Integrand size = 12, antiderivative size = 138 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b}{x}+\frac {b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1+a) x^2}-\frac {b^2 \arctan \left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{3/2}} \] Output:
-1/2*a/x^2-b/x+1/2*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x-1/2*(b*x+a -1)^(1/2)*(b*x+a+1)^(3/2)/(1+a)/x^2-b^2*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2) /(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(3/2)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\frac {1}{2} \left (-\frac {a}{x^2}-\frac {2 b}{x}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-1+a^2+a b x\right )}{\left (-1+a^2\right ) x^2}-\frac {i b^2 \log \left (\frac {4 i \sqrt {1-a^2} \left (-1+a^2+a b x-i \sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{b^2 x}\right )}{\left (1-a^2\right )^{3/2}}\right ) \] Input:
Integrate[E^ArcCosh[a + b*x]/x^3,x]
Output:
(-(a/x^2) - (2*b)/x - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-1 + a^2 + a* b*x))/((-1 + a^2)*x^2) - (I*b^2*Log[((4*I)*Sqrt[1 - a^2]*(-1 + a^2 + a*b*x - I*Sqrt[1 - a^2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]))/(b^2*x)])/(1 - a ^2)^(3/2))/2
Time = 0.66 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6435, 2010, 2009}
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.
\(\displaystyle \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6435 |
\(\displaystyle \int \frac {\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{x^3}+\frac {a}{x^3}+\frac {b}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^2 \arctan \left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{3/2}}+\frac {b \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right ) x}-\frac {\sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (a+1) x^2}-\frac {a}{2 x^2}-\frac {b}{x}\) |
Input:
Int[E^ArcCosh[a + b*x]/x^3,x]
Output:
-1/2*a/x^2 - b/x + (b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*(1 - a^2)*x ) - (Sqrt[-1 + a + b*x]*(1 + a + b*x)^(3/2))/(2*(1 + a)*x^2) - (b^2*ArcTan [(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a ^2)^(3/2)
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[E^(ArcCosh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[-1 + u ]*Sqrt[1 + u])^n, x] /; RationalQ[m] && IntegerQ[n] && PolyQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(114)=228\).
Time = 0.19 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.71
method | result | size |
default | \(\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (\sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) b^{2} x^{2}-a^{3} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{2} x^{2}}-\frac {a}{2 x^{2}}-\frac {b}{x}\) | \(236\) |
Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^3,x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*((a^2-1)^(1/2)*ln(2*(a*b*x+(a^2-1)^(1/ 2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*b^2*x^2-a^3*b*x*(b^2*x^2+2*a*b* x+a^2-1)^(1/2)-a^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a*b*x*(b^2*x^2+2*a*b*x+a^ 2-1)^(1/2)+2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^2-(b^2*x^2+2*a*b*x+a^2-1)^(1/ 2))/(b^2*x^2+2*a*b*x+a^2-1)^(1/2)/(a^2-1)^2/x^2-1/2*a/x^2-b/x
Time = 0.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.45 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\left [\frac {\sqrt {a^{2} - 1} b^{2} x^{2} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - a^{5} - {\left (a^{3} - a\right )} b^{2} x^{2} + 2 \, a^{3} - 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b x - {\left (a^{4} + {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}, -\frac {2 \, \sqrt {-a^{2} + 1} b^{2} x^{2} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) + a^{5} + {\left (a^{3} - a\right )} b^{2} x^{2} - 2 \, a^{3} + 2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} b x + {\left (a^{4} + {\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + a}{2 \, {\left (a^{4} - 2 \, a^{2} + 1\right )} x^{2}}\right ] \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^3,x, algorithm="fricas ")
Output:
[1/2*(sqrt(a^2 - 1)*b^2*x^2*log((a^2*b*x + a^3 + (a^2 + sqrt(a^2 - 1)*a - 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + (a*b*x + a^2 - 1)*sqrt(a^2 - 1) - a)/x) - a^5 - (a^3 - a)*b^2*x^2 + 2*a^3 - 2*(a^4 - 2*a^2 + 1)*b*x - (a^4 + (a^3 - a)*b*x - 2*a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) - a)/((a^ 4 - 2*a^2 + 1)*x^2), -1/2*(2*sqrt(-a^2 + 1)*b^2*x^2*arctan(-(sqrt(-a^2 + 1 )*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) + a ^5 + (a^3 - a)*b^2*x^2 - 2*a^3 + 2*(a^4 - 2*a^2 + 1)*b*x + (a^4 + (a^3 - a )*b*x - 2*a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + a)/((a^4 - 2*a^2 + 1)*x^2)]
\[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\int \frac {a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}}{x^{3}}\, dx \] Input:
integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))/x**3,x)
Output:
Integral((a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1))/x**3, x)
Exception generated. \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^3,x, algorithm="maxima ")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (112) = 224\).
