\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx\) [228]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5797, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^2 d}-\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {\log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \]

[In]

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

[Out]

-1/3*(a + b*ArcSinh[c*x])^3/(b*c^2*d) + ((a + b*ArcSinh[c*x])^2*Log[1 + E^(2*ArcSinh[c*x])])/(c^2*d) + (b*(a +
b*ArcSinh[c*x])*PolyLog[2, -E^(2*ArcSinh[c*x])])/(c^2*d) - (b^2*PolyLog[3, -E^(2*ArcSinh[c*x])])/(2*c^2*d)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3799

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

Rule 5797

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

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {b^2 \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^2 d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {b^2 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \\ & = -\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(281\) vs. \(2(105)=210\).

Time = 0.21 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.68 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {-6 a b \text {arcsinh}(c x)^2-2 b^2 \text {arcsinh}(c x)^3+12 a b \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+6 b^2 \text {arcsinh}(c x)^2 \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 a b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+6 b^2 \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+3 a^2 \log \left (1+c^2 x^2\right )+12 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-12 b^2 \operatorname {PolyLog}\left (3,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-12 b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{6 c^2 d} \]

[In]

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

[Out]

(-6*a*b*ArcSinh[c*x]^2 - 2*b^2*ArcSinh[c*x]^3 + 12*a*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6
*b^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*a*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh
[c*x])/c] + 6*b^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 3*a^2*Log[1 + c^2*x^2] + 12*b*(a + b
*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*b*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^
ArcSinh[c*x])/c] - 12*b^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 12*b^2*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[
c*x])/c])/(6*c^2*d)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.71

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(180\)
default	\(\frac {\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(180\)
parts	\(\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{3}}{3}+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}\)	\(185\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*b^2*log(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d) + 1/2*a^2*log(c^2*d*x^2 + d)/(c^2*d) - integra
te(-(2*a*b*c^2*x^2 - (b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) - (b^2*c*x*log(c^2*x^2 + 1) - 2*a*b*c*x)*sqrt(c^2*x^
2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c^4*d*x^3 + c^2*d*x + (c^3*d*x^2 + c*d)*sqrt(c^2*x^2 + 1)), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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