Integrand size = 10, antiderivative size = 92 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {b \sqrt {1+(a+b x)^2}}{2 \left (1+a^2\right ) x}-\frac {\text {arcsinh}(a+b x)}{2 x^2}+\frac {a b^2 \text {arctanh}\left (\frac {1+a (a+b x)}{\sqrt {1+a^2} \sqrt {1+(a+b x)^2}}\right )}{2 \left (1+a^2\right )^{3/2}} \]
-1/2*arcsinh(b*x+a)/x^2+1/2*a*b^2*arctanh((1+a*(b*x+a))/(a^2+1)^(1/2)/(1+( b*x+a)^2)^(1/2))/(a^2+1)^(3/2)-1/2*b*(1+(b*x+a)^2)^(1/2)/(a^2+1)/x
Time = 0.16 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {\text {arcsinh}(a+b x)+\frac {b x \left (\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}+a b x \log (x)-a b x \log \left (1+a^2+a b x+\sqrt {1+a^2} \sqrt {1+a^2+2 a b x+b^2 x^2}\right )\right )}{\left (1+a^2\right )^{3/2}}}{2 x^2} \]
-1/2*(ArcSinh[a + b*x] + (b*x*(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2* x^2] + a*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]]))/(1 + a^2)^(3/2))/x^2
Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6274, 25, 27, 6243, 491, 488, 219}
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.
\(\displaystyle \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int -\frac {\text {arcsinh}(a+b x)}{x^3}d(a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -b^2 \int -\frac {\text {arcsinh}(a+b x)}{b^3 x^3}d(a+b x)\) |
\(\Big \downarrow \) 6243 |
\(\displaystyle -b^2 \left (\frac {\text {arcsinh}(a+b x)}{2 b^2 x^2}-\frac {1}{2} \int \frac {1}{b^2 x^2 \sqrt {(a+b x)^2+1}}d(a+b x)\right )\) |
\(\Big \downarrow \) 491 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {\sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}-\frac {a \int -\frac {1}{b x \sqrt {(a+b x)^2+1}}d(a+b x)}{a^2+1}\right )+\frac {\text {arcsinh}(a+b x)}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {a \int \frac {1}{a^2-\frac {(-a (a+b x)-1)^2}{(a+b x)^2+1}+1}d\frac {-a (a+b x)-1}{\sqrt {(a+b x)^2+1}}}{a^2+1}+\frac {\sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}\right )+\frac {\text {arcsinh}(a+b x)}{2 b^2 x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -b^2 \left (\frac {1}{2} \left (\frac {a \text {arctanh}\left (\frac {-a (a+b x)-1}{\sqrt {a^2+1} \sqrt {(a+b x)^2+1}}\right )}{\left (a^2+1\right )^{3/2}}+\frac {\sqrt {(a+b x)^2+1}}{\left (a^2+1\right ) b x}\right )+\frac {\text {arcsinh}(a+b x)}{2 b^2 x^2}\right )\) |
-(b^2*(ArcSinh[a + b*x]/(2*b^2*x^2) + (Sqrt[1 + (a + b*x)^2]/((1 + a^2)*b* x) + (a*ArcTanh[(-1 - a*(a + b*x))/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])]) /(1 + a^2)^(3/2))/2))
3.1.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b*(c/(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 3, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.02 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.12
method | result | size |
parts | \(-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 x^{2}}+\frac {b \left (-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{\left (a^{2}+1\right ) x}+\frac {a b \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2}\) | \(103\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(112\) |
default | \(b^{2} \left (-\frac {\operatorname {arcsinh}\left (b x +a \right )}{2 b^{2} x^{2}}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 \left (a^{2}+1\right ) b x}+\frac {a \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b x}\right )}{2 \left (a^{2}+1\right )^{\frac {3}{2}}}\right )\) | \(112\) |
-1/2*arcsinh(b*x+a)/x^2+1/2*b*(-1/(a^2+1)/x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+ a*b/(a^2+1)^(3/2)*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))
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.57 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\frac {\sqrt {a^{2} + 1} a b^{2} x^{2} \log \left (-\frac {a^{2} b x + a^{3} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + \sqrt {a^{2} + 1} a + 1\right )} + {\left (a b x + a^{2} + 1\right )} \sqrt {a^{2} + 1} + a}{x}\right ) - {\left (a^{2} + 1\right )} b^{2} x^{2} + {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} b x - {\left (a^{4} - {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2} + 2 \, a^{2} + 1\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \]
1/2*(sqrt(a^2 + 1)*a*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^2 + 1)*b^2*x^2 + (a^4 + 2*a^2 + 1)*x^2*log(-b*x - a + sqrt(b^2 *x^2 + 2*a*b*x + a^2 + 1)) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*b *x - (a^4 - (a^4 + 2*a^2 + 1)*x^2 + 2*a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)))/((a^4 + 2*a^2 + 1)*x^2)
\[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x^{3}}\, dx \]
Time = 0.18 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.59 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\frac {1}{2} \, {\left (\frac {a b \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (a^{2} + 1\right )} x}\right )} b - \frac {\operatorname {arsinh}\left (b x + a\right )}{2 \, x^{2}} \]
1/2*(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)))/(a^2 + 1)^(3/2) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/ ((a^2 + 1)*x))*b - 1/2*arcsinh(b*x + a)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (78) = 156\).
Time = 0.36 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.16 \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=-\frac {1}{2} \, {\left (\frac {a b \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} - \frac {2 \, {\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} a b + a^{2} {\left | b \right |} + {\left | b \right |}\right )}}{{\left ({\left (x {\left | b \right |} - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}^{2} - a^{2} - 1\right )} {\left (a^{2} + 1\right )}}\right )} b - \frac {\log \left (b x + a + \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}{2 \, x^{2}} \]
-1/2*(a*b*log(abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 2*sq rt(a^2 + 1))/abs(-2*x*abs(b) + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 2*sqr t(a^2 + 1)))/(a^2 + 1)^(3/2) - 2*((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*a*b + a^2*abs(b) + abs(b))/(((x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a ^2 + 1))^2 - a^2 - 1)*(a^2 + 1)))*b - 1/2*log(b*x + a + sqrt((b*x + a)^2 + 1))/x^2
Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x^3} \,d x \]