\(\int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx\) [484]

Optimal result

Integrand size = 15, antiderivative size = 203 \[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=\frac {1}{8} \text {arcsinh}(2 x)^2-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1-\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1+\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1-\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1+\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right ) \] Output:

1/8*arcsinh(2*x)^2-1/8*arcsinh(2*x)*ln(1-(1-2^(1/2))*(2*x+(4*x^2+1)^(1/2)) 
)-1/8*arcsinh(2*x)*ln(1+(1-2^(1/2))*(2*x+(4*x^2+1)^(1/2)))-1/8*arcsinh(2*x 
)*ln(1-(1+2^(1/2))*(2*x+(4*x^2+1)^(1/2)))-1/8*arcsinh(2*x)*ln(1+(1+2^(1/2) 
)*(2*x+(4*x^2+1)^(1/2)))-1/8*polylog(2,-(1-2^(1/2))*(2*x+(4*x^2+1)^(1/2))) 
-1/8*polylog(2,(1-2^(1/2))*(2*x+(4*x^2+1)^(1/2)))-1/8*polylog(2,-(1+2^(1/2 
))*(2*x+(4*x^2+1)^(1/2)))-1/8*polylog(2,(1+2^(1/2))*(2*x+(4*x^2+1)^(1/2)))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.01 \[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=\frac {1}{8} \text {arcsinh}(2 x)^2-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1+\frac {e^{\text {arcsinh}(2 x)}}{1-\sqrt {2}}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1-\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1+\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1+\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right ) \] Input:

Integrate[(x*ArcSinh[2*x])/(1 - 4*x^2),x]
 

Output:

ArcSinh[2*x]^2/8 - (ArcSinh[2*x]*Log[1 + E^ArcSinh[2*x]/(1 - Sqrt[2])])/8 
- (ArcSinh[2*x]*Log[1 - (1 - Sqrt[2])*E^ArcSinh[2*x]])/8 - (ArcSinh[2*x]*L 
og[1 + (1 - Sqrt[2])*E^ArcSinh[2*x]])/8 - (ArcSinh[2*x]*Log[1 + (1 + Sqrt[ 
2])*E^ArcSinh[2*x]])/8 - PolyLog[2, -((1 - Sqrt[2])*E^ArcSinh[2*x])]/8 - P 
olyLog[2, (1 - Sqrt[2])*E^ArcSinh[2*x]]/8 - PolyLog[2, -((1 + Sqrt[2])*E^A 
rcSinh[2*x])]/8 - PolyLog[2, (1 + Sqrt[2])*E^ArcSinh[2*x]]/8
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6238, 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.

\(\displaystyle \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx\)

\(\Big \downarrow \) 6238

\(\displaystyle \int \left (-\frac {\text {arcsinh}(2 x)}{4 (2 x-1)}-\frac {\text {arcsinh}(2 x)}{4 (2 x+1)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )\right )-\frac {1}{8} \operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )+\frac {1}{8} \text {arcsinh}(2 x)^2-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1-\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (\left (1-\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}+1\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (1-\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}\right )-\frac {1}{8} \text {arcsinh}(2 x) \log \left (\left (1+\sqrt {2}\right ) e^{\text {arcsinh}(2 x)}+1\right )\)

Input:

Int[(x*ArcSinh[2*x])/(1 - 4*x^2),x]
 

Output:

ArcSinh[2*x]^2/8 - (ArcSinh[2*x]*Log[1 - (1 - Sqrt[2])*E^ArcSinh[2*x]])/8 
- (ArcSinh[2*x]*Log[1 + (1 - Sqrt[2])*E^ArcSinh[2*x]])/8 - (ArcSinh[2*x]*L 
og[1 - (1 + Sqrt[2])*E^ArcSinh[2*x]])/8 - (ArcSinh[2*x]*Log[1 + (1 + Sqrt[ 
2])*E^ArcSinh[2*x]])/8 - PolyLog[2, -((1 - Sqrt[2])*E^ArcSinh[2*x])]/8 - P 
olyLog[2, (1 - Sqrt[2])*E^ArcSinh[2*x]]/8 - PolyLog[2, -((1 + Sqrt[2])*E^A 
rcSinh[2*x])]/8 - PolyLog[2, (1 + Sqrt[2])*E^ArcSinh[2*x]]/8
 

