\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [304]

Optimal result

Integrand size = 26, antiderivative size = 188 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \]

[Out]

-(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(1/2)+4*b*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+
1)^(1/2)/c^2/d/(c^2*d*x^2+d)^(1/2)-2*I*b^2*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/c^2/d/(c^2*
d*x^2+d)^(1/2)+2*I*b^2*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/c^2/d/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5798, 5789, 4265, 2317, 2438} \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 b \sqrt {c^2 x^2+1} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {2 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

-((a + b*ArcSinh[c*x])^2/(c^2*d*Sqrt[d + c^2*d*x^2])) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^A
rcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^2
*d*Sqrt[d + c^2*d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2
])

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5789

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]

Rule 5798

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a^2+2 a b \text {arcsinh}(c x)+b^2 \text {arcsinh}(c x)^2-4 a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

-((a^2 + 2*a*b*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 - 4*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I
)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[
1 + I/E^ArcSinh[c*x]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x
^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.69

method	result	size
default	\(-\frac {a^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )+\operatorname {arcsinh}\left (c x \right )^{2}\right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-i \sqrt {c^{2} x^{2}+1}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+i \sqrt {c^{2} x^{2}+1}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )+\operatorname {arcsinh}\left (c x \right )\right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}\)	\(318\)
parts	\(-\frac {a^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )+\operatorname {arcsinh}\left (c x \right )^{2}\right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-i \sqrt {c^{2} x^{2}+1}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+i \sqrt {c^{2} x^{2}+1}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )+\operatorname {arcsinh}\left (c x \right )\right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}\)	\(318\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*x*arcsinh(c*x)^2 + 2*a*b*x*arcsinh(c*x) + a^2*x)/(c^4*d^2*x^4 + 2*c^2*d^2*x^
2 + d^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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