\(\int \frac {\text {arcsinh}(c x)}{d+e x} \, dx\) [1]

Optimal result

Integrand size = 12, antiderivative size = 170 \[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=-\frac {\text {arcsinh}(c x)^2}{2 e}+\frac {\text {arcsinh}(c x) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \] Output:

-1/2*arcsinh(c*x)^2/e+arcsinh(c*x)*ln(1+e*(c*x+(c^2*x^2+1)^(1/2))/(c*d-(c^ 
2*d^2+e^2)^(1/2)))/e+arcsinh(c*x)*ln(1+e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+(c^2 
*d^2+e^2)^(1/2)))/e+polylog(2,-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2+e^2 
)^(1/2)))/e+polylog(2,-e*(c*x+(c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2)) 
)/e
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99 \[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=-\frac {\text {arcsinh}(c x)^2}{2 e}+\frac {\text {arcsinh}(c x) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\text {arcsinh}(c x) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {\operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \] Input:

Integrate[ArcSinh[c*x]/(d + e*x),x]
 

Output:

-1/2*ArcSinh[c*x]^2/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sq 
rt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d + S 
qrt[c^2*d^2 + e^2])])/e + PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2 
*d^2 + e^2])]/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^ 
2]))]/e
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 170, 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 = {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.

\(\displaystyle \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx\)

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcSinh[c*x]/(d + e*x),x]
 

Output:

-1/2*ArcSinh[c*x]^2/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sq 
rt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d + S 
qrt[c^2*d^2 + e^2])])/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2* 
d^2 + e^2]))]/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^ 
2]))]/e
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {-\frac {c \operatorname {arcsinh}\left (x c \right )^{2}}{2 e}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}}{c}\)	\(272\)
default	\(\frac {-\frac {c \operatorname {arcsinh}\left (x c \right )^{2}}{2 e}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {c \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}}{c}\)	\(272\)

Input:

int(arcsinh(x*c)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/c*(-1/2*c*arcsinh(x*c)^2/e+c/e*arcsinh(x*c)*ln((-c*d-e*(x*c+(c^2*x^2+1)^ 
(1/2))+(c^2*d^2+e^2)^(1/2))/(-c*d+(c^2*d^2+e^2)^(1/2)))+c/e*arcsinh(x*c)*l 
n((c*d+e*(x*c+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(c*d+(c^2*d^2+e^2)^( 
1/2)))+c/e*dilog((-c*d-e*(x*c+(c^2*x^2+1)^(1/2))+(c^2*d^2+e^2)^(1/2))/(-c* 
d+(c^2*d^2+e^2)^(1/2)))+c/e*dilog((c*d+e*(x*c+(c^2*x^2+1)^(1/2))+(c^2*d^2+ 
e^2)^(1/2))/(c*d+(c^2*d^2+e^2)^(1/2))))
 

Fricas [F]

\[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )}{e x + d} \,d x } \] Input:

integrate(arcsinh(c*x)/(e*x+d),x, algorithm="fricas")
 

Output:

integral(arcsinh(c*x)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=\int \frac {\operatorname {asinh}{\left (c x \right )}}{d + e x}\, dx \] Input:

integrate(asinh(c*x)/(e*x+d),x)
 

Output:

Integral(asinh(c*x)/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )}{e x + d} \,d x } \] Input:

integrate(arcsinh(c*x)/(e*x+d),x, algorithm="maxima")
 

Output:

integrate(arcsinh(c*x)/(e*x + d), x)
 

Giac [F]

\[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=\int { \frac {\operatorname {arsinh}\left (c x\right )}{e x + d} \,d x } \] Input:

integrate(arcsinh(c*x)/(e*x+d),x, algorithm="giac")
 

Output:

integrate(arcsinh(c*x)/(e*x + d), x)
 

Mupad [F(-1)]

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

int(asinh(c*x)/(d + e*x),x)
 

Output:

int(asinh(c*x)/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {\text {arcsinh}(c x)}{d+e x} \, dx=\int \frac {\mathit {asinh} \left (c x \right )}{e x +d}d x \] Input:

int(asinh(c*x)/(e*x+d),x)
 

Output:

int(asinh(c*x)/(d + e*x),x)