Integrand size = 12, antiderivative size = 83 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=-\frac {a}{x}-\frac {\sqrt {1+(a+b x)^2}}{x}+b \text {arcsinh}(a+b x)-\frac {a b \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{\sqrt {1+a^2}}+b \log (x) \] Output:
-a/x-(1+(b*x+a)^2)^(1/2)/x+b*arcsinh(b*x+a)-a*b*arctanh((1+a*(b*x+a))/(a^2 +1)^(1/2)/(1+(b*x+a)^2)^(1/2))/(a^2+1)^(1/2)+b*ln(x)
Time = 0.19 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=b \text {arcsinh}(a+b x)-\frac {a+\sqrt {1+a^2+2 a b x+b^2 x^2}+\left (-1-\frac {a}{\sqrt {1+a^2}}\right ) b x \log (x)+\frac {a b x \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\sqrt {1+a^2}}}{x} \] Input:
Integrate[E^ArcSinh[a + b*x]/x^2,x]
Output:
b*ArcSinh[a + b*x] - (a + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (-1 - a/Sqrt [1 + a^2])*b*x*Log[x] + (a*b*x*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[1 + a^2])/x
Time = 0.57 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.19, 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.
\(\displaystyle \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 6293 |
\(\displaystyle \int \frac {\sqrt {(a+b x)^2+1}+a+b x}{x^2}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{x^2}+\frac {a}{x^2}+\frac {b}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a b \text {arctanh}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{\sqrt {a^2+1}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{x}+b \text {arcsinh}(a+b x)-\frac {a}{x}+b \log (x)\) |
Input:
Int[E^ArcSinh[a + b*x]/x^2,x]
Output:
-(a/x) - Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/x + b*ArcSinh[a + b*x] - (a*b*A rcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) ])/Sqrt[1 + a^2] + b*Log[x]
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. \(283\) vs. \(2(75)=150\).
Time = 0.18 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.42
method | result | size |
default | \(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}-\frac {a}{x}+b \ln \left (x \right )\) | \(284\) |
Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^2,x,method=_RETURNVERBOSE)
Output:
-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))-a/x+b*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (75) = 150\).
Time = 0.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.20 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\frac {\sqrt {a^{2} + 1} a b x \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - {\left (a^{2} + 1\right )} b x \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (a^{2} + 1\right )} b x \log \left (x\right ) - a^{3} - {\left (a^{2} + 1\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} - a}{{\left (a^{2} + 1\right )} x} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^2,x, algorithm="fricas")
Output:
(sqrt(a^2 + 1)*a*b*x*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^2 + 1)*b*x*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (a^2 + 1)*b*x*log(x) - a^3 - (a^2 + 1)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a ^2 + 1) - a)/((a^2 + 1)*x)
\[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{2}}\, dx \] Input:
integrate((b*x+a+(1+(b*x+a)**2)**(1/2))/x**2,x)
Output:
Integral((a + b*x + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1))/x**2, x)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (75) = 150\).
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=-\frac {a b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{\sqrt {a^{2} + 1}} + b \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right ) + b \log \left (x\right ) - \frac {a}{x} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{x} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^2,x, algorithm="maxima")
Output:
-a*b*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) + b*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^ 2 + 4*(a^2 + 1)*b^2)) + b*log(x) - a/x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) /x
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (75) = 150\).
Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.83 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\frac {a b \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1}} - \frac {b^{2} \log \left ({\left | -a b - {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} {\left | b \right |} \right |}\right )}{{\left | b \right |}} + b \log \left ({\left | x \right |}\right ) - \frac {a}{x} + \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b^{5} + a^{2} b^{4} {\left | b \right |} + b^{4} {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )} b^{4}} \] Input:
integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^2,x, algorithm="giac")
Output:
a*b*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)))/sqrt(a^2 + 1) - b^2*log(abs(-a*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b *x + a^2 + 1))*abs(b)))/abs(b) + b*log(abs(x)) - a/x + 2*((x*abs(b) - sqrt (b^2*x^2 + 2*a*b*x + a^2 + 1))*a*b^5 + a^2*b^4*abs(b) + b^4*abs(b))/(((x*a bs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - a^2 - 1)*b^4)
Time = 24.07 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.24 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=b\,\ln \left (x\right )-\frac {a}{x}+\ln \left (\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}+\frac {x\,b^2+a\,b}{\sqrt {b^2}}\right )\,\sqrt {b^2}-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x\,\left (a^2+1\right )}+\frac {a^3\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )}{{\left (a^2+1\right )}^{3/2}}-\frac {a^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x\,\left (a^2+1\right )}+\frac {a\,b\,\mathrm {atanh}\left (\frac {a^2+b\,x\,a+1}{\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}\right )}{{\left (a^2+1\right )}^{3/2}}-\frac {2\,a\,b\,\ln \left (a\,b+\frac {a^2+1}{x}+\frac {\sqrt {a^2+1}\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{x}\right )}{\sqrt {a^2+1}} \] Input:
int((a + ((a + b*x)^2 + 1)^(1/2) + b*x)/x^2,x)
Output:
b*log(x) - a/x + log((a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2) + (a*b + b^2*x)/( b^2)^(1/2))*(b^2)^(1/2) - (a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2)/(x*(a^2 + 1) ) + (a^3*b*atanh((a^2 + a*b*x + 1)/((a^2 + 1)^(1/2)*(a^2 + b^2*x^2 + 2*a*b *x + 1)^(1/2))))/(a^2 + 1)^(3/2) - (a^2*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2 ))/(x*(a^2 + 1)) + (a*b*atanh((a^2 + a*b*x + 1)/((a^2 + 1)^(1/2)*(a^2 + b^ 2*x^2 + 2*a*b*x + 1)^(1/2))))/(a^2 + 1)^(3/2) - (2*a*b*log(a*b + (a^2 + 1) /x + ((a^2 + 1)^(1/2)*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(1/2))/x))/(a^2 + 1)^( 1/2)
Time = 0.15 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^2} \, dx=\frac {2 \sqrt {a^{2}+1}\, \mathit {atan} \left (\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, i +b i x}{\sqrt {a^{2}+1}}\right ) a b i x -\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) a^{2} b x +\mathrm {log}\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+a +b x \right ) b x +\mathrm {log}\left (x \right ) a^{2} b x +\mathrm {log}\left (x \right ) b x -a^{3}-a}{x \left (a^{2}+1\right )} \] Input:
int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^2,x)
Output:
(2*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*i*x - sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a**2 - sqrt(a **2 + 2*a*b*x + b**2*x**2 + 1) + log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)*a**2*b*x + log(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + a + b*x)* b*x + log(x)*a**2*b*x + log(x)*b*x - a**3 - a)/(x*(a**2 + 1))