\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx\) [134]

Optimal result

Integrand size = 26, antiderivative size = 105 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b \sqrt {1+c^2 x^2}}+\frac {b c \sqrt {d+c^2 d x^2} \log (x)}{\sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x^2} \, dx=-\frac {a \sqrt {d \left (1+c^2 x^2\right )}}{x}+\frac {b c \sqrt {d \left (1+c^2 x^2\right )} \left (-\frac {2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}+\text {arcsinh}(c x)^2+2 \log (c x)\right )}{2 \sqrt {1+c^2 x^2}}+a c \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6220, 14, 6198}

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.

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 6198

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 6198

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(93)=186\).

Time = 0.90 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.39

method	result	size
default	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a \,c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a \,c^{2} d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\)	\(251\)
parts	\(-\frac {a \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{d x}+a \,c^{2} x \sqrt {c^{2} d \,x^{2}+d}+\frac {a \,c^{2} d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} c}{2 \sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) c}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{x \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}}\right )\)	\(251\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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