Defintions of rubi rules used

rule 2009

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

rule 6238

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^ 
2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {\operatorname {arcsinh}\left (2 x \right )^{2}}{8}-\frac {\operatorname {arcsinh}\left (2 x \right ) \ln \left (\frac {3+2 \sqrt {2}-\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{3+2 \sqrt {2}}\right )}{8}-\frac {\operatorname {arcsinh}\left (2 x \right ) \ln \left (\frac {-3+2 \sqrt {2}+\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{-3+2 \sqrt {2}}\right )}{8}-\frac {\operatorname {dilog}\left (\frac {3+2 \sqrt {2}-\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{3+2 \sqrt {2}}\right )}{16}-\frac {\operatorname {dilog}\left (\frac {-3+2 \sqrt {2}+\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{-3+2 \sqrt {2}}\right )}{16}\)	\(162\)
default	\(\frac {\operatorname {arcsinh}\left (2 x \right )^{2}}{8}-\frac {\operatorname {arcsinh}\left (2 x \right ) \ln \left (\frac {3+2 \sqrt {2}-\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{3+2 \sqrt {2}}\right )}{8}-\frac {\operatorname {arcsinh}\left (2 x \right ) \ln \left (\frac {-3+2 \sqrt {2}+\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{-3+2 \sqrt {2}}\right )}{8}-\frac {\operatorname {dilog}\left (\frac {3+2 \sqrt {2}-\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{3+2 \sqrt {2}}\right )}{16}-\frac {\operatorname {dilog}\left (\frac {-3+2 \sqrt {2}+\left (2 x +\sqrt {4 x^{2}+1}\right )^{2}}{-3+2 \sqrt {2}}\right )}{16}\)	\(162\)

Input:

int(x*arcsinh(2*x)/(-4*x^2+1),x,method=_RETURNVERBOSE)
 

Output:

1/8*arcsinh(2*x)^2-1/8*arcsinh(2*x)*ln((3+2*2^(1/2)-(2*x+(4*x^2+1)^(1/2))^ 
2)/(3+2*2^(1/2)))-1/8*arcsinh(2*x)*ln((-3+2*2^(1/2)+(2*x+(4*x^2+1)^(1/2))^ 
2)/(-3+2*2^(1/2)))-1/16*dilog((3+2*2^(1/2)-(2*x+(4*x^2+1)^(1/2))^2)/(3+2*2 
^(1/2)))-1/16*dilog((-3+2*2^(1/2)+(2*x+(4*x^2+1)^(1/2))^2)/(-3+2*2^(1/2)))
 

Fricas [F]

\[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=\int { -\frac {x \operatorname {arsinh}\left (2 \, x\right )}{4 \, x^{2} - 1} \,d x } \] Input:

integrate(x*arcsinh(2*x)/(-4*x^2+1),x, algorithm="fricas")
 

Output:

integral(-x*arcsinh(2*x)/(4*x^2 - 1), x)
 

Sympy [F]

\[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=- \int \frac {x \operatorname {asinh}{\left (2 x \right )}}{4 x^{2} - 1}\, dx \] Input:

integrate(x*asinh(2*x)/(-4*x**2+1),x)
 

Output:

-Integral(x*asinh(2*x)/(4*x**2 - 1), x)
 

Maxima [F]

\[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=\int { -\frac {x \operatorname {arsinh}\left (2 \, x\right )}{4 \, x^{2} - 1} \,d x } \] Input:

integrate(x*arcsinh(2*x)/(-4*x^2+1),x, algorithm="maxima")
 

Output:

-integrate(x*arcsinh(2*x)/(4*x^2 - 1), x)
 

Giac [F]

\[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=\int { -\frac {x \operatorname {arsinh}\left (2 \, x\right )}{4 \, x^{2} - 1} \,d x } \] Input:

integrate(x*arcsinh(2*x)/(-4*x^2+1),x, algorithm="giac")
 

Output:

integrate(-x*arcsinh(2*x)/(4*x^2 - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=-\int \frac {x\,\mathrm {asinh}\left (2\,x\right )}{4\,x^2-1} \,d x \] Input:

int(-(x*asinh(2*x))/(4*x^2 - 1),x)
 

Output:

-int((x*asinh(2*x))/(4*x^2 - 1), x)
 

Reduce [F]

\[ \int \frac {x \text {arcsinh}(2 x)}{1-4 x^2} \, dx=-\left (\int \frac {\mathit {asinh} \left (2 x \right ) x}{4 x^{2}-1}d x \right ) \] Input:

int(x*asinh(2*x)/(-4*x^2+1),x)
 

Output:

 - int((asinh(2*x)*x)/(4*x**2 - 1),x)