3.2.51 \(\int \frac {c e+d e x}{(a+b \cosh ^{-1}(c+d x))^4} \, dx\) [151]

Optimal. Leaf size=218 \[ -\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d} \]

[Out]

1/6*e/b^2/d/(a+b*arccosh(d*x+c))^2-1/3*e*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))^2+2/3*e*Chi(2*(a+b*arccosh(d*x+c
))/b)*cosh(2*a/b)/b^4/d-2/3*e*Shi(2*(a+b*arccosh(d*x+c))/b)*sinh(2*a/b)/b^4/d-1/3*e*(d*x+c)*(d*x+c-1)^(1/2)*(d
*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^3-2/3*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+
c))

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Rubi [A]
time = 0.34, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5996, 12, 5886, 5951, 5885, 3384, 3379, 3382, 5893} \begin {gather*} \frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

-1/3*(e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x])^3) + e/(6*b^2*d*(a + b*A
rcCosh[c + d*x])^2) - (e*(c + d*x)^2)/(3*b^2*d*(a + b*ArcCosh[c + d*x])^2) - (2*e*Sqrt[-1 + c + d*x]*(c + d*x)
*Sqrt[1 + c + d*x])/(3*b^3*d*(a + b*ArcCosh[c + d*x])) + (2*e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c +
d*x]))/b])/(3*b^4*d) - (2*e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c + d*x]))/b])/(3*b^4*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 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 5885

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5886

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

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5951

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rule 5996

Int[((a_.) + ArcCosh[(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*ArcCosh[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}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac {(2 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (2 e \cosh \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,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (2 e \sinh \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,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e (c+d x)^2}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {2 e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 195, normalized size = 0.89 \begin {gather*} \frac {e \left (-\frac {2 b^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {b^2 \left (1-2 (c+d x)^2\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {4 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{a+b \cosh ^{-1}(c+d x)}-4 \log \left (a+b \cosh ^{-1}(c+d x)\right )+4 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\log \left (a+b \cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^4,x]

[Out]

(e*((-2*b^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x])^3 + (b^2*(1 - 2*(c + d*x)
^2))/(a + b*ArcCosh[c + d*x])^2 - (4*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(a + b*ArcCosh[c + d*x]
) - 4*Log[a + b*ArcCosh[c + d*x]] + 4*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c + d*x])] + Log[a + b*ArcC
osh[c + d*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c + d*x])])))/(6*b^4*d)

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Maple [A]
time = 0.05, size = 353, normalized size = 1.62

method	result	size
derivativedivides	\(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+4 a b \,\mathrm {arccosh}\left (d x +c \right )-b^{2} \mathrm {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\)	\(353\)
default	\(\frac {\frac {\left (-2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )+2 \left (d x +c \right )^{2}-1\right ) e \left (2 b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+4 a b \,\mathrm {arccosh}\left (d x +c \right )-b^{2} \mathrm {arccosh}\left (d x +c \right )+2 a^{2}-a b +b^{2}\right )}{12 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (d x +c \right )+\frac {2 a}{b}\right )}{3 b^{4}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{12 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}-1+2 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )\right )}{6 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (d x +c \right )-\frac {2 a}{b}\right )}{3 b^{4}}}{d}\)	\(353\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/12*(-2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)+2*(d*x+c)^2-1)*e*(2*b^2*arccosh(d*x+c)^2+4*a*b*arccosh(d
*x+c)-b^2*arccosh(d*x+c)+2*a^2-a*b+b^2)/b^3/(b^3*arccosh(d*x+c)^3+3*a*b^2*arccosh(d*x+c)^2+3*a^2*b*arccosh(d*x
+c)+a^3)-1/3*e/b^4*exp(2*a/b)*Ei(1,2*arccosh(d*x+c)+2*a/b)-1/12/b*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)
^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))^3-1/12/b^2*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a
+b*arccosh(d*x+c))^2-1/6/b^3*e*(2*(d*x+c)^2-1+2*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c))/(a+b*arccosh(d*x+c))-
1/3/b^4*e*exp(-2*a/b)*Ei(1,-2*arccosh(d*x+c)-2*a/b))

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")

[Out]

Timed out

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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((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")

[Out]

integral((d*x + c)*e/(b^4*arccosh(d*x + c)^4 + 4*a*b^3*arccosh(d*x + c)^3 + 6*a^2*b^2*arccosh(d*x + c)^2 + 4*a
^3*b*arccosh(d*x + c) + a^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e \left (\int \frac {c}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*acosh(d*x+c))**4,x)

[Out]

e*(Integral(c/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*a*b**3*acosh(c + d*x)**3 + b
**4*acosh(c + d*x)**4), x) + Integral(d*x/(a**4 + 4*a**3*b*acosh(c + d*x) + 6*a**2*b**2*acosh(c + d*x)**2 + 4*
a*b**3*acosh(c + d*x)**3 + b**4*acosh(c + d*x)**4), 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((d*e*x+c*e)/(a+b*arccosh(d*x+c))^4,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)/(b*arccosh(d*x + c) + a)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^4,x)

[Out]

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^4, x)

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