3.170 \(\int \frac {(c e+d e x)^2}{(a+b \sinh ^{-1}(c+d x))^3} \, dx\)

Optimal. Leaf size=246 \[ -\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}+\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

[Out]

-e^2*(d*x+c)/b^2/d/(a+b*arcsinh(d*x+c))-3/2*e^2*(d*x+c)^3/b^2/d/(a+b*arcsinh(d*x+c))-1/8*e^2*Chi((a+b*arcsinh(
d*x+c))/b)*cosh(a/b)/b^3/d+9/8*e^2*Chi(3*(a+b*arcsinh(d*x+c))/b)*cosh(3*a/b)/b^3/d+1/8*e^2*Shi((a+b*arcsinh(d*
x+c))/b)*sinh(a/b)/b^3/d-9/8*e^2*Shi(3*(a+b*arcsinh(d*x+c))/b)*sinh(3*a/b)/b^3/d-1/2*e^2*(d*x+c)^2*(1+(d*x+c)^
2)^(1/2)/b/d/(a+b*arcsinh(d*x+c))^2

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Rubi [A]  time = 0.61, antiderivative size = 305, normalized size of antiderivative = 1.24, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5865, 12, 5667, 5774, 5669, 5448, 3303, 3298, 3301, 5657} \[ -\frac {9 e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {9 e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(2*b*d*(a + b*ArcSinh[c + d*x])^2) - (e^2*(c + d*x))/(b^2*d*(a + b*Ar
cSinh[c + d*x])) - (3*e^2*(c + d*x)^3)/(2*b^2*d*(a + b*ArcSinh[c + d*x])) - (9*e^2*Cosh[a/b]*CoshIntegral[a/b
+ ArcSinh[c + d*x]])/(8*b^3*d) + (9*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*a)/b + 3*ArcSinh[c + d*x]])/(8*b^3*d) +
(e^2*Cosh[a/b]*CoshIntegral[(a + b*ArcSinh[c + d*x])/b])/(b^3*d) + (9*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh
[c + d*x]])/(8*b^3*d) - (9*e^2*Sinh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcSinh[c + d*x]])/(8*b^3*d) - (e^2*Sinh
[a/b]*SinhIntegral[(a + b*ArcSinh[c + d*x])/b])/(b^3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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 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 5657

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cosh[a/b - x/b], x], x,
a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 5667

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 + c^2*x^2]*(a + b*ArcSi
nh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcSinh[c*x])^(n +
1))/Sqrt[1 + c^2*x^2], x], x] - Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1))/Sqrt[1 + c
^2*x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5669

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

Rule 5774

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

Rule 5865

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

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

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Mathematica [A]  time = 0.73, size = 216, normalized size = 0.88 \[ \frac {e^2 \left (-\frac {4 b^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+9 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )+8 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-8 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\frac {4 b \left (-3 (c+d x)^3-2 (c+d x)\right )}{a+b \sinh ^{-1}(c+d x)}\right )}{8 b^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*((-4*b^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (4*b*(-2*(c + d*x) - 3*(c + d*x)
^3))/(a + b*ArcSinh[c + d*x]) + 8*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 8*Sinh[a/b]*SinhIntegral[a/
b + ArcSinh[c + d*x]] + 9*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a
/b + ArcSinh[c + d*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b +
ArcSinh[c + d*x])])))/(8*b^3*d)

