\(\int \frac {a+b \text {arcsinh}(c x)}{x^2 (d+c^2 d x^2)^{3/2}} \, dx\) [170]

Optimal result

Integrand size = 26, antiderivative size = 141 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {a+b \text {arcsinh}(c x)}{d x \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{d^2 x}+\frac {b c \sqrt {1+c^2 x^2} \log (x)}{d \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 d \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+c^2 d x^2} \left (2 a \sqrt {1+c^2 x^2}+4 a c^2 x^2 \sqrt {1+c^2 x^2}+2 b \sqrt {1+c^2 x^2} \left (1+2 c^2 x^2\right ) \text {arcsinh}(c x)+b c x \left (1+c^2 x^2\right ) \log \left (1+\frac {1}{c^2 x^2}\right )-2 b c x \log \left (1+c^2 x^2\right )-2 b c^3 x^3 \log \left (1+c^2 x^2\right )\right )}{2 d^2 x \left (1+c^2 x^2\right )^{3/2}} \] Input:

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

Output:

-1/2*(Sqrt[d + c^2*d*x^2]*(2*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^ 
2*x^2] + 2*b*Sqrt[1 + c^2*x^2]*(1 + 2*c^2*x^2)*ArcSinh[c*x] + b*c*x*(1 + c 
^2*x^2)*Log[1 + 1/(c^2*x^2)] - 2*b*c*x*Log[1 + c^2*x^2] - 2*b*c^3*x^3*Log[ 
1 + c^2*x^2]))/(d^2*x*(1 + c^2*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6219, 25, 27, 354, 86, 2009}

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.

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

\(\Big \downarrow \) 6219

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((a + b*ArcSinh[c*x])/(d*x*Sqrt[d + c^2*d*x^2])) - (2*c^2*x*(a + b*ArcSin 
h[c*x]))/(d*Sqrt[d + c^2*d*x^2]) + (b*c*Sqrt[d + c^2*d*x^2]*(Log[x^2] + Lo 
g[1 + c^2*x^2]))/(2*d^2*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

rule 25

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

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.70

method	result	size
default	\(a \left (-\frac {1}{d x \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 c^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}\right )-\frac {b \left (2 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{4} c^{4}-2 \sqrt {c^{2} x^{2}+1}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x c +\operatorname {arcsinh}\left (x c \right )\right ) \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{x \,d^{2} \left (c^{2} x^{2}+1\right )}\)	\(239\)
parts	\(a \left (-\frac {1}{d x \sqrt {c^{2} d \,x^{2}+d}}-\frac {2 c^{2} x}{d \sqrt {c^{2} d \,x^{2}+d}}\right )-\frac {b \left (2 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{4} c^{4}-2 \sqrt {c^{2} x^{2}+1}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{3} c^{3}+2 \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right ) x c +\operatorname {arcsinh}\left (x c \right )\right ) \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{x \,d^{2} \left (c^{2} x^{2}+1\right )}\)	\(239\)

Input:

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

Output:

a*(-1/d/x/(c^2*d*x^2+d)^(1/2)-2*c^2/d*x/(c^2*d*x^2+d)^(1/2))-b*(2*ln((x*c+ 
(c^2*x^2+1)^(1/2))^4-1)*x^4*c^4-2*(c^2*x^2+1)^(1/2)*ln((x*c+(c^2*x^2+1)^(1 
/2))^4-1)*x^3*c^3+2*ln((x*c+(c^2*x^2+1)^(1/2))^4-1)*x^2*c^2-(c^2*x^2+1)^(1 
/2)*ln((x*c+(c^2*x^2+1)^(1/2))^4-1)*x*c+arcsinh(x*c))*(2*c^2*x^2+2*(c^2*x^ 
2+1)^(1/2)*x*c+1)*(d*(c^2*x^2+1))^(1/2)/x/d^2/(c^2*x^2+1)
 

Fricas [F]

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

integrate((a+b*arcsinh(c*x))/x^2/(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^6 + 2*c^2*d^2 
*x^4 + d^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {1}{2} \, b c {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{d^{\frac {3}{2}}} + \frac {2 \, \log \left (x\right )}{d^{\frac {3}{2}}}\right )} - {\left (\frac {2 \, c^{2} x}{\sqrt {c^{2} d x^{2} + d} d} + \frac {1}{\sqrt {c^{2} d x^{2} + d} d x}\right )} b \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, c^{2} x}{\sqrt {c^{2} d x^{2} + d} d} + \frac {1}{\sqrt {c^{2} d x^{2} + d} d x}\right )} a \] Input:

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

Output:

1/2*b*c*(log(c^2*x^2 + 1)/d^(3/2) + 2*log(x)/d^(3/2)) - (2*c^2*x/(sqrt(c^2 
*d*x^2 + d)*d) + 1/(sqrt(c^2*d*x^2 + d)*d*x))*b*arcsinh(c*x) - (2*c^2*x/(s 
qrt(c^2*d*x^2 + d)*d) + 1/(sqrt(c^2*d*x^2 + d)*d*x))*a
 

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) b \,c^{2} x^{3}+\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{4}+\sqrt {c^{2} x^{2}+1}\, x^{2}}d x \right ) b x -2 a \,c^{3} x^{3}-2 a c x}{\sqrt {d}\, d x \left (c^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 2*sqrt(c**2*x**2 + 1)*a*c**2*x**2 - sqrt(c**2*x**2 + 1)*a + int(asinh( 
c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1)*x**2),x)*b*c**2* 
x**3 + int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**4 + sqrt(c**2*x**2 + 1) 
*x**2),x)*b*x - 2*a*c**3*x**3 - 2*a*c*x)/(sqrt(d)*d*x*(c**2*x**2 + 1))