3.5.20 \(\int \frac {x^2 (1+c^2 x^2)^{3/2}}{(a+b \sinh ^{-1}(c x))^2} \, dx\) [420]

Optimal. Leaf size=219 \[ -\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3} \]

[Out]

-x^2*(c^2*x^2+1)^2/b/c/(a+b*arcsinh(c*x))-1/16*cosh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c^3+1/4*cosh(4*a/b)
*Shi(4*(a+b*arcsinh(c*x))/b)/b^2/c^3+3/16*cosh(6*a/b)*Shi(6*(a+b*arcsinh(c*x))/b)/b^2/c^3+1/16*Chi(2*(a+b*arcs
inh(c*x))/b)*sinh(2*a/b)/b^2/c^3-1/4*Chi(4*(a+b*arcsinh(c*x))/b)*sinh(4*a/b)/b^2/c^3-3/16*Chi(6*(a+b*arcsinh(c
*x))/b)*sinh(6*a/b)/b^2/c^3

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Rubi [A]
time = 0.42, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5814, 5819, 5556, 3384, 3379, 3382} \begin {gather*} \frac {\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {3 \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (c^2 x^2+1\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-((x^2*(1 + c^2*x^2)^2)/(b*c*(a + b*ArcSinh[c*x]))) + (CoshIntegral[(2*(a + b*ArcSinh[c*x]))/b]*Sinh[(2*a)/b])
/(16*b^2*c^3) - (CoshIntegral[(4*(a + b*ArcSinh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c^3) - (3*CoshIntegral[(6*(a +
b*ArcSinh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3) - (Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(1
6*b^2*c^3) + (Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcSinh[c*x]))/b])/(4*b^2*c^3) + (3*Cosh[(6*a)/b]*SinhInteg
ral[(6*(a + b*ArcSinh[c*x]))/b])/(16*b^2*c^3)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5814

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

Rule 5819

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

Rubi steps

\begin {align*} \int \frac {x^2 \left (1+c^2 x^2\right )^{3/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \int \frac {x \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac {(6 c) \int \frac {x^3 \left (1+c^2 x^2\right )}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {6 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh ^3(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {2 \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {6 \text {Subst}\left (\int \left (-\frac {3 \sinh (2 x)}{32 (a+b x)}+\frac {\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac {9 \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac {\left (9 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {\left (3 \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {\left (9 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (3 \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac {x^2 \left (1+c^2 x^2\right )^2}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c^3}-\frac {3 \text {Chi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \sinh ^{-1}(c x)\right )}{16 b^2 c^3}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 306, normalized size = 1.40 \begin {gather*} -\frac {16 b c^2 x^2+32 b c^4 x^4+16 b c^6 x^6-\left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+4 \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \sinh ^{-1}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+b \sinh ^{-1}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 b \sinh ^{-1}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-3 b \sinh ^{-1}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{16 b^2 c^3 \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-1/16*(16*b*c^2*x^2 + 32*b*c^4*x^4 + 16*b*c^6*x^6 - (a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSinh[c*x])]*
Sinh[(2*a)/b] + 4*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6
*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + a
*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSi
nh[c*x])] - 4*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 4*b*ArcSinh[c*x]*Cosh[(4*a)/b]*SinhIntegr
al[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - 3*b*ArcSinh[c*x]*Cosh[(6
*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(b^2*c^3*(a + b*ArcSinh[c*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(703\) vs. \(2(207)=414\).
time = 9.24, size = 704, normalized size = 3.21