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arsinh}\left (d x + c\right ) + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.24, size = 507, normalized size = 2.06 \[ \frac {-\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (3 b \arcsinh \left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \arcsinh \left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}+\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b \arcsinh \left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \arcsinh \left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/16*(-4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+4*(d*x+c)^3-(1+(d*x+c)^2)^(1/2)+3*d*x+3*c)*e^2*(3*b*arcsinh(d*x+c
)+3*a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c)+a^2)-9/16*e^2/b^3*exp(3*a/b)*Ei(1,3*arcsinh(d*x+c)+3*a
/b)+1/16*(-(1+(d*x+c)^2)^(1/2)+d*x+c)*e^2*(b*arcsinh(d*x+c)+a-b)/b^2/(b^2*arcsinh(d*x+c)^2+2*a*b*arcsinh(d*x+c
)+a^2)+1/16*e^2/b^3*exp(a/b)*Ei(1,arcsinh(d*x+c)+a/b)+1/16/b*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+
c))^2+1/16/b^2*e^2*(d*x+c+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x+c))+1/16/b^3*e^2*exp(-a/b)*Ei(1,-arcsinh(d*x+c
)-a/b)-1/16/b*e^2*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d*x
+c))^2-3/16/b^2*e^2*(4*(d*x+c)^3+3*d*x+3*c+4*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+(1+(d*x+c)^2)^(1/2))/(a+b*arcsinh(d
*x+c))-9/16/b^3*e^2*exp(-3*a/b)*Ei(1,-3*arcsinh(d*x+c)-3*a/b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((3*a*d^9*e^2 + b*d^9*e^2)*x^9 + 9*(3*a*c*d^8*e^2 + b*c*d^8*e^2)*x^8 + 3*(3*(12*c^2*d^7*e^2 + d^7*e^2)*a
+ (12*c^2*d^7*e^2 + d^7*e^2)*b)*x^7 + 21*(3*(4*c^3*d^6*e^2 + c*d^6*e^2)*a + (4*c^3*d^6*e^2 + c*d^6*e^2)*b)*x^6
+ 3*(3*(42*c^4*d^5*e^2 + 21*c^2*d^5*e^2 + d^5*e^2)*a + (42*c^4*d^5*e^2 + 21*c^2*d^5*e^2 + d^5*e^2)*b)*x^5 + 3
*(3*(42*c^5*d^4*e^2 + 35*c^3*d^4*e^2 + 5*c*d^4*e^2)*a + (42*c^5*d^4*e^2 + 35*c^3*d^4*e^2 + 5*c*d^4*e^2)*b)*x^4
+ (3*(84*c^6*d^3*e^2 + 105*c^4*d^3*e^2 + 30*c^2*d^3*e^2 + d^3*e^2)*a + (84*c^6*d^3*e^2 + 105*c^4*d^3*e^2 + 30
*c^2*d^3*e^2 + d^3*e^2)*b)*x^3 + 3*(3*(12*c^7*d^2*e^2 + 21*c^5*d^2*e^2 + 10*c^3*d^2*e^2 + c*d^2*e^2)*a + (12*c
^7*d^2*e^2 + 21*c^5*d^2*e^2 + 10*c^3*d^2*e^2 + c*d^2*e^2)*b)*x^2 + ((3*a*d^6*e^2 + b*d^6*e^2)*x^6 + 6*(3*a*c*d
^5*e^2 + b*c*d^5*e^2)*x^5 + ((45*c^2*d^4*e^2 + 4*d^4*e^2)*a + (15*c^2*d^4*e^2 + d^4*e^2)*b)*x^4 + 4*((15*c^3*d
