3.4.17 \(\int \frac {(a+b \sinh ^{-1}(c x))^2}{x^2 (d+c^2 d x^2)^{5/2}} \, dx\) [317]

Optimal. Leaf size=421 \[ \frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {16 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}} \]

[Out]

-(a+b*arcsinh(c*x))^2/d/x/(c^2*d*x^2+d)^(3/2)-4/3*c^2*x*(a+b*arcsinh(c*x))^2/d/(c^2*d*x^2+d)^(3/2)+1/3*b^2*c^2
*x/d^2/(c^2*d*x^2+d)^(1/2)-8/3*c^2*x*(a+b*arcsinh(c*x))^2/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*arcsinh(c*x))/d
^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-8/3*c*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)-
4*b*c*(a+b*arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+16/3*b*c
*(a+b*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+5/3*b^2*c*polylo
g(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)+b^2*c*polylog(2,(c*x+(c^2*x^2+1)^(1/
2))^2)*(c^2*x^2+1)^(1/2)/d^2/(c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5809, 5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 5811, 5799, 5569, 4267} \begin {gather*} -\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {8 c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {16 b c \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 b c \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {c^2 d x^2+d}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (c^2 d x^2+d\right )^{3/2}}+\frac {5 b^2 c \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 c \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2 c^2 x}{3 d^2 \sqrt {c^2 d x^2+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)),x]

[Out]

(b^2*c^2*x)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (b*c*(a + b*ArcSinh[c*x]))/(3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x
^2]) - (a + b*ArcSinh[c*x])^2/(d*x*(d + c^2*d*x^2)^(3/2)) - (4*c^2*x*(a + b*ArcSinh[c*x])^2)/(3*d*(d + c^2*d*x
^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (8*c*Sqrt[1 + c^2*x^2]*(a + b*ArcS
inh[c*x])^2)/(3*d^2*Sqrt[d + c^2*d*x^2]) - (4*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[
c*x])])/(d^2*Sqrt[d + c^2*d*x^2]) + (16*b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])]
)/(3*d^2*Sqrt[d + c^2*d*x^2]) + (5*b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(2*ArcSinh[c*x])])/(3*d^2*Sqrt[d + c^
2*d*x^2]) + (b^2*c*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(2*ArcSinh[c*x])])/(d^2*Sqrt[d + c^2*d*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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 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 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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5788

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2
)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

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 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]

Rule 5799

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

Rule 5809

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] + (-Dist[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] - Dist[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 5811

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/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(p + 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, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\left (4 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {\left (8 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 x}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (32 b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {16 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {16 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b^2 c \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2 c^2 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {8 c^2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {16 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 c \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.24, size = 408, normalized size = 0.97 \begin {gather*} -\frac {3 a^2+12 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4-b^2 c^4 x^4+a b c x \sqrt {1+c^2 x^2}+6 a b \sinh ^{-1}(c x)+24 a b c^2 x^2 \sinh ^{-1}(c x)+16 a b c^4 x^4 \sinh ^{-1}(c x)+b^2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+3 b^2 \sinh ^{-1}(c x)^2+12 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-8 b^2 c x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2-6 b^2 c x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-10 b^2 c x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )-6 a b c x \left (1+c^2 x^2\right )^{3/2} \log (c x)-5 a b c x \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )+5 b^2 c x \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+3 b^2 c x \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 d x \left (d+c^2 d x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)),x]

[Out]

-1/3*(3*a^2 + 12*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 + a*b*c*x*Sqrt[1 + c^2*x^2] + 6*a*b*A
rcSinh[c*x] + 24*a*b*c^2*x^2*ArcSinh[c*x] + 16*a*b*c^4*x^4*ArcSinh[c*x] + b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*
x] + 3*b^2*ArcSinh[c*x]^2 + 12*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*ArcSinh[c*x]^2 - 8*b^2*c*x*(1 + c^2*
x^2)^(3/2)*ArcSinh[c*x]^2 - 6*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 10*b^2*c
*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 6*a*b*c*x*(1 + c^2*x^2)^(3/2)*Log[c*x] - 5*
a*b*c*x*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2] + 5*b^2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])]
+ 3*b^2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, E^(-2*ArcSinh[c*x])])/(d*x*(d + c^2*d*x^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3512\) vs. \(2(411)=822\).
time = 2.26, size = 3513, normalized size = 8.34

method	result	size
default	\(\text {Expression too large to display}\)	\(3513\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