Time = 0.22 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.36 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\frac {\frac {2 \, b^{3} \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{2} - 1\right )} \sqrt {-a^{2} + 1}} + \frac {4 \, {\left (2 \, a^{2} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 4 \, a^{3} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 2 \, a b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 8 \, a^{2} b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4 \, b^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 8 \, a b^{3}\right )}}{{\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{2} {\left (a^{2} - 1\right )}} - \frac {2 \, {\left (b x + a + 1\right )} b^{3} - a b^{3} - 2 \, b^{3}}{b^{2} x^{2}}}{2 \, b} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^3,x, algorithm="giac")
Output:
1/2*(2*b^3*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sq rt(-a^2 + 1))/((a^2 - 1)*sqrt(-a^2 + 1)) + 4*(2*a^2*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 4*a^3*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1) )^4 - b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 2*a*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 8*a^2*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 4*b^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 8*a*b^3)/((( sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 4*a*(sqrt(b*x + a + 1) - sqrt(b *x + a - 1))^2 + 4)^2*(a^2 - 1)) - (2*(b*x + a + 1)*b^3 - a*b^3 - 2*b^3)/( b^2*x^2))/b
Time = 50.72 (sec) , antiderivative size = 958, normalized size of antiderivative = 6.94 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx =\text {Too large to display} \] Input:
int((a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x)/x^3,x)
Output:
(b^2*log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)))*(a - 1)^(1/2)*(a + 1)^(1/2))/(2*a^4 - 4*a^2 + 2) - ((a*b^2*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5)/(8*(a^2 - 1)*((a + 1)^(1/2) - (a + b *x + 1)^(1/2))^5) - (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3*((3*a*b^2)/8 - (7*a^3*b^2)/8))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3*(a^4 - 2*a^2 + 1)) - (b^2*(a - 1)^(1/2)*(a + 1)^(1/2))/(32*(a^2 - 1)) + (a*b^2*((a - 1)^( 1/2) - (a + b*x - 1)^(1/2)))/(4*(a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^( 1/2))) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2*(b^2/16 - (11*a^2*b^2)/1 6)*(a - 1)^(1/2)*(a + 1)^(1/2))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2*( a^4 - 2*a^2 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(a - 1)^(1/2) *(a + 1)^(1/2)*((15*b^2)/32 + (9*a^2*b^2)/16 - (17*a^4*b^2)/32))/(((a + 1) ^(1/2) - (a + b*x + 1)^(1/2))^4*(3*a^2 - 3*a^4 + a^6 - 1)))/(((a - 1)^(1/2 ) - (a + b*x - 1)^(1/2))^2/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2 + ((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6/((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6 + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*(6*a^2 - 2))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4) - (4*a*((a - 1)^(1/2) - (a + b*x - 1)^( 1/2))^3*(a - 1)^(1/2)*(a + 1)^(1/2))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3) - (4*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^5*(a - 1)^(1/2 )*(a + 1)^(1/2))/((a^2 - 1)*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5)) - (( (a - 1)^(1/2) - (a + b*x - 1)^(1/2))*((a*b^2)/(2*(a - 1)*(a + 1)) - (3*...
Time = 0.19 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.47 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^3} \, dx=\frac {2 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\sqrt {a +1}\, \sqrt {-a +1}\, i +\sqrt {b x +a +1}\, \sqrt {b x +a -1}+b x \right ) a \,b^{2} i \,x^{2}-2 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\frac {-\sqrt {\sqrt {a +1}\, \sqrt {-a +1}\, i +a}\, \sqrt {2}+\sqrt {b x +a -1}+\sqrt {b x +a +1}}{\sqrt {2}}\right ) a \,b^{2} i \,x^{2}-2 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\frac {\sqrt {\sqrt {a +1}\, \sqrt {-a +1}\, i +a}\, \sqrt {2}+\sqrt {b x +a -1}+\sqrt {b x +a +1}}{\sqrt {2}}\right ) a \,b^{2} i \,x^{2}-2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{5}-2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{4} b x +4 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{3}+2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{2} b x -2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a -2 a^{6}-4 a^{5} b x +4 a^{4}+8 a^{3} b x +a^{2} b^{2} x^{2}-2 a^{2}-4 a b x -b^{2} x^{2}}{4 a \,x^{2} \left (a^{4}-2 a^{2}+1\right )} \] Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^3,x)
Output:
(2*sqrt(a + 1)*sqrt( - a + 1)*log(sqrt(a + 1)*sqrt( - a + 1)*i + sqrt(a + b*x + 1)*sqrt(a + b*x - 1) + b*x)*a*b**2*i*x**2 - 2*sqrt(a + 1)*sqrt( - a + 1)*log(( - sqrt(sqrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sqrt(a + b*x + 1))/sqrt(2))*a*b**2*i*x**2 - 2*sqrt(a + 1)*sqrt( - a + 1)*log((sqrt(sqrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sqrt(a + b*x + 1))/sqrt(2))*a*b**2*i*x**2 - 2*sqrt(a + b*x + 1)*sqr t(a + b*x - 1)*a**5 - 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**4*b*x + 4*s qrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**3 + 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**2*b*x - 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a - 2*a**6 - 4*a**5 *b*x + 4*a**4 + 8*a**3*b*x + a**2*b**2*x**2 - 2*a**2 - 4*a*b*x - b**2*x**2 )/(4*a*x**2*(a**4 - 2*a**2 + 1))