\(\int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx\) [8]

Optimal result

Integrand size = 16, antiderivative size = 187 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{2 b e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {(a+b \text {arcsinh}(c x)) \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \] Output:

-1/2*(a+b*arcsinh(c*x))^2/b/e+(a+b*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+(a+b*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+b*polylog(2,-e*(c*x+(c^2*x^2+1)^(1 
/2))/(c*d-(c^2*d^2+e^2)^(1/2)))/e+b*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 = 0.05 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+e x} \, dx=\frac {-\left ((a+b \text {arcsinh}(c x)) \left (a+b \text {arcsinh}(c x)-2 b \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )-2 b \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )\right )\right )+2 b^2 \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+2 b^2 \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{2 b e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, 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.

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

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/2*(a + b*ArcSinh[c*x])^2/(b*e) + ((a + b*ArcSinh[c*x])*Log[1 + (e*E^Arc 
Sinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + ((a + b*ArcSinh[c*x])*Log[1 + 
 (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + (b*PolyLog[2, -((e*E 
^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))])/e + (b*PolyLog[2, -((e*E^Arc 
Sinh[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 = 1.66 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.49

method	result	size
parts	\(\frac {a \ln \left (e x +d \right )}{e}-\frac {b \operatorname {arcsinh}\left (x c \right )^{2}}{2 e}+\frac {b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \,\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {b \operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\)	\(279\)
derivativedivides	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2 e}+\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\right )}{c}\)	\(286\)
default	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2 e}+\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {arcsinh}\left (x c \right ) \ln \left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-c d -e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{-c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}+\frac {\operatorname {dilog}\left (\frac {c d +e \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\sqrt {c^{2} d^{2}+e^{2}}}{c d +\sqrt {c^{2} d^{2}+e^{2}}}\right )}{e}\right )}{c}\)	\(286\)

Input:

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

Output:

a*ln(e*x+d)/e-1/2*b*arcsinh(x*c)^2/e+b/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)))+b/e*arcsin 
h(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)))+b/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)))+b/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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(e*x + d), x) + a*log(e*x + d)/e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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