\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\) [135]

Optimal result

Integrand size = 26, antiderivative size = 201 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 a \sqrt {d+c^2 d x^2}}{x^2}+4 a c^2 \sqrt {d} \log (x)-4 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b c^2 \sqrt {d+c^2 d x^2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}\right ) \] Input:

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

Output:

((-4*a*Sqrt[d + c^2*d*x^2])/x^2 + 4*a*c^2*Sqrt[d]*Log[x] - 4*a*c^2*Sqrt[d] 
*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*c^2*Sqrt[d + c^2*d*x^2]*(-2*Cot 
h[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*L 
og[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4* 
PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh 
[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2]) 
/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.89 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6220, 15, 6231, 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.

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 6220

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e* 
x^2]/Sqrt[1 + c^2*x^2]]   Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x] - Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] 
Int[(f*x)^(m + 2)*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x]) /; Fr 
eeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[m, -1]
 

rule 6231

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

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.66

method	result	size
default	\(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(334\)
parts	\(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(334\)

Input:

int((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))/x^3,x,method=_RETURNVERBOSE)
 

Output:

a*(-1/2/d/x^2*(c^2*d*x^2+d)^(3/2)+1/2*c^2*((c^2*d*x^2+d)^(1/2)-d^(1/2)*ln( 
(2*d+2*d^(1/2)*(c^2*d*x^2+d)^(1/2))/x)))+b*(-1/2*(arcsinh(x*c)*c^2*x^2+(c^ 
2*x^2+1)^(1/2)*x*c+arcsinh(x*c))*(d*(c^2*x^2+1))^(1/2)/x^2/(c^2*x^2+1)+1/2 
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1) 
^(1/2))*c^2+1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,x*c+(c^2 
*x^2+1)^(1/2))*c^2-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c 
)*ln(1+x*c+(c^2*x^2+1)^(1/2))*c^2-1/2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1 
/2)*polylog(2,-x*c-(c^2*x^2+1)^(1/2))*c^2)
 

Fricas [F]

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

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="fricas" 
)
 

Output:

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

Sympy [F]

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

integrate((c**2*d*x**2+d)**(1/2)*(a+b*asinh(c*x))/x**3,x)
 

Output:

Integral(sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))/x**3, x)
 

Maxima [F]

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

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="maxima" 
)
 

Output:

-1/2*(c^2*sqrt(d)*arcsinh(1/(c*abs(x))) - sqrt(c^2*d*x^2 + d)*c^2 + (c^2*d 
*x^2 + d)^(3/2)/(d*x^2))*a + b*integrate(sqrt(c^2*d*x^2 + d)*log(c*x + sqr 
t(c^2*x^2 + 1))/x^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/x^3,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
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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