^3*e^2 + 4*c*d^3*e^2)*a + (5*c^3*d^3*e^2 + c*d^3*e^2)*b)*x^3 + ((45*c^4*d^2*e^2 + 24*c^2*d^2*e^2 + d^2*e^2)*a
+ 3*(5*c^4*d^2*e^2 + 2*c^2*d^2*e^2)*b)*x^2 + (3*c^6*e^2 + 4*c^4*e^2 + c^2*e^2)*a + (c^6*e^2 + c^4*e^2)*b + 2*(
(9*c^5*d*e^2 + 8*c^3*d*e^2 + c*d*e^2)*a + (3*c^5*d*e^2 + 2*c^3*d*e^2)*b)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2
) + (3*(3*a*d^7*e^2 + b*d^7*e^2)*x^7 + 21*(3*a*c*d^6*e^2 + b*c*d^6*e^2)*x^6 + ((189*c^2*d^5*e^2 + 17*d^5*e^2)*
a + (63*c^2*d^5*e^2 + 5*d^5*e^2)*b)*x^5 + 5*((63*c^3*d^4*e^2 + 17*c*d^4*e^2)*a + (21*c^3*d^4*e^2 + 5*c*d^4*e^2
)*b)*x^4 + (5*(63*c^4*d^3*e^2 + 34*c^2*d^3*e^2 + 2*d^3*e^2)*a + (105*c^4*d^3*e^2 + 50*c^2*d^3*e^2 + 2*d^3*e^2)
*b)*x^3 + ((189*c^5*d^2*e^2 + 170*c^3*d^2*e^2 + 30*c*d^2*e^2)*a + (63*c^5*d^2*e^2 + 50*c^3*d^2*e^2 + 6*c*d^2*e
^2)*b)*x^2 + (9*c^7*e^2 + 17*c^5*e^2 + 10*c^3*e^2 + 2*c*e^2)*a + (3*c^7*e^2 + 5*c^5*e^2 + 2*c^3*e^2)*b + ((63*
c^6*d*e^2 + 85*c^4*d*e^2 + 30*c^2*d*e^2 + 2*d*e^2)*a + (21*c^6*d*e^2 + 25*c^4*d*e^2 + 6*c^2*d*e^2)*b)*x)*(d^2*
x^2 + 2*c*d*x + c^2 + 1) + 3*(c^9*e^2 + 3*c^7*e^2 + 3*c^5*e^2 + c^3*e^2)*a + (c^9*e^2 + 3*c^7*e^2 + 3*c^5*e^2
+ c^3*e^2)*b + 3*(3*(3*c^8*d*e^2 + 7*c^6*d*e^2 + 5*c^4*d*e^2 + c^2*d*e^2)*a + (3*c^8*d*e^2 + 7*c^6*d*e^2 + 5*c
^4*d*e^2 + c^2*d*e^2)*b)*x + (3*b*d^9*e^2*x^9 + 27*b*c*d^8*e^2*x^8 + 9*(12*c^2*d^7*e^2 + d^7*e^2)*b*x^7 + 63*(
4*c^3*d^6*e^2 + c*d^6*e^2)*b*x^6 + 9*(42*c^4*d^5*e^2 + 21*c^2*d^5*e^2 + d^5*e^2)*b*x^5 + 9*(42*c^5*d^4*e^2 + 3
5*c^3*d^4*e^2 + 5*c*d^4*e^2)*b*x^4 + 3*(84*c^6*d^3*e^2 + 105*c^4*d^3*e^2 + 30*c^2*d^3*e^2 + d^3*e^2)*b*x^3 + 9
*(12*c^7*d^2*e^2 + 21*c^5*d^2*e^2 + 10*c^3*d^2*e^2 + c*d^2*e^2)*b*x^2 + 9*(3*c^8*d*e^2 + 7*c^6*d*e^2 + 5*c^4*d
*e^2 + c^2*d*e^2)*b*x + (3*b*d^6*e^2*x^6 + 18*b*c*d^5*e^2*x^5 + (45*c^2*d^4*e^2 + 4*d^4*e^2)*b*x^4 + 4*(15*c^3
*d^3*e^2 + 4*c*d^3*e^2)*b*x^3 + (45*c^4*d^2*e^2 + 24*c^2*d^2*e^2 + d^2*e^2)*b*x^2 + 2*(9*c^5*d*e^2 + 8*c^3*d*e
^2 + c*d*e^2)*b*x + (3*c^6*e^2 + 4*c^4*e^2 + c^2*e^2)*b)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (9*b*d^7*e^2*x^
7 + 63*b*c*d^6*e^2*x^6 + (189*c^2*d^5*e^2 + 17*d^5*e^2)*b*x^5 + 5*(63*c^3*d^4*e^2 + 17*c*d^4*e^2)*b*x^4 + 5*(6
3*c^4*d^3*e^2 + 34*c^2*d^3*e^2 + 2*d^3*e^2)*b*x^3 + (189*c^5*d^2*e^2 + 170*c^3*d^2*e^2 + 30*c*d^2*e^2)*b*x^2 +
