3.1.88 \(\int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {\pi +c^2 \pi x^2}} \, dx\) [88]

3.1.88.1 Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {b c}{2 \sqrt {\pi } x}-\frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 \pi x^2}+\frac {c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {\pi }}+\frac {b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {\pi }}-\frac {b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {\pi }} \]

output

-1/2*b*c/x/Pi^(1/2)+c^2*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/ 
Pi^(1/2)+1/2*b*c^2*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/Pi^(1/2)-1/2*b*c^2*po 
lylog(2,c*x+(c^2*x^2+1)^(1/2))/Pi^(1/2)-1/2*(a+b*arcsinh(c*x))*(Pi*c^2*x^2 
+Pi)^(1/2)/Pi/x^2
 

3.1.88.2 Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.61 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {-\frac {4 a \sqrt {1+c^2 x^2}}{x^2}-4 a c^2 \log (x)+4 a c^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+b c^2 \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{8 \sqrt {\pi }} \]

input

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

output

((-4*a*Sqrt[1 + c^2*x^2])/x^2 - 4*a*c^2*Log[x] + 4*a*c^2*Log[Pi*(1 + Sqrt[ 
1 + c^2*x^2])] + b*c^2*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSin 
h[c*x]/2]^2 - 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*L 
og[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcSinh[c*x])] + 4*PolyLog[2 
, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSin 
h[c*x]/2]))/(8*Sqrt[Pi])
 

3.1.88.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6224, 15, 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.

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^3 \sqrt {\pi c^2 x^2+\pi }} \, dx\)

\(\Big \downarrow \) 6224

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

-1/2*(b*c)/(Sqrt[Pi]*x) - (Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(2* 
Pi*x^2) - ((I/2)*c^2*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + 
 I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]))/Sqrt[P 
i]
 

3.1.88.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 6224

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/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 
1)))   Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim 
p[b*c*(n/(f*(m + 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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -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]
 

3.1.88.4 Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.88

method	result	size
default	\(a \left (-\frac {\sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,x^{2}}+\frac {c^{2} \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi }}\right )+b \left (-\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}+\frac {c^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}\right )\)	\(216\)
parts	\(a \left (-\frac {\sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,x^{2}}+\frac {c^{2} \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{2 \sqrt {\pi }}\right )+b \left (-\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )}{2 \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}+\frac {c^{2} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}-\frac {c^{2} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{2 \sqrt {\pi }}\right )\)	\(216\)

input

int((a+b*arcsinh(c*x))/x^3/(Pi*c^2*x^2+Pi)^(1/2),x,method=_RETURNVERBOSE)
 

output

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

3.1.88.5 Fricas [F]

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

input

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(1/2),x, algorithm="frica 
s")
 

output

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

3.1.88.6 Sympy [F]

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

input

integrate((a+b*asinh(c*x))/x**3/(pi*c**2*x**2+pi)**(1/2),x)
 

output

(Integral(a/(x**3*sqrt(c**2*x**2 + 1)), x) + Integral(b*asinh(c*x)/(x**3*s 
qrt(c**2*x**2 + 1)), x))/sqrt(pi)
 

3.1.88.7 Maxima [F]

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

input

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(1/2),x, algorithm="maxim 
a")
 

output

1/2*(c^2*arcsinh(1/(c*abs(x)))/sqrt(pi) - sqrt(pi + pi*c^2*x^2)/(pi*x^2))* 
a + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(sqrt(pi + pi*c^2*x^2)*x^3), 
x)
 

3.1.88.8 Giac [F]

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

input

integrate((a+b*arcsinh(c*x))/x^3/(pi*c^2*x^2+pi)^(1/2),x, algorithm="giac" 
)
 

output

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

3.1.88.9 Mupad [F(-1)]

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

input

int((a + b*asinh(c*x))/(x^3*(Pi + Pi*c^2*x^2)^(1/2)),x)
 

output

int((a + b*asinh(c*x))/(x^3*(Pi + Pi*c^2*x^2)^(1/2)), x)