method	result	size
default	\(\frac {1}{16 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}-\frac {32 x^{6} c^{6}-32 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+48 c^{4} x^{4}-32 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+18 c^{2} x^{2}-6 \sqrt {c^{2} x^{2}+1}\, c x +1}{64 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}+\frac {3 \,{\mathrm e}^{\frac {6 a}{b}} \expIntegral \left (1, 6 \arcsinh \left (c x \right )+\frac {6 a}{b}\right )}{32 c^{3} b^{2}}-\frac {8 c^{4} x^{4}-8 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 c^{2} x^{2}-4 \sqrt {c^{2} x^{2}+1}\, c x +1}{32 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}+\frac {{\mathrm e}^{\frac {4 a}{b}} \expIntegral \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{8 c^{3} b^{2}}+\frac {2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, c x +1}{64 c^{3} \left (a +b \arcsinh \left (c x \right )\right ) b}-\frac {{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{32 c^{3} b^{2}}+\frac {2 b \,c^{2} x^{2}+2 x b c \sqrt {c^{2} x^{2}+1}+2 \,{\mathrm e}^{-\frac {2 a}{b}} \arcsinh \left (c x \right ) \expIntegral \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right ) b +2 \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right ) a +b}{64 c^{3} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {8 b \,c^{4} x^{4}+8 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}+8 b \,c^{2} x^{2}+4 x b c \sqrt {c^{2} x^{2}+1}+4 \,{\mathrm e}^{-\frac {4 a}{b}} \arcsinh \left (c x \right ) \expIntegral \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right ) b +4 \,{\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right ) a +b}{32 c^{3} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {32 b \,c^{6} x^{6}+32 \sqrt {c^{2} x^{2}+1}\, b \,c^{5} x^{5}+48 b \,c^{4} x^{4}+32 \sqrt {c^{2} x^{2}+1}\, b \,c^{3} x^{3}+18 b \,c^{2} x^{2}+6 x b c \sqrt {c^{2} x^{2}+1}+6 \,{\mathrm e}^{-\frac {6 a}{b}} \arcsinh \left (c x \right ) \expIntegral \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right ) b +6 \,{\mathrm e}^{-\frac {6 a}{b}} \expIntegral \left (1, -6 \arcsinh \left (c x \right )-\frac {6 a}{b}\right ) a +b}{64 c^{3} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}\)	\(704\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/c^3/(a+b*arcsinh(c*x))/b-1/64*(32*x^6*c^6-32*(c^2*x^2+1)^(1/2)*x^5*c^5+48*c^4*x^4-32*(c^2*x^2+1)^(1/2)*x^
3*c^3+18*c^2*x^2-6*(c^2*x^2+1)^(1/2)*c*x+1)/c^3/(a+b*arcsinh(c*x))/b+3/32/c^3/b^2*exp(6*a/b)*Ei(1,6*arcsinh(c*
x)+6*a/b)-1/32*(8*c^4*x^4-8*(c^2*x^2+1)^(1/2)*x^3*c^3+8*c^2*x^2-4*(c^2*x^2+1)^(1/2)*c*x+1)/c^3/(a+b*arcsinh(c*
x))/b+1/8/c^3/b^2*exp(4*a/b)*Ei(1,4*arcsinh(c*x)+4*a/b)+1/64*(2*c^2*x^2-2*(c^2*x^2+1)^(1/2)*c*x+1)/c^3/(a+b*ar
csinh(c*x))/b-1/32/c^3/b^2*exp(2*a/b)*Ei(1,2*arcsinh(c*x)+2*a/b)+1/64/c^3/b^2*(2*b*c^2*x^2+2*x*b*c*(c^2*x^2+1)
^(1/2)+2*exp(-2*a/b)*arcsinh(c*x)*Ei(1,-2*arcsinh(c*x)-2*a/b)*b+2*exp(-2*a/b)*Ei(1,-2*arcsinh(c*x)-2*a/b)*a+b)
/(a+b*arcsinh(c*x))-1/32/c^3/b^2*(8*b*c^4*x^4+8*(c^2*x^2+1)^(1/2)*b*c^3*x^3+8*b*c^2*x^2+4*x*b*c*(c^2*x^2+1)^(1
/2)+4*exp(-4*a/b)*arcsinh(c*x)*Ei(1,-4*arcsinh(c*x)-4*a/b)*b+4*exp(-4*a/b)*Ei(1,-4*arcsinh(c*x)-4*a/b)*a+b)/(a
+b*arcsinh(c*x))-1/64/c^3/b^2*(32*b*c^6*x^6+32*(c^2*x^2+1)^(1/2)*b*c^5*x^5+48*b*c^4*x^4+32*(c^2*x^2+1)^(1/2)*b
*c^3*x^3+18*b*c^2*x^2+6*x*b*c*(c^2*x^2+1)^(1/2)+6*exp(-6*a/b)*arcsinh(c*x)*Ei(1,-6*arcsinh(c*x)-6*a/b)*b+6*exp
(-6*a/b)*Ei(1,-6*arcsinh(c*x)-6*a/b)*a+b)/(a+b*arcsinh(c*x))

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

[Out]

-((c^4*x^6 + 2*c^2*x^4 + x^2)*(c^2*x^2 + 1) + (c^5*x^7 + 2*c^3*x^5 + c*x^3)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 +
sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x
^2 + 1))) + integrate(((6*c^5*x^6 + 7*c^3*x^4 + c*x^2)*(c^2*x^2 + 1)^(3/2) + 2*(6*c^6*x^7 + 11*c^4*x^5 + 6*c^2
*x^3 + x)*(c^2*x^2 + 1) + 3*(2*c^7*x^8 + 5*c^5*x^6 + 4*c^3*x^4 + c*x^2)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^4 + (c^2
*x^2 + 1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^2 + 2*b^2*c^3*x^2 + b^2
*c + 2*(b^2*c^4*x^3 + b^2*c^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)
*sqrt(c^2*x^2 + 1)), 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(x^2*(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^2*x^4 + x^2)*sqrt(c^2*x^2 + 1)/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c^2*x^2 + 1)^(3/2)*x^2/(b*arcsinh(c*x) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (c^2\,x^2+1\right )}^{3/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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