(63*c^6*d*e^2 + 85*c^4*d*e^2 + 30*c^2*d*e^2 + 2*d*e^2)*b*x + (9*c^7*e^2 + 17*c^5*e^2 + 10*c^3*e^2 + 2*c*e^2)*
b)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(c^9*e^2 + 3*c^7*e^2 + 3*c^5*e^2 + c^3*e^2)*b + (9*b*d^8*e^2*x^8 + 72*b*c
*d^7*e^2*x^7 + 2*(126*c^2*d^6*e^2 + 11*d^6*e^2)*b*x^6 + 12*(42*c^3*d^5*e^2 + 11*c*d^5*e^2)*b*x^5 + 6*(105*c^4*
d^4*e^2 + 55*c^2*d^4*e^2 + 3*d^4*e^2)*b*x^4 + 8*(63*c^5*d^3*e^2 + 55*c^3*d^3*e^2 + 9*c*d^3*e^2)*b*x^3 + (252*c
^6*d^2*e^2 + 330*c^4*d^2*e^2 + 108*c^2*d^2*e^2 + 5*d^2*e^2)*b*x^2 + 2*(36*c^7*d*e^2 + 66*c^5*d*e^2 + 36*c^3*d*
e^2 + 5*c*d*e^2)*b*x + (9*c^8*e^2 + 22*c^6*e^2 + 18*c^4*e^2 + 5*c^2*e^2)*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))
*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + (3*(3*a*d^8*e^2 + b*d^8*e^2)*x^8 + 24*(3*a*c*d^7*e^2 + b*c
*d^7*e^2)*x^7 + (2*(126*c^2*d^6*e^2 + 11*d^6*e^2)*a + 7*(12*c^2*d^6*e^2 + d^6*e^2)*b)*x^6 + 6*(2*(42*c^3*d^5*e
^2 + 11*c*d^5*e^2)*a + 7*(4*c^3*d^5*e^2 + c*d^5*e^2)*b)*x^5 + (6*(105*c^4*d^4*e^2 + 55*c^2*d^4*e^2 + 3*d^4*e^2
)*a + 5*(42*c^4*d^4*e^2 + 21*c^2*d^4*e^2 + d^4*e^2)*b)*x^4 + 4*(2*(63*c^5*d^3*e^2 + 55*c^3*d^3*e^2 + 9*c*d^3*e
^2)*a + (42*c^5*d^3*e^2 + 35*c^3*d^3*e^2 + 5*c*d^3*e^2)*b)*x^3 + ((252*c^6*d^2*e^2 + 330*c^4*d^2*e^2 + 108*c^2
*d^2*e^2 + 5*d^2*e^2)*a + (84*c^6*d^2*e^2 + 105*c^4*d^2*e^2 + 30*c^2*d^2*e^2 + d^2*e^2)*b)*x^2 + (9*c^8*e^2 +
22*c^6*e^2 + 18*c^4*e^2 + 5*c^2*e^2)*a + (3*c^8*e^2 + 7*c^6*e^2 + 5*c^4*e^2 + c^2*e^2)*b + 2*((36*c^7*d*e^2 +
66*c^5*d*e^2 + 36*c^3*d*e^2 + 5*c*d*e^2)*a + (12*c^7*d*e^2 + 21*c^5*d*e^2 + 10*c^3*d*e^2 + c*d*e^2)*b)*x)*sqrt
(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a^2*b^2*d^7*x^6 + 6*a^2*b^2*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*a^2*b^2*x^4 + 4*(5
*c^3*d^4 + 3*c*d^4)*a^2*b^2*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 + c*d^2
)*a^2*b^2*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a^2*b^2 + (b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 3*(5*c^2*d^5 + d^5)*b
^4*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*b^4*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*b^4*x^2 + 6*(c^5*d^2 + 2*c^3*d^2 +
c*d^2)*b^4*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*b^4 + (b^4*d^4*x^3 + 3*b^4*c*d^3*x^2 + 3*b^4*c^2*d^2*x + b^4*c^
3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(b^4*d^5*x^4 + 4*b^4*c*d^4*x^3 + (6*c^2*d^3 + d^3)*b^4*x^2 + 2*(2
