Integrand size = 12, antiderivative size = 100 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b}{x}-\frac {(1+a (a+b x)) \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x^2}-\frac {b^2 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \] Output:
-1/2*a/x^2-b/x-1/2*(1+a*(b*x+a))*(1+(b*x+a)^2)^(1/2)/(a^2+1)/x^2-1/2*b^2*a rctanh((1+a*(b*x+a))/(a^2+1)^(1/2)/(1+(b*x+a)^2)^(1/2))/(a^2+1)^(3/2)
Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {1}{2} \left (-\frac {a}{x^2}-\frac {2 b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{\left (1+a^2\right ) x^2}+\frac {b^2 \log (x)}{\left (1+a^2\right )^{3/2}}-\frac {b^2 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{3/2}}\right ) \] Input:
Integrate[E^ArcSinh[a + b*x]/x^3,x]
Output:
(-(a/x^2) - (2*b)/x - ((1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] )/((1 + a^2)*x^2) + (b^2*Log[x])/(1 + a^2)^(3/2) - (b^2*Log[1 + a^2 + a*b* x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(1 + a^2)^(3/2))/2
Time = 0.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6293, 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.
\(\displaystyle \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx\) |
\(\Big \downarrow \) 6293 |
\(\displaystyle \int \frac {\sqrt {(a+b x)^2+1}+a+b x}{x^3}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{x^3}+\frac {a}{x^3}+\frac {b}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^2 \text {arctanh}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac {\left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right ) x^2}-\frac {a}{2 x^2}-\frac {b}{x}\) |
Input:
Int[E^ArcSinh[a + b*x]/x^3,x]
Output:
-1/2*a/x^2 - b/x - ((1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/( 2*(1 + a^2)*x^2) - (b^2*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])])/(2*(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^(ArcSinh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[1 + u^ 2])^n, x] /; RationalQ[m] && IntegerQ[n] && PolyQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(86)=172\).
Time = 0.19 (sec) , antiderivative size = 459, normalized size of antiderivative = 4.59
method | result | size |
default | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{2 a^{2}+2}-\frac {a}{2 x^{2}}-\frac {b}{x}\) | \(459\) |
Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2/(a^2+1)/x^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-1/2*a*b/(a^2+1)*(-1/(a^2+1) /x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+a*b/(a^2+1)*((b^2*x^2+2*a*b*x+a^2+1)^(1/2 )+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2 )-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1 )^(1/2))/x))+2*b^2/(a^2+1)*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1 )^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2* x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)))+1/2*b^2/(a^2+1)*((b^2*x^2+2*a*b*x+ a^2+1)^(1/2)+a*b*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2)) /(b^2)^(1/2)-(a^2+1)^(1/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2* a*b*x+a^2+1)^(1/2))/x))-1/2*a/x^2-b/x
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (86) = 172\).
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.81 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {\sqrt {a^{2} + 1} b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - a^{5} - {\left (a^{3} + a\right )} b^{2} x^{2} - 2 \, a^{3} - 2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} b x - {\left (a^{4} + {\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - a}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(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 + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 - sqrt(a^2 + 1)*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^2*x^2 + 2*a*b*x + a^2 + 1) - a)/((a^4 + 2*a^2 + 1)*x^2)
\[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{3}}\, dx \] Input:
integrate((b*x+a+(1+(b*x+a)**2)**(1/2))/x**3,x)
Output:
Integral((a + b*x + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1))/x**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (86) = 172\).
Time = 0.05 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.13 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {a^{2} b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, \sqrt {a^{2} + 1}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}}{2 \, {\left (a^{2} + 1\right )}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{2 \, {\left (a^{2} + 1\right )} x} - \frac {b}{x} - \frac {a}{2 \, x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (a^{2} + 1\right )} x^{2}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x, algorithm="maxima")
Output:
1/2*a^2*b^2*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4 *(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(3/2) - 1/2*b^2*arcsinh(2*a*b*x/(sqrt(- 4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1 )*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/sqrt(a^2 + 1) + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2/(a^2 + 1) + 1/2*sqrt(b^2*x ^2 + 2*a*b*x + a^2 + 1)*a*b/((a^2 + 1)*x) - b/x - 1/2*a/x^2 - 1/2*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/((a^2 + 1)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (86) = 172\).
Time = 0.16 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.84 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {b^{2} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, b x + a}{2 \, x^{2}} + \frac {2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{2} b^{2} + 2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{4} b^{2} + 4 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{3} b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} b^{2} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{2} b^{2} + 4 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b^{2}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{2} {\left (a^{2} + 1\right )}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x, algorithm="giac")
Output:
1/2*b^2*log(abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 2*sqrt (a^2 + 1))/abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*sqrt( a^2 + 1)))/(a^2 + 1)^(3/2) - 1/2*(2*b*x + a)/x^2 + (2*(x*abs(b) - sqrt(b^2 *x^2 + 2*a*b*x + a^2 + 1))^3*a^2*b^2 + 2*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b* x + a^2 + 1))*a^4*b^2 + 4*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 *a^3*b*abs(b) + (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3*b^2 + 3*( x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a^2*b^2 + 4*(x*abs(b) - sqrt (b^2*x^2 + 2*a*b*x + a^2 + 1))^2*a*b*abs(b) + (x*abs(b) - sqrt(b^2*x^2 + 2 *a*b*x + a^2 + 1))*b^2)/(((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - a^2 - 1)^2*(a^2 + 1))
Timed out. \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^3} \,d x \] Input:
int((a + ((a + b*x)^2 + 1)^(1/2) + b*x)/x^3,x)
Output:
int((a + ((a + b*x)^2 + 1)^(1/2) + b*x)/x^3, x)
Time = 0.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {4 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a \,b^{2} i \,x^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{5}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{4} b x -4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a -2 a^{6}-4 a^{5} b x -4 a^{4}-8 a^{3} b x -a^{2} b^{2} x^{2}-2 a^{2}-4 a b x -b^{2} x^{2}}{4 a \,x^{2} \left (a^{4}+2 a^{2}+1\right )} \] Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x)
Output:
(4*sqrt(a**2 + 1)*atan((sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*i + b*i*x)/sq rt(a**2 + 1))*a*b**2*i*x**2 - 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**5 - 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**4*b*x - 4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**3 - 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2*b*x - 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 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))