\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [7]

Optimal result

Integrand size = 22, antiderivative size = 66 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^2} \, dx=-b c d \sqrt {1+c^2 x^2}-\frac {d (a+b \text {arcsinh}(c x))}{x}+c^2 d x (a+b \text {arcsinh}(c x))-b c d \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {a d}{x}+a c^2 d x-b c d \sqrt {1+c^2 x^2}-\frac {b d \text {arcsinh}(c x)}{x}+b c^2 d x \text {arcsinh}(c x)-b c d \text {arctanh}\left (\sqrt {1+c^2 x^2}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6218, 25, 27, 354, 90, 73, 221}

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.

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

\(\Big \downarrow \) 6218

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((d*(a + b*ArcSinh[c*x]))/x) + c^2*d*x*(a + b*ArcSinh[c*x]) + (b*c*d*(-2* 
Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/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 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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

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 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 6218

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

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02

method	result	size
parts	\(a d \left (c^{2} x -\frac {1}{x}\right )+b d c \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\)	\(67\)
derivativedivides	\(c \left (a d \left (x c -\frac {1}{x c}\right )+b d \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(69\)
default	\(c \left (a d \left (x c -\frac {1}{x c}\right )+b d \left (x c \,\operatorname {arcsinh}\left (x c \right )-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\sqrt {c^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(69\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (62) = 124\).

Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.36 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {a c^{2} d x^{2} - b c d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} + 1\right ) + b c d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1} - 1\right ) - \sqrt {c^{2} x^{2} + 1} b c d x - {\left (b c^{2} - b\right )} d x \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) - a d + {\left (b c^{2} d x^{2} - {\left (b c^{2} - b\right )} d x - b d\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x} \] Input:

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

Output:

(a*c^2*d*x^2 - b*c*d*x*log(-c*x + sqrt(c^2*x^2 + 1) + 1) + b*c*d*x*log(-c* 
x + sqrt(c^2*x^2 + 1) - 1) - sqrt(c^2*x^2 + 1)*b*c*d*x - (b*c^2 - b)*d*x*l 
og(-c*x + sqrt(c^2*x^2 + 1)) - a*d + (b*c^2*d*x^2 - (b*c^2 - b)*d*x - b*d) 
*log(c*x + sqrt(c^2*x^2 + 1)))/x
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^2} \, dx=a c^{2} d x + {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b c d - {\left (c \operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\operatorname {arsinh}\left (c x\right )}{x}\right )} b d - \frac {a d}{x} \] Input:

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

Output:

a*c^2*d*x + (c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*b*c*d - (c*arcsinh(1/(c 
*abs(x))) + arcsinh(c*x)/x)*b*d - a*d/x
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.39 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))}{x^2} \, dx=\frac {d \left (\mathit {asinh} \left (c x \right ) b \,c^{2} x^{2}-\mathit {asinh} \left (c x \right ) b -\sqrt {c^{2} x^{2}+1}\, b c x +\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) b c x -\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) b c x +a \,c^{2} x^{2}-a \right )}{x} \] Input:

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

Output:

(d*(asinh(c*x)*b*c**2*x**2 - asinh(c*x)*b - sqrt(c**2*x**2 + 1)*b*c*x + lo 
g(sqrt(c**2*x**2 + 1) + c*x - 1)*b*c*x - log(sqrt(c**2*x**2 + 1) + c*x + 1 
)*b*c*x + a*c**2*x**2 - a))/x