3.427 \(\int \frac{x^3 (1+c^2 x^2)^{5/2}}{(a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=277 \[ -\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 b^2 c^4}-\frac{3 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}+\frac{21 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}+\frac{9 \cosh \left (\frac{9 a}{b}\right ) \text{Chi}\left (\frac{9 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{128 b^2 c^4}+\frac{3 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}-\frac{21 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac{9 \sinh \left (\frac{9 a}{b}\right ) \text{Shi}\left (\frac{9 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac{x^3 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-((x^3*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c*x])/b])/(128*
b^2*c^4) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) + (21*Cosh[(7*a)/b]*CoshInt
egral[(7*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4) + (9*Cosh[(9*a)/b]*CoshIntegral[(9*(a + b*ArcSinh[c*x]))/b])/
(256*b^2*c^4) + (3*Sinh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(128*b^2*c^4) + (3*Sinh[(3*a)/b]*SinhIntegr
al[(3*(a + b*ArcSinh[c*x]))/b])/(32*b^2*c^4) - (21*Sinh[(7*a)/b]*SinhIntegral[(7*(a + b*ArcSinh[c*x]))/b])/(25
6*b^2*c^4) - (9*Sinh[(9*a)/b]*SinhIntegral[(9*(a + b*ArcSinh[c*x]))/b])/(256*b^2*c^4)

________________________________________________________________________________________

Rubi [A]  time = 1.21539, antiderivative size = 273, normalized size of antiderivative = 0.99, number of steps used = 34, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5777, 5779, 5448, 3303, 3298, 3301} \[ -\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}-\frac{3 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}+\frac{21 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{9 \cosh \left (\frac{9 a}{b}\right ) \text{Chi}\left (\frac{9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \sinh \left (\frac{9 a}{b}\right ) \text{Shi}\left (\frac{9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{x^3 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-((x^3*(1 + c^2*x^2)^3)/(b*c*(a + b*ArcSinh[c*x]))) - (3*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]])/(128*b^2*
c^4) - (3*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSinh[c*x]])/(32*b^2*c^4) + (21*Cosh[(7*a)/b]*CoshIntegral[
(7*a)/b + 7*ArcSinh[c*x]])/(256*b^2*c^4) + (9*Cosh[(9*a)/b]*CoshIntegral[(9*a)/b + 9*ArcSinh[c*x]])/(256*b^2*c
^4) + (3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(128*b^2*c^4) + (3*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3
*ArcSinh[c*x]])/(32*b^2*c^4) - (21*Sinh[(7*a)/b]*SinhIntegral[(7*a)/b + 7*ArcSinh[c*x]])/(256*b^2*c^4) - (9*Si
nh[(9*a)/b]*SinhIntegral[(9*a)/b + 9*ArcSinh[c*x]])/(256*b^2*c^4)

Rule 5777

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*d^IntP
art[p]*(d + e*x^2)^FracPart[p])/(b*c*(n + 1)*(1 + c^2*x^2)^FracPart[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)*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(b*f*(
n + 1)*(1 + c^2*x^2)^FracPart[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[m, -3] && IGtQ[2*p, 0]

Rule 5779

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

Rule 5448

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 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int \frac{x^3 \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \int \frac{x^2 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac{(9 c) \int \frac{x^4 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh ^5(x) \sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{5 \cosh (x)}{64 (a+b x)}+\frac{\cosh (3 x)}{64 (a+b x)}+\frac{3 \cosh (5 x)}{64 (a+b x)}+\frac{\cosh (7 x)}{64 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac{9 \operatorname{Subst}\left (\int \left (\frac{3 \cosh (x)}{128 (a+b x)}-\frac{\cosh (3 x)}{64 (a+b x)}-\frac{\cosh (5 x)}{64 (a+b x)}+\frac{\cosh (7 x)}{256 (a+b x)}+\frac{\cosh (9 x)}{256 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (9 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{27 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}-\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac{\left (27 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}-\frac{\left (15 \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (3 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (9 \cosh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac{\left (3 \cosh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (9 \cosh \left (\frac{9 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}-\frac{\left (27 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}+\frac{\left (15 \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (3 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac{\left (9 \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \sinh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}-\frac{\left (3 \sinh \left (\frac{7 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac{\left (9 \sinh \left (\frac{9 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}\\ &=-\frac{x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac{3 \cosh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}-\frac{3 \cosh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}+\frac{21 \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{9 \cosh \left (\frac{9 a}{b}\right ) \text{Chi}\left (\frac{9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac{3 \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}+\frac{3 \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}-\frac{21 \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (\frac{7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac{9 \sinh \left (\frac{9 a}{b}\right ) \text{Shi}\left (\frac{9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}\\ \end{align*}