*c^3*d^2 + c*d^2)*b^4*x + (c^4*d + c^2*d)*b^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 3*(b^4*d^6*x^5 + 5*b^4*c*d^5*x^
4 + 2*(5*c^2*d^4 + d^4)*b^4*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*b^4*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*b^4*x + (c^5
*d + 2*c^3*d + c*d)*b^4)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2
+ (a^2*b^2*d^4*x^3 + 3*a^2*b^2*c*d^3*x^2 + 3*a^2*b^2*c^2*d^2*x + a^2*b^2*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)
^(3/2) + 3*(a^2*b^2*d^5*x^4 + 4*a^2*b^2*c*d^4*x^3 + (6*c^2*d^3 + d^3)*a^2*b^2*x^2 + 2*(2*c^3*d^2 + c*d^2)*a^2*
b^2*x + (c^4*d + c^2*d)*a^2*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(a*b^3*d^7*x^6 + 6*a*b^3*c*d^6*x^5 + 3*(5*c
^2*d^5 + d^5)*a*b^3*x^4 + 4*(5*c^3*d^4 + 3*c*d^4)*a*b^3*x^3 + 3*(5*c^4*d^3 + 6*c^2*d^3 + d^3)*a*b^3*x^2 + 6*(c
^5*d^2 + 2*c^3*d^2 + c*d^2)*a*b^3*x + (c^6*d + 3*c^4*d + 3*c^2*d + d)*a*b^3 + (a*b^3*d^4*x^3 + 3*a*b^3*c*d^3*x
^2 + 3*a*b^3*c^2*d^2*x + a*b^3*c^3*d)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 3*(a*b^3*d^5*x^4 + 4*a*b^3*c*d^4*x
^3 + (6*c^2*d^3 + d^3)*a*b^3*x^2 + 2*(2*c^3*d^2 + c*d^2)*a*b^3*x + (c^4*d + c^2*d)*a*b^3)*(d^2*x^2 + 2*c*d*x +
c^2 + 1) + 3*(a*b^3*d^6*x^5 + 5*a*b^3*c*d^5*x^4 + 2*(5*c^2*d^4 + d^4)*a*b^3*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a*b
^3*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a*b^3*x + (c^5*d + 2*c^3*d + c*d)*a*b^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 +
1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 3*(a^2*b^2*d^6*x^5 + 5*a^2*b^2*c*d^5*x^4 + 2*(5*c^2*d^
4 + d^4)*a^2*b^2*x^3 + 2*(5*c^3*d^3 + 3*c*d^3)*a^2*b^2*x^2 + (5*c^4*d^2 + 6*c^2*d^2 + d^2)*a^2*b^2*x + (c^5*d
+ 2*c^3*d + c*d)*a^2*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + integrate(1/2*(9*d^10*e^2*x^10 + 90*c*d^9*e^2*x
^9 + 9*c^10*e^2 + 36*c^8*e^2 + 9*(45*c^2*d^8*e^2 + 4*d^8*e^2)*x^8 + 54*c^6*e^2 + 72*(15*c^3*d^7*e^2 + 4*c*d^7*
e^2)*x^7 + 18*(105*c^4*d^6*e^2 + 56*c^2*d^6*e^2 + 3*d^6*e^2)*x^6 + 36*c^4*e^2 + 36*(63*c^5*d^5*e^2 + 56*c^3*d^
5*e^2 + 9*c*d^5*e^2)*x^5 + 18*(105*c^6*d^4*e^2 + 140*c^4*d^4*e^2 + 45*c^2*d^4*e^2 + 2*d^4*e^2)*x^4 + 9*c^2*e^2
+ 72*(15*c^7*d^3*e^2 + 28*c^5*d^3*e^2 + 15*c^3*d^3*e^2 + 2*c*d^3*e^2)*x^3 + (9*d^6*e^2*x^6 + 54*c*d^5*e^2*x^5
+ 9*c^6*e^2 + 4*c^4*e^2 + (135*c^2*d^4*e^2 + 4*d^4*e^2)*x^4 - c^2*e^2 + 4*(45*c^3*d^3*e^2 + 4*c*d^3*e^2)*x^3
