3.1.62 \(\int \frac {\text {arcsinh}(a+b x)}{x} \, dx\) [62]

3.1.62.1 Optimal result

Integrand size = 10, antiderivative size = 131 \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

output

-1/2*arcsinh(b*x+a)^2+arcsinh(b*x+a)*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-( 
a^2+1)^(1/2)))+arcsinh(b*x+a)*ln(1-(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^ 
(1/2)))+polylog(2,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a-(a^2+1)^(1/2)))+polylog(2 
,(b*x+a+(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))
 

3.1.62.2 Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=-\frac {1}{2} \text {arcsinh}(a+b x)^2+\text {arcsinh}(a+b x) \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\text {arcsinh}(a+b x) \log \left (1+\frac {e^{\text {arcsinh}(a+b x)}}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(a+b x)}}{-a+\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {1+a^2}}\right ) \]

input

Integrate[ArcSinh[a + b*x]/x,x]
 

output

-1/2*ArcSinh[a + b*x]^2 + ArcSinh[a + b*x]*Log[1 + E^ArcSinh[a + b*x]/((-( 
a/b) - Sqrt[1 + a^2]/b)*b)] + ArcSinh[a + b*x]*Log[1 + E^ArcSinh[a + b*x]/ 
((-(a/b) + Sqrt[1 + a^2]/b)*b)] + PolyLog[2, -(E^ArcSinh[a + b*x]/(-a + Sq 
rt[1 + a^2]))] + PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]
 

3.1.62.3 Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6274, 25, 27, 6242, 6095, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {\text {arcsinh}(a+b x)}{x} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {\text {arcsinh}(a+b x)}{x}d(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\text {arcsinh}(a+b x)}{x}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int -\frac {\text {arcsinh}(a+b x)}{b x}d(a+b x)\)

\(\Big \downarrow \) 6242

\(\displaystyle -\int -\frac {\sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)}{b x}d\text {arcsinh}(a+b x)\)

\(\Big \downarrow \) 6095

\(\displaystyle -\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{a-e^{\text {arcsinh}(a+b x)}-\sqrt {a^2+1}}d\text {arcsinh}(a+b x)-\int \frac {e^{\text {arcsinh}(a+b x)} \text {arcsinh}(a+b x)}{a-e^{\text {arcsinh}(a+b x)}+\sqrt {a^2+1}}d\text {arcsinh}(a+b x)-\frac {1}{2} \text {arcsinh}(a+b x)^2\)

\(\Big \downarrow \) 2620

\(\displaystyle -\int \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)-\int \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )d\text {arcsinh}(a+b x)+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \text {arcsinh}(a+b x)^2\)

\(\Big \downarrow \) 2715

\(\displaystyle -\int e^{-\text {arcsinh}(a+b x)} \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}-\int e^{-\text {arcsinh}(a+b x)} \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )de^{\text {arcsinh}(a+b x)}+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \text {arcsinh}(a+b x)^2\)

\(\Big \downarrow \) 2838

\(\displaystyle \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(a+b x)}}{a+\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {arcsinh}(a+b x) \log \left (1-\frac {e^{\text {arcsinh}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \text {arcsinh}(a+b x)^2\)

input

Int[ArcSinh[a + b*x]/x,x]
 

output

-1/2*ArcSinh[a + b*x]^2 + ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a - 
 Sqrt[1 + a^2])] + ArcSinh[a + b*x]*Log[1 - E^ArcSinh[a + b*x]/(a + Sqrt[1 
 + a^2])] + PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + PolyLog[2 
, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]
 

3.1.62.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6242

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo 
l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x 
]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
 

rule 6274

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]
 

3.1.62.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(387\) vs. \(2(153)=306\).

Time = 0.66 (sec) , antiderivative size = 388, normalized size of antiderivative = 2.96

method	result	size
derivativedivides	\(-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{2}+\frac {\left (a^{2}+1+\sqrt {a^{2}+1}\, a \right ) \operatorname {arcsinh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) \sqrt {a^{2}+1}\, a +\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{a^{2}+1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\)	\(388\)
default	\(-\frac {\operatorname {arcsinh}\left (b x +a \right )^{2}}{2}+\frac {\left (a^{2}+1+\sqrt {a^{2}+1}\, a \right ) \operatorname {arcsinh}\left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) \sqrt {a^{2}+1}\, a +\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{a^{2}+1}+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )+\operatorname {dilog}\left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )+\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {a \,\operatorname {arcsinh}\left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}\)	\(388\)

input

int(arcsinh(b*x+a)/x,x,method=_RETURNVERBOSE)
 

output

-1/2*arcsinh(b*x+a)^2+(a^2+1+(a^2+1)^(1/2)*a)/(a^2+1)*arcsinh(b*x+a)*(2*ln 
(((a^2+1)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))*a^2+ln(((a^2+1 
)^(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-2*ln(((a^2+1)^(1/2)-b* 
x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))*(a^2+1)^(1/2)*a+ln(((a^2+1)^(1/2 
)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2))))+dilog(((a^2+1)^(1/2)+b*x+( 
1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))+dilog(((a^2+1)^(1/2)-b*x-(1+(b*x+a 
)^2)^(1/2))/(a+(a^2+1)^(1/2)))+a*arcsinh(b*x+a)/(a^2+1)^(1/2)*ln(((a^2+1)^ 
(1/2)-b*x-(1+(b*x+a)^2)^(1/2))/(a+(a^2+1)^(1/2)))-a*arcsinh(b*x+a)/(a^2+1) 
^(1/2)*ln(((a^2+1)^(1/2)+b*x+(1+(b*x+a)^2)^(1/2))/(-a+(a^2+1)^(1/2)))
 

3.1.62.5 Fricas [F]

\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsinh(b*x+a)/x,x, algorithm="fricas")
 

output

integral(arcsinh(b*x + a)/x, x)
 

3.1.62.6 Sympy [F]

\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x}\, dx \]

input

integrate(asinh(b*x+a)/x,x)
 

output

Integral(asinh(a + b*x)/x, x)
 

3.1.62.7 Maxima [F]

\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsinh(b*x+a)/x,x, algorithm="maxima")
 

output

integrate(arcsinh(b*x + a)/x, x)
 

3.1.62.8 Giac [F]

\[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsinh(b*x+a)/x,x, algorithm="giac")
 

output

integrate(arcsinh(b*x + a)/x, x)
 

3.1.62.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a+b x)}{x} \, dx=\int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x} \,d x \]

input

int(asinh(a + b*x)/x,x)
 

output

int(asinh(a + b*x)/x, x)