3.1.6 \(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))}{x} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 22, antiderivative size = 111 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}-\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))+\frac {d (a+b \text {arcsinh}(c x))^2}{2 b}+d (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]

output

-1/4*b*d*arcsinh(c*x)+1/2*d*(c^2*x^2+1)*(a+b*arcsinh(c*x))+1/2*d*(a+b*arcs 
inh(c*x))^2/b+d*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b 
*d*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)-1/4*b*c*d*x*(c^2*x^2+1)^(1/2)
 

3.1.6.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{2} a c^2 d x^2-\frac {1}{4} b c d x \sqrt {1+c^2 x^2}+\frac {1}{4} b d \text {arcsinh}(c x)+\frac {1}{2} b c^2 d x^2 \text {arcsinh}(c x)-\frac {1}{2} b d \text {arcsinh}(c x)^2+b d \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) \]

input

Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x,x]
 

output

(a*c^2*d*x^2)/2 - (b*c*d*x*Sqrt[1 + c^2*x^2])/4 + (b*d*ArcSinh[c*x])/4 + ( 
b*c^2*d*x^2*ArcSinh[c*x])/2 - (b*d*ArcSinh[c*x]^2)/2 + b*d*ArcSinh[c*x]*Lo 
g[1 - E^(2*ArcSinh[c*x])] + a*d*Log[x] + (b*d*PolyLog[2, E^(2*ArcSinh[c*x] 
)])/2
 

3.1.6.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.38, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6216, 211, 222, 6190, 25, 3042, 26, 4201, 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.

\(\displaystyle \int \frac {\left (c^2 d x^2+d\right ) (a+b \text {arcsinh}(c x))}{x} \, dx\)

\(\Big \downarrow \) 6216

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {i d \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c d \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {i d \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c d \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\)

input

Int[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x,x]
 

output

(d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/2 - (b*c*d*((x*Sqrt[1 + c^2*x^2])/2 
 + ArcSinh[c*x]/(2*c)))/2 + (I*d*((-1/2*I)*(a + b*ArcSinh[c*x])^2 + (2*I)* 
(-1/2*(b*(a + b*ArcSinh[c*x])*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSin 
h[c*x]))/b)]) + (b^2*PolyLog[2, -a - b*ArcSinh[c*x]])/4)))/b
 

3.1.6.3.1 Defintions of rubi rules used

rule 25

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

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6216

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), 
 x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSinh[c*x])/(2*p)), x] + (Simp[d 
   Int[(d + e*x^2)^(p - 1)*((a + b*ArcSinh[c*x])/x), x], x] - Simp[b*c*(d^p 
/(2*p))   Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] 
 && EqQ[e, c^2*d] && IGtQ[p, 0]
 

3.1.6.4 Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.43

method	result	size
parts	\(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\)	\(159\)
derivativedivides	\(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\)	\(161\)
default	\(d a \left (\frac {c^{2} x^{2}}{2}+\ln \left (c x \right )\right )-\frac {d b \operatorname {arcsinh}\left (c x \right )^{2}}{2}+\frac {d b \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c d x \sqrt {c^{2} x^{2}+1}}{4}+\frac {b d \,\operatorname {arcsinh}\left (c x \right )}{4}+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+d b \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+d b \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\)	\(161\)

input

int((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x,x,method=_RETURNVERBOSE)
 

output

d*a*(1/2*c^2*x^2+ln(x))-1/2*d*b*arcsinh(c*x)^2+1/2*d*b*arcsinh(c*x)*c^2*x^ 
2-1/4*b*c*d*x*(c^2*x^2+1)^(1/2)+1/4*b*d*arcsinh(c*x)+d*b*arcsinh(c*x)*ln(1 
+c*x+(c^2*x^2+1)^(1/2))+d*b*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+d*b*arcsinh( 
c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+d*b*polylog(2,c*x+(c^2*x^2+1)^(1/2))
 

3.1.6.5 Fricas [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]

input

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x,x, algorithm="fricas")
 

output

integral((a*c^2*d*x^2 + a*d + (b*c^2*d*x^2 + b*d)*arcsinh(c*x))/x, x)
 

3.1.6.6 Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=d \left (\int \frac {a}{x}\, dx + \int a c^{2} x\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{2} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]

input

integrate((c**2*d*x**2+d)*(a+b*asinh(c*x))/x,x)
 

output

d*(Integral(a/x, x) + Integral(a*c**2*x, x) + Integral(b*asinh(c*x)/x, x) 
+ Integral(b*c**2*x*asinh(c*x), x))
 

3.1.6.7 Maxima [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]

input

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x,x, algorithm="maxima")
 

output

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

3.1.6.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]

input

integrate((c^2*d*x^2+d)*(a+b*arcsinh(c*x))/x,x, algorithm="giac")
 

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

3.1.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right )}{x} \,d x \]

input

int(((a + b*asinh(c*x))*(d + c^2*d*x^2))/x,x)
 

output

int(((a + b*asinh(c*x))*(d + c^2*d*x^2))/x, x)