+ (135*c^4*d^2*e^2 + 24*c^2*d^2*e^2 - d^2*e^2)*x^2 + 2*(27*c^5*d*e^2 + 8*c^3*d*e^2 - c*d*e^2)*x)*(d^2*x^2 + 2*
c*d*x + c^2 + 1)^2 + 9*(45*c^8*d^2*e^2 + 112*c^6*d^2*e^2 + 90*c^4*d^2*e^2 + 24*c^2*d^2*e^2 + d^2*e^2)*x^2 + (3
6*d^7*e^2*x^7 + 252*c*d^6*e^2*x^6 + 36*c^7*e^2 + 48*c^5*e^2 + 12*(63*c^2*d^5*e^2 + 4*d^5*e^2)*x^5 + 13*c^3*e^2
+ 60*(21*c^3*d^4*e^2 + 4*c*d^4*e^2)*x^4 + (1260*c^4*d^3*e^2 + 480*c^2*d^3*e^2 + 13*d^3*e^2)*x^3 - 2*c*e^2 + 3
*(252*c^5*d^2*e^2 + 160*c^3*d^2*e^2 + 13*c*d^2*e^2)*x^2 + (252*c^6*d*e^2 + 240*c^4*d*e^2 + 39*c^2*d*e^2 - 2*d*
e^2)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + (54*d^8*e^2*x^8 + 432*c*d^7*e^2*x^7 + 54*c^8*e^2 + 120*c^6*e^2 +
24*(63*c^2*d^6*e^2 + 5*d^6*e^2)*x^6 + 83*c^4*e^2 + 144*(21*c^3*d^5*e^2 + 5*c*d^5*e^2)*x^5 + (3780*c^4*d^4*e^2
+ 1800*c^2*d^4*e^2 + 83*d^4*e^2)*x^4 + 19*c^2*e^2 + 4*(756*c^5*d^3*e^2 + 600*c^3*d^3*e^2 + 83*c*d^3*e^2)*x^3
+ (1512*c^6*d^2*e^2 + 1800*c^4*d^2*e^2 + 498*c^2*d^2*e^2 + 19*d^2*e^2)*x^2 + 2*e^2 + 2*(216*c^7*d*e^2 + 360*c^
5*d*e^2 + 166*c^3*d*e^2 + 19*c*d*e^2)*x)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 18*(5*c^9*d*e^2 + 16*c^7*d*e^2 + 18*c
^5*d*e^2 + 8*c^3*d*e^2 + c*d*e^2)*x + (36*d^9*e^2*x^9 + 324*c*d^8*e^2*x^8 + 36*c^9*e^2 + 112*c^7*e^2 + 16*(81*
c^2*d^7*e^2 + 7*d^7*e^2)*x^7 + 123*c^5*e^2 + 112*(27*c^3*d^6*e^2 + 7*c*d^6*e^2)*x^6 + 3*(1512*c^4*d^5*e^2 + 78
4*c^2*d^5*e^2 + 41*d^5*e^2)*x^5 + 57*c^3*e^2 + (4536*c^5*d^4*e^2 + 3920*c^3*d^4*e^2 + 615*c*d^4*e^2)*x^4 + (30
24*c^6*d^3*e^2 + 3920*c^4*d^3*e^2 + 1230*c^2*d^3*e^2 + 57*d^3*e^2)*x^3 + 10*c*e^2 + 3*(432*c^7*d^2*e^2 + 784*c
^5*d^2*e^2 + 410*c^3*d^2*e^2 + 57*c*d^2*e^2)*x^2 + (324*c^8*d*e^2 + 784*c^6*d*e^2 + 615*c^4*d*e^2 + 171*c^2*d*
e^2 + 10*d*e^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b^2*d^8*x^8 + 8*a*b^2*c*d^7*x^7 + 4*(7*c^2*d^6 + d^6)
*a*b^2*x^6 + 8*(7*c^3*d^5 + 3*c*d^5)*a*b^2*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*a*b^2*x^4 + 8*(7*c^5*d^3
+ 10*c^3*d^3 + 3*c*d^3)*a*b^2*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*a*b^2*x^2 + 8*(c^7*d + 3*c^5*
d + 3*c^3*d + c*d)*a*b^2*x + (c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)*a*b^2 + (a*b^2*d^4*x^4 + 4*a*b^2*c*d^3*x^3 + 6*
a*b^2*c^2*d^2*x^2 + 4*a*b^2*c^3*d*x + a*b^2*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(a*b^2*d^5*x^5 + 5*a*b^2*