Mathematica [A]  time = 1.33807, size = 408, normalized size = 1.47 \[ -\frac{6 \cosh \left (\frac{a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )+24 \cosh \left (\frac{3 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-21 a \cosh \left (\frac{7 a}{b}\right ) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-21 b \cosh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-9 a \cosh \left (\frac{9 a}{b}\right ) \text{Chi}\left (9 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-9 b \cosh \left (\frac{9 a}{b}\right ) \sinh ^{-1}(c x) \text{Chi}\left (9 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-6 a \sinh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-6 b \sinh \left (\frac{a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c x)\right )-24 a \sinh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )-24 b \sinh \left (\frac{3 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (3 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+21 a \sinh \left (\frac{7 a}{b}\right ) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+21 b \sinh \left (\frac{7 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (7 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+9 a \sinh \left (\frac{9 a}{b}\right ) \text{Shi}\left (9 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+9 b \sinh \left (\frac{9 a}{b}\right ) \sinh ^{-1}(c x) \text{Shi}\left (9 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )\right )+256 b c^9 x^9+768 b c^7 x^7+768 b c^5 x^5+256 b c^3 x^3}{256 b^2 c^4 \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]

[Out]

-(256*b*c^3*x^3 + 768*b*c^5*x^5 + 768*b*c^7*x^7 + 256*b*c^9*x^9 + 6*(a + b*ArcSinh[c*x])*Cosh[a/b]*CoshIntegra
l[a/b + ArcSinh[c*x]] + 24*(a + b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 21*a*Cosh
[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 21*b*ArcSinh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh
[c*x])] - 9*a*Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcSinh[c*x])] - 9*b*ArcSinh[c*x]*Cosh[(9*a)/b]*CoshIntegral
[9*(a/b + ArcSinh[c*x])] - 6*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 6*b*ArcSinh[c*x]*Sinh[a/b]*SinhInt
egral[a/b + ArcSinh[c*x]] - 24*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 24*b*ArcSinh[c*x]*Sinh[(
3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 21*a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 21*b*
ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 9*a*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcS
inh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])])/(256*b^2*c^4*(a + b*ArcSinh[
c*x]))

________________________________________________________________________________________

Maple [B]  time = 0.452, size = 1070, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

[Out]