136/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^2*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*c
^3-8/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^
3+8*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*arcsinh(c*x)*(c^2*x^2+1)*c^2+64/3*b^2*
(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^7*arcsinh(c*x)*(c^2*x^2+1)*c^8+160/3*b^2*(d*(c
^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*arcsinh(c*x)*(c^2*x^2+1)*c^6+64/3*b^2*(d*(c^2*x^2
+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^4*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*c^5+40*b^2*(d*(c^2*x^2
+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*arcsinh(c*x)*(c^2*x^2+1)*c^4-88*a*b*(d*(c^2*x^2+1))^(1/
2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*arcsinh(c*x)*c^2+48*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x
^4+26*c^2*x^2+9)/d^3*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c-32/3*a*b/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3*arc
sinh(c*x)*c-64/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^9*c^10-224/3*a*b*(d*(c^2*
x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^7*c^8-280/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^
4*x^4+26*c^2*x^2+9)/d^3*x^5*c^6-48*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*c^4-8
*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*c^2-5*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^
6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*c^2-9*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3/x*arc
sinh(c*x)^2+2*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c-3*b^2*(d*(c^2
*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*c*(c^2*x^2+1)^(1/2)-16/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*
x^2+1))^(1/2)/d^3*arcsinh(c*x)^2*c-32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^9*
c^10-40*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^7*c^8-160/3*b^2*(d*(c^2*x^2+1))^(1
/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*c^6-29*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*
x^2+9)/d^3*x^3*c^4+2*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c+5/3*b
^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)*c+64/3*a*b*(d*(c^2*x^2+1)
)^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^7*(c^2*x^2+1)*c^8+160/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6
+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*(c^2*x^2+1)*c^6-128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2
*x^2+9)/d^3*x^5*arcsinh(c*x)*c^6+40*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*(c^2
*x^2+1)*c^4-3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*c*(c^2*x^2+1)^(1/2)-18*a*b*(d*
(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3/x*arcsinh(c*x)+10/3*a*b/(c^2*x^2+1)^(1/2)*(d*(c^2*x
^2+1))^(1/2)/d^3*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*c+2*a*b/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3*ln((c*x+(
c^2*x^2+1)^(1/2))^2-1)*c+128/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^4*arcsinh(c
*x)*(c^2*x^2+1)^(1/2)*c^5+24*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*arcsinh(c*x)^2*
(c^2*x^2+1)^(1/2)*c-3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*arcsinh(c*x)*(c^2*x^2+
1)^(1/2)*c+88/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*(c^2*x^2+1)*c^6-280/3*b^
2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^5*arcsinh(c*x)*c^6-8/3*b^2*(d*(c^2*x^2+1))^(
1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^4*(c^2*x^2+1)^(1/2)*c^5+10/3*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+
1))^(1/2)/d^3*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)*c+2*b^2/(c^2*x^2+1)^(1/2)*(d*(c^2*x^2+1))^(1/2)/d^3
*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c-8*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^
3*x*arcsinh(c*x)*c^2+8*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*(c^2*x^2+1)*c^2+272
/3*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*c^3-11
2*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*arcsinh(c*x)*c^4-8/3*a*b*(d*(c^2*x^2+1
))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^2*(c^2*x^2+1)^(1/2)*c^3+8*a*b*(d*(c^2*x^2+1))^(1/2)/(8*c^6*
x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*(c^2*x^2+1)*c^2-64/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2
*x^2+9)/d^3*x^9*arcsinh(c*x)*c^10-56*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*arc
sinh(c*x)^2*c^4+80/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*(c^2*x^2+1)*c^4-48*
b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^3*arcsinh(c*x)*c^4-17/3*b^2*(d*(c^2*x^2+1)
)^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x^2*(c^2*x^2+1)^(1/2)*c^3-44*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*
x^6+25*c^4*x^4+26*c^2*x^2+9)/d^3*x*arcsinh(c*x)^2*c^2+32/3*b^2*(d*(c^2*x^2+1))^(1/2)/(8*c^6*x^6+25*c^4*x^4+26*
c^2*x^2+9)/d^3*x^7*(c^2*x^2+1)*c^8-224/3*b^2*(d...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*a^2*(8*c^2*x/(sqrt(c^2*d*x^2 + d)*d^2) + 4*c^2*x/((c^2*d*x^2 + d)^(3/2)*d) + 3/((c^2*d*x^2 + d)^(3/2)*d*x
)) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/((c^2*d*x^2 + d)^(5/2)*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 +
1))/((c^2*d*x^2 + d)^(5/2)*x^2), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^6*d^3*x^8 + 3*c^4*d^3*x^6 + 3*
c^2*d^3*x^4 + d^3*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2/x**2/(c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*asinh(c*x))**2/(x**2*(d*(c**2*x**2 + 1))**(5/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^(5/2)*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^(5/2)), x)

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