c*d^4*x^4 + (10*c^2*d^3 + d^3)*a*b^2*x^3 + (10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (5*c^4*d + 3*c^2*d)*a*b^2*x + (c
^5 + c^3)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(a*b^2*d^6*x^6 + 6*a*b^2*c*d^5*x^5 + (15*c^2*d^4 + 2*
d^4)*a*b^2*x^4 + 4*(5*c^3*d^3 + 2*c*d^3)*a*b^2*x^3 + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*a*b^2*x^2 + 2*(3*c^5*d +
4*c^3*d + c*d)*a*b^2*x + (c^6 + 2*c^4 + c^2)*a*b^2)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^3*d^8*x^8 + 8*b^3*c*d^7
*x^7 + 4*(7*c^2*d^6 + d^6)*b^3*x^6 + 8*(7*c^3*d^5 + 3*c*d^5)*b^3*x^5 + 2*(35*c^4*d^4 + 30*c^2*d^4 + 3*d^4)*b^3
*x^4 + 8*(7*c^5*d^3 + 10*c^3*d^3 + 3*c*d^3)*b^3*x^3 + 4*(7*c^6*d^2 + 15*c^4*d^2 + 9*c^2*d^2 + d^2)*b^3*x^2 + 8
*(c^7*d + 3*c^5*d + 3*c^3*d + c*d)*b^3*x + (c^8 + 4*c^6 + 6*c^4 + 4*c^2 + 1)*b^3 + (b^3*d^4*x^4 + 4*b^3*c*d^3*
x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*c^3*d*x + b^3*c^4)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*(b^3*d^5*x^5 + 5*b^3*c*
d^4*x^4 + (10*c^2*d^3 + d^3)*b^3*x^3 + (10*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (5*c^4*d + 3*c^2*d)*b^3*x + (c^5 + c^3
)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 6*(b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + (15*c^2*d^4 + 2*d^4)*b^3*x^4 +
4*(5*c^3*d^3 + 2*c*d^3)*b^3*x^3 + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*b^3*x^2 + 2*(3*c^5*d + 4*c^3*d + c*d)*b^3*x
+ (c^6 + 2*c^4 + c^2)*b^3)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 4*(b^3*d^7*x^7 + 7*b^3*c*d^6*x^6 + 3*(7*c^2*d^5 +
d^5)*b^3*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*b^3*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)*b^3*x^3 + 3*(7*c^5*d^2 + 10
*c^3*d^2 + 3*c*d^2)*b^3*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*b^3*x + (c^7 + 3*c^5 + 3*c^3 + c)*b^3)*sqrt(d
^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 4*(a*b^2*d^7*x^7 + 7*a*b^2*c*d
^6*x^6 + 3*(7*c^2*d^5 + d^5)*a*b^2*x^5 + 5*(7*c^3*d^4 + 3*c*d^4)*a*b^2*x^4 + (35*c^4*d^3 + 30*c^2*d^3 + 3*d^3)
*a*b^2*x^3 + 3*(7*c^5*d^2 + 10*c^3*d^2 + 3*c*d^2)*a*b^2*x^2 + (7*c^6*d + 15*c^4*d + 9*c^2*d + d)*a*b^2*x + (c^
7 + 3*c^5 + 3*c^3 + c)*a*b^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{2} \left (\int \frac {c^{2}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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