-1/512*(256*c^9*x^9-256*c^8*x^8*(c^2*x^2+1)^(1/2)+576*c^7*x^7-448*c^6*x^6*(c^2*x^2+1)^(1/2)+432*c^5*x^5-240*c^
4*x^4*(c^2*x^2+1)^(1/2)+120*c^3*x^3-40*c^2*x^2*(c^2*x^2+1)^(1/2)+9*c*x-(c^2*x^2+1)^(1/2))/c^4/(a+b*arcsinh(c*x
))/b-9/512/c^4/b^2*exp(9*a/b)*Ei(1,9*arcsinh(c*x)+9*a/b)-3/512*(64*c^7*x^7-64*c^6*x^6*(c^2*x^2+1)^(1/2)+112*c^
5*x^5-80*c^4*x^4*(c^2*x^2+1)^(1/2)+56*c^3*x^3-24*c^2*x^2*(c^2*x^2+1)^(1/2)+7*c*x-(c^2*x^2+1)^(1/2))/c^4/(a+b*a
rcsinh(c*x))/b-21/512/c^4/b^2*exp(7*a/b)*Ei(1,7*arcsinh(c*x)+7*a/b)+1/64*(4*c^3*x^3-4*c^2*x^2*(c^2*x^2+1)^(1/2
)+3*c*x-(c^2*x^2+1)^(1/2))/c^4/b/(a+b*arcsinh(c*x))+3/64/c^4/b^2*exp(3*a/b)*Ei(1,3*arcsinh(c*x)+3*a/b)+3/256*(
c*x-(c^2*x^2+1)^(1/2))/c^4/b/(a+b*arcsinh(c*x))+3/256/c^4/b^2*exp(a/b)*Ei(1,arcsinh(c*x)+a/b)+3/256/c^4/b^2*(a
rcsinh(c*x)*Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*b+Ei(1,-arcsinh(c*x)-a/b)*exp(-a/b)*a+x*b*c+(c^2*x^2+1)^(1/2)*b)
/(a+b*arcsinh(c*x))+1/64/c^4/b^2*(4*x^3*b*c^3+4*(c^2*x^2+1)^(1/2)*x^2*b*c^2+3*arcsinh(c*x)*exp(-3*a/b)*Ei(1,-3
*arcsinh(c*x)-3*a/b)*b+3*exp(-3*a/b)*Ei(1,-3*arcsinh(c*x)-3*a/b)*a+3*x*b*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c
*x))-3/512/c^4/b^2*(64*x^7*b*c^7+64*(c^2*x^2+1)^(1/2)*x^6*b*c^6+112*x^5*b*c^5+80*(c^2*x^2+1)^(1/2)*x^4*b*c^4+5
6*x^3*b*c^3+24*(c^2*x^2+1)^(1/2)*x^2*b*c^2+7*arcsinh(c*x)*Ei(1,-7*arcsinh(c*x)-7*a/b)*exp(-7*a/b)*b+7*Ei(1,-7*
arcsinh(c*x)-7*a/b)*exp(-7*a/b)*a+7*x*b*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))-1/512/c^4/b^2*(256*x^9*b*c^9
+256*(c^2*x^2+1)^(1/2)*x^8*b*c^8+576*x^7*b*c^7+448*(c^2*x^2+1)^(1/2)*x^6*b*c^6+432*x^5*b*c^5+240*(c^2*x^2+1)^(
1/2)*x^4*b*c^4+120*x^3*b*c^3+40*(c^2*x^2+1)^(1/2)*x^2*b*c^2+9*arcsinh(c*x)*exp(-9*a/b)*Ei(1,-9*arcsinh(c*x)-9*
a/b)*b+9*exp(-9*a/b)*Ei(1,-9*arcsinh(c*x)-9*a/b)*a+9*x*b*c+(c^2*x^2+1)^(1/2)*b)/(a+b*arcsinh(c*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (c^{6} x^{9} + 3 \, c^{4} x^{7} + 3 \, c^{2} x^{5} + x^{3}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (c^{7} x^{10} + 3 \, c^{5} x^{8} + 3 \, c^{3} x^{6} + c x^{4}\right )} \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 +{\left (b^{2} c^{3} x^{2} + \sqrt{c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )} + \int \frac{{\left (9 \, c^{7} x^{9} + 20 \, c^{5} x^{7} + 13 \, c^{3} x^{5} + 2 \, c x^{3}\right )}{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 3 \,{\left (6 \, c^{8} x^{10} + 17 \, c^{6} x^{8} + 17 \, c^{4} x^{6} + 7 \, c^{2} x^{4} + x^{2}\right )}{\left (c^{2} x^{2} + 1\right )} +{\left (9 \, c^{9} x^{11} + 31 \, c^{7} x^{9} + 39 \, c^{5} x^{7} + 21 \, c^{3} x^{5} + 4 \, c x^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{a b c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c +{\left (b^{2} c^{5} x^{4} +{\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^9 + 3*c^4*x^7 + 3*c^2*x^5 + x^3)*(c^2*x^2 + 1) + (c^7*x^10 + 3*c^5*x^8 + 3*c^3*x^6 + c*x^4)*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(((9*c^7*x^9 + 20*c^5*x^7 + 13*c^3*x^5 + 2*c*x^3)*(c^2*x^2 + 1)^
(3/2) + 3*(6*c^8*x^10 + 17*c^6*x^8 + 17*c^4*x^6 + 7*c^2*x^4 + x^2)*(c^2*x^2 + 1) + (9*c^9*x^11 + 31*c^7*x^9 +
39*c^5*x^7 + 21*c^3*x^5 + 4*c*x^3)*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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} x^{7} + 2 \, c^{2} x^{5} + x^{3}\right )} \sqrt{c^{2} x^{2} + 1}}{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^7 + 2*c^2*x^5 + x^3)*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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} x^{2} + 1\right )}^{\frac{5}{2}} x^{3}}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate((c^2*x^2 + 1)^(5/2)*x^3/(b*arcsinh(c*x) + a)^2, x)