184*a^3*c^4*e^6 + 15728640*a^4*c^3*e^8 + 65 53600*a^5*c^2*e^10 - (((e*x - 1)^(1/2) - 1i)^4*(100663296*a^2*c^5*e^4 + 15 6237824*a^3*c^4*e^6 + 33554432*a^4*c^3*e^8 - 22282240*a^5*c^2*e^10))/((e*x + 1)^(1/2) - 1)^4 - (((e*x - 1)^(1/2) - 1i)^2*(100663296*a^2*c^5*e^4 + 19 7132288*a^3*c^4*e^6 + 116391936*a^4*c^3*e^8 + 18350080*a^5*c^2*e^10))/((e* x + 1)^(1/2) - 1)^2)/(2*a^(1/2)*(c + a*e^2)^(1/2)))/(2*a^(1/2)*(c + a*e^2) ^(1/2)) - 3670016*a^2*c^3*e^6 - 2949120*a^3*c^2*e^8 + (((e*x - 1)^(1/2) - 1i)^4*(25165824*a*c^4*e^4 + 7340032*a^2*c^3*e^6 - 12124160*a^3*c^2*e^8))/( (e*x + 1)^(1/2) - 1)^4 + (((e*x - 1)^(1/2) - 1i)^2*(25165824*a*c^4*e^4 + 2 6738688*a^2*c^3*e^6 + 7208960*a^3*c^2*e^8))/((e*x + 1)^(1/2) - 1)^2)/(2*a^ (1/2)*(c + a*e^2)^(1/2)) + (1048576*a*c^3*e^5*((e*x - 1)^(1/2) - 1i))/((e* x + 1)^(1/2) - 1))*1i)/(2*a^(1/2)*(c + a*e^2)^(1/2)) + ((((13631488*a*c^3* e^5 + 1310720*a^2*c^2*e^7)*((e*x - 1)^(1/2) - 1i)^3)/((e*x + 1)^(1/2) - 1) ^3 - (((((e*x - 1)^(1/2) - 1i)*(4194304*a^2*c^4*e^5 + 4718592*a^3*c^3*e^7) )/((e*x + 1)^(1/2) - 1) + (((e*x - 1)^(1/2) - 1i)^3*(54525952*a^2*c^4*e^5 + 60293120*a^3*c^3*e^7 + 5242880*a^4*c^2*e^9))/((e*x + 1)^(1/2) - 1)^3 ...
\[ \int \frac {1}{\sqrt {-1+e x} \sqrt {1+e x} \left (a+c x^2\right )} \, dx=\int \frac {1}{\sqrt {e x +1}\, \sqrt {e x -1}\, a +\sqrt {e x +1}\, \sqrt {e x -1}\, c \,x^{2}}d x \] Input:
int(1/(e*x-1)^(1/2)/(e*x+1)^(1/2)/(c*x^2+a),x)
Output:
int(1/(sqrt(e*x + 1)*sqrt(e*x - 1)*a + sqrt(e*x + 1)*sqrt(e*x - 1)*c*x**2) ,x)