\(\int \frac {a+b \text {arcsinh}(c x)}{x^2 (d+c^2 d x^2)} \, dx\) [34]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {a+b \text {arcsinh}(c x)}{d x}-\frac {2 c (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {i b c \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d} \] Output:

-(a+b*arcsinh(c*x))/d/x-2*c*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2 
))/d-b*c*arctanh((c^2*x^2+1)^(1/2))/d+I*b*c*polylog(2,-I*(c*x+(c^2*x^2+1)^ 
(1/2)))/d-I*b*c*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {a+b \text {arcsinh}(c x)+a c x \arctan (c x)+b c x \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+b \sqrt {-c^2} x \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-b \sqrt {-c^2} x \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-b \sqrt {-c^2} x \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+b \sqrt {-c^2} x \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{d x} \] Input:

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

Output:

-((a + b*ArcSinh[c*x] + a*c*x*ArcTan[c*x] + b*c*x*ArcTanh[Sqrt[1 + c^2*x^2 
]] + b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 
b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - b*Sqr 
t[-c^2]*x*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + b*Sqrt[-c^2]*x*PolyL 
og[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(d*x))
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6224, 27, 243, 73, 221, 6204, 3042, 4668, 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.

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

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle -\frac {c^2 \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{d}+\frac {b \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c d}-\frac {a+b \text {arcsinh}(c x)}{d x}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-((a + b*ArcSinh[c*x])/(d*x)) - (b*c*ArcTanh[Sqrt[1 + c^2*x^2]])/d - (c*(2 
*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSi 
nh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]]))/d
 

Defintions of rubi rules used

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4668

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

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

rule 6224

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 
1)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 
1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.74

method	result	size
parts	\(\frac {a \left (-c \arctan \left (x c \right )-\frac {1}{x}\right )}{d}+\frac {b c \left (-\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\)	\(176\)
derivativedivides	\(c \left (\frac {a \left (-\arctan \left (x c \right )-\frac {1}{x c}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\right )\)	\(179\)
default	\(c \left (\frac {a \left (-\arctan \left (x c \right )-\frac {1}{x c}\right )}{d}+\frac {b \left (-\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}\right )\)	\(179\)

Input:

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

Output:

a/d*(-c*arctan(x*c)-1/x)+b/d*c*(-arcsinh(x*c)*arctan(x*c)-arcsinh(x*c)/x/c 
-arctanh(1/(c^2*x^2+1)^(1/2))-arctan(x*c)*ln(1+I*(1+I*x*c)/(c^2*x^2+1)^(1/ 
2))+arctan(x*c)*ln(1-I*(1+I*x*c)/(c^2*x^2+1)^(1/2))+I*dilog(1+I*(1+I*x*c)/ 
(c^2*x^2+1)^(1/2))-I*dilog(1-I*(1+I*x*c)/(c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

integral((b*arcsinh(c*x) + a)/(c^2*d*x^4 + d*x^2), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx}{d} \] Input:

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

Output:

(Integral(a/(c**2*x**4 + x**2), x) + Integral(b*asinh(c*x)/(c**2*x**4 + x* 
*2), x))/d
 

Maxima [F]

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

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

Output:

-a*(c*arctan(c*x)/d + 1/(d*x)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/ 
(c^2*d*x^4 + d*x^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx=\frac {-\mathit {atan} \left (c x \right ) a c x +\left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{4}+x^{2}}d x \right ) b x -a}{d x} \] Input:

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

Output:

( - atan(c*x)*a*c*x + int(asinh(c*x)/(c**2*x**4 + x**2),x)*b*x - a)/(d*x)