\(\int \frac {a+b \text {arcsinh}(c x)}{x (d+c^2 d x^2)^{3/2}} \, dx\) [169]

Optimal result

Integrand size = 26, antiderivative size = 198 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\frac {a}{d}+\frac {b \text {arcsinh}(c x)}{d}}{\sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{d \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \text {arcsinh}(c x)}{x \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\frac {a \sqrt {d+c^2 d x^2}}{1+c^2 x^2}+a \sqrt {d} \log (x)-a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {b d \left (\text {arcsinh}(c x)-2 \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {d+c^2 d x^2}}}{d^2} \] Input:

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.74, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6226, 216, 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.

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

\(\Big \downarrow \) 6226

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 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 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 6226

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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 
))   Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ 
b*c*(n/(2*f*(p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !G 
tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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 = 1.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {a}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+2 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2}}\)	\(271\)
parts	\(\frac {a}{d \sqrt {c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+2 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+\operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \arctan \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {dilog}\left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2}}\)	\(271\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^4*d^2*x^5 + 2*c^2*d^2 
*x^3 + d^2*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c**2*x**2 + 1)*a + int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**3 + s 
qrt(c**2*x**2 + 1)*x),x)*b*c**2*x**2 + int(asinh(c*x)/(sqrt(c**2*x**2 + 1) 
*c**2*x**3 + sqrt(c**2*x**2 + 1)*x),x)*b + log(sqrt(c**2*x**2 + 1) + c*x - 
 1)*a*c**2*x**2 + log(sqrt(c**2*x**2 + 1) + c*x - 1)*a - log(sqrt(c**2*x** 
2 + 1) + c*x + 1)*a*c**2*x**2 - log(sqrt(c**2*x**2 + 1) + c*x + 1)*a)/(sqr 
t(d)*d*(c**2*x**2 + 1))