\(\int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx\) [355]

Optimal result

Integrand size = 12, antiderivative size = 116 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=-\frac {a}{2 x^2}-\frac {b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}-\frac {b^2 \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \]

[Out]

-1/2*a/x^2-b/x-1/2*b^2*arctanh((a*b*x+a^2+1)/(a^2+1)^(1/2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(a^2+1)^(3/2)-1/2*(a
*b*x+a^2+1)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/(a^2+1)/x^2

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5878, 14, 734, 738, 212} \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=-\frac {b^2 \text {arctanh}\left (\frac {a^2+a b x+1}{\sqrt {a^2+1} \sqrt {a^2+2 a b x+b^2 x^2+1}}\right )}{2 \left (a^2+1\right )^{3/2}}-\frac {\left (a^2+a b x+1\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}}{2 \left (a^2+1\right ) x^2}-\frac {a}{2 x^2}-\frac {b}{x} \]

[In]

Int[E^ArcSinh[a + b*x]/x^3,x]

[Out]

-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))

Rule 14

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]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))
*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[p*((b^2
- 4*a*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m
+ 2*p + 2, 0] && GtQ[p, 0]

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 5878

Int[E^(ArcSinh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[1 + u^2])^n, x] /; RationalQ[m] && Intege
rQ[n] && PolyQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b x+\sqrt {1+(a+b x)^2}}{x^3} \, dx \\ & = \int \left (\frac {a}{x^3}+\frac {b}{x^2}+\frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3}\right ) \, dx \\ & = -\frac {a}{2 x^2}-\frac {b}{x}+\int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{x^3} \, dx \\ & = -\frac {a}{2 x^2}-\frac {b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}+\frac {b^2 \int \frac {1}{x \sqrt {1+a^2+2 a b x+b^2 x^2}} \, dx}{2 \left (1+a^2\right )} \\ & = -\frac {a}{2 x^2}-\frac {b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}-\frac {b^2 \text {Subst}\left (\int \frac {1}{4 \left (1+a^2\right )-x^2} \, dx,x,\frac {2 \left (1+a^2\right )+2 a b x}{\sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{1+a^2} \\ & = -\frac {a}{2 x^2}-\frac {b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{2 \left (1+a^2\right ) x^2}-\frac {b^2 \text {arctanh}\left (\frac {1+a^2+a b x}{\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {1}{2} \left (-\frac {a}{x^2}-\frac {2 b}{x}-\frac {\left (1+a^2+a b x\right ) \sqrt {1+a^2+2 a b x+b^2 x^2}}{\left (1+a^2\right ) x^2}+\frac {b^2 \log (x)}{\left (1+a^2\right )^{3/2}}-\frac {b^2 \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )}{\left (1+a^2\right )^{3/2}}\right ) \]

[In]

Integrate[E^ArcSinh[a + b*x]/x^3,x]

[Out]

(-(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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(458\) vs. \(2(102)=204\).

Time = 0.75 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.96

method	result	size
default	\(-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{2 \left (a^{2}+1\right ) x^{2}}-\frac {a b \left (-\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{\left (a^{2}+1\right ) x}+\frac {a b \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{a^{2}+1}+\frac {2 b^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{a^{2}+1}\right )}{2 \left (a^{2}+1\right )}+\frac {b^{2} \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )\right )}{2 a^{2}+2}-\frac {b}{x}-\frac {a}{2 x^{2}}\)	\(459\)

[In]

int((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x,method=_RETURNVERBOSE)

[Out]

-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))-b/x-1/2*a/x^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.56 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {\sqrt {a^{2} + 1} b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} - \sqrt {a^{2} + 1} a + 1\right )} - {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - a^{5} - {\left (a^{3} + a\right )} b^{2} x^{2} - 2 \, a^{3} - 2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} b x - {\left (a^{4} + {\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - a}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \]

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x, algorithm="fricas")

[Out]

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)

Sympy [F]

\[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\int \frac {a + b x + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{x^{3}}\, dx \]

[In]

integrate((b*x+a+(1+(b*x+a)**2)**(1/2))/x**3,x)

[Out]

Integral((a + b*x + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1))/x**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (102) = 204\).

Time = 0.38 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.70 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {a^{2} b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {b^{2} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{2 \, \sqrt {a^{2} + 1}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}}{2 \, {\left (a^{2} + 1\right )}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a b}{2 \, {\left (a^{2} + 1\right )} x} - \frac {b}{x} - \frac {a}{2 \, x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 \, {\left (a^{2} + 1\right )} x^{2}} \]

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x, algorithm="maxima")

[Out]

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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (102) = 204\).

Time = 0.34 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.31 \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\frac {b^{2} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{2 \, {\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, b x + a}{2 \, x^{2}} + \frac {2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} a^{2} b^{2} + 2 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{4} b^{2} + 4 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a^{3} b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{3} b^{2} + 3 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a^{2} b^{2} + 4 \, {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} a b {\left | b \right |} + {\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b^{2}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )}^{2} {\left (a^{2} + 1\right )}} \]

[In]

integrate((b*x+a+(1+(b*x+a)^2)^(1/2))/x^3,x, algorithm="giac")

[Out]

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))

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\text {arcsinh}(a+b x)}}{x^3} \, dx=\int \frac {a+\sqrt {{\left (a+b\,x\right )}^2+1}+b\,x}{x^3} \,d x \]

[In]

int((a + ((a + b*x)^2 + 1)^(1/2) + b*x)/x^3,x)

[Out]

int((a + ((a + b*x)^2 + 1)^(1/2) + b*x)/x^3, x)