\(\int \frac {a+b \text {arcsinh}(c x)}{x^3 (d+c^2 d x^2)} \, dx\) [35]

Optimal result

Integrand size = 24, antiderivative size = 113 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{2 d x}-\frac {a+b \text {arcsinh}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(240\) vs. \(2(113)=226\).

Time = 0.18 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.12 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {-\frac {b c \sqrt {1+c^2 x^2}}{x}-b c^2 \text {arcsinh}(c x)^2-\frac {a+b \text {arcsinh}(c x)}{x^2}+\frac {c^2 (a+b \text {arcsinh}(c x))^2}{b}+2 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+a c^2 \log \left (1+c^2 x^2\right )+2 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+2 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{2 d} \] Input:

Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)),x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6224, 27, 242, 6214, 5984, 3042, 26, 4670, 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.

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 d x^2+d\right )} \, dx\)

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 6214

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

\(\Big \downarrow \) 5984

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)),x]
 

Output:

-1/2*(b*c*Sqrt[1 + c^2*x^2])/(d*x) - (a + b*ArcSinh[c*x])/(2*d*x^2) - ((2* 
I)*c^2*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyL 
og[2, -E^(2*ArcSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])]))/d
 

Defintions of rubi rules used

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 27

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

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

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 4670

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

rule 5984

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n   Int[(c + d*x)^m*Csch[2*a + 2*b*x 
]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
 

rule 6214

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), 
 x_Symbol] :> Simp[1/d   Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar 
cSinh[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 = 1.14 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.92

method	result	size
derivativedivides	\(c^{2} \left (\frac {a \left (\frac {\ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 c^{2} x^{2}}-\ln \left (x c \right )\right )}{d}+\frac {b \left (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(217\)
default	\(c^{2} \left (\frac {a \left (\frac {\ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 c^{2} x^{2}}-\ln \left (x c \right )\right )}{d}+\frac {b \left (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(217\)
parts	\(\frac {a \left (-\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )+\frac {c^{2} \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b \,c^{2} \left (-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\)	\(217\)

Input:

int((a+b*arcsinh(x*c))/x^3/(c^2*d*x^2+d),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

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

Fricas [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="fricas")
 

Output:

integral((b*arcsinh(c*x) + a)/(c^2*d*x^5 + d*x^3), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx}{d} \] Input:

integrate((a+b*asinh(c*x))/x**3/(c**2*d*x**2+d),x)
 

Output:

(Integral(a/(c**2*x**5 + x**3), x) + Integral(b*asinh(c*x)/(c**2*x**5 + x* 
*3), x))/d
 

Maxima [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="maxima")
 

Output:

1/2*(c^2*log(c^2*x^2 + 1)/d - 2*c^2*log(x)/d - 1/(d*x^2))*a + b*integrate( 
log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^5 + d*x^3), x)
 

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \] Input:

integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d),x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)*x^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,\left (d\,c^2\,x^2+d\right )} \,d x \] Input:

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)),x)
 

Output:

int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {2 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{5}+x^{3}}d x \right ) b \,x^{2}+\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{2} x^{2}-2 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-a}{2 d \,x^{2}} \] Input:

int((a+b*asinh(c*x))/x^3/(c^2*d*x^2+d),x)
 

Output:

(2*int(asinh(c*x)/(c**2*x**5 + x**3),x)*b*x**2 + log(c**2*x**2 + 1)*a*c**2 
*x**2 - 2*log(x)*a*c**2*x**2 - a)/(2*d*x**2)