3.2.38 \(\int \frac {(c e+d e x)^2}{(a+b \cosh ^{-1}(c+d x))^2} \, dx\) [138]

Optimal. Leaf size=191 \[ -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 12, 5885, 3384, 3379, 3382} \begin {gather*} \frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{4 b^2 d}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{4 b^2 d}-\frac {e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])/(b*d*(a + b*ArcCosh[c + d*x]))) + (e^2*Cosh[a/b]*Cosh
Integral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) + (3*e^2*Cosh[(3*a)/b]*CoshIntegral[(3*(a + b*ArcCosh[c + d*x]
))/b])/(4*b^2*d) - (e^2*Sinh[a/b]*SinhIntegral[(a + b*ArcCosh[c + d*x])/b])/(4*b^2*d) - (3*e^2*Sinh[(3*a)/b]*S
inhIntegral[(3*(a + b*ArcCosh[c + d*x]))/b])/(4*b^2*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 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)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}-\frac {3 \cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}+\frac {\left (3 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}-\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}-\frac {\left (3 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{4 b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{4 b^2 d}+\frac {3 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{4 b^2 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{4 b^2 d}-\frac {3 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{4 b^2 d}\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 150, normalized size = 0.79 \begin {gather*} \frac {e^2 \left (-\frac {4 b (c+d x)^2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{a+b \cosh ^{-1}(c+d x)}+\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )}{4 b^2 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(179)=358\).
time = 71.38, size = 374, normalized size = 1.96

method	result	size
derivativedivides	\(\frac {\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2}}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2}}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\)	\(374\)
default	\(\frac {\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2}}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{8 b^{2}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2}}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{8 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {3 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{8 b^{2}}}{d}\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/8*(-4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^2+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+4*(d*x+c)^3-3*d*x-3*c)*
e^2/b/(a+b*arccosh(d*x+c))-3/8*e^2/b^2*exp(3*a/b)*Ei(1,3*arccosh(d*x+c)+3*a/b)+1/8*(-(d*x+c-1)^(1/2)*(d*x+c+1)
^(1/2)+d*x+c)*e^2/b/(a+b*arccosh(d*x+c))-1/8*e^2/b^2*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/8/b*e^2*(d*x+c+(d*x+c
-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/8/b^2*e^2*exp(-a/b)*Ei(1,-arccosh(d*x+c)-a/b)-1/8/b*e^2*(4*(
d*x+c)^3-3*d*x-3*c+4*(d*x+c+1)^(1/2)*(d*x+c-1)^(1/2)*(d*x+c)^2-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d
*x+c))-3/8/b^2*e^2*exp(-3*a/b)*Ei(1,-3*arccosh(d*x+c)-3*a/b))

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

[Out]

-(d^5*x^5*e^2 + 5*c*d^4*x^4*e^2 + (10*c^2*d^3 - d^3)*x^3*e^2 + (10*c^3*d^2 - 3*c*d^2)*x^2*e^2 + (5*c^4*d - 3*c
^2*d)*x*e^2 + (d^4*x^4*e^2 + 4*c*d^3*x^3*e^2 + (6*c^2*d^2 - d^2)*x^2*e^2 + 2*(2*c^3*d - c*d)*x*e^2 + (c^4 - c^
2)*e^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^5 - c^3)*e^2)/(a*b*d^3*x^2 + 2*a*b*c*d^2*x + (c^2*d - d)*a*b
+ (a*b*d^2*x + a*b*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^3*x^2 + 2*b^2*c*d^2*x + (c^2*d - d)*b^2 +
(b^2*d^2*x + b^2*c*d)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)
) + integrate((3*d^6*x^6*e^2 + 18*c*d^5*x^5*e^2 + 3*(15*c^2*d^4 - 2*d^4)*x^4*e^2 + 12*(5*c^3*d^3 - 2*c*d^3)*x^
3*e^2 + 3*(15*c^4*d^2 - 12*c^2*d^2 + d^2)*x^2*e^2 + (3*d^4*x^4*e^2 + 12*c*d^3*x^3*e^2 + (18*c^2*d^2 - d^2)*x^2
*e^2 + 2*(6*c^3*d - c*d)*x*e^2 + (3*c^4 - c^2)*e^2)*(d*x + c + 1)*(d*x + c - 1) + 6*(3*c^5*d - 4*c^3*d + c*d)*
x*e^2 + (6*d^5*x^5*e^2 + 30*c*d^4*x^4*e^2 + (60*c^2*d^3 - 7*d^3)*x^3*e^2 + 3*(20*c^3*d^2 - 7*c*d^2)*x^2*e^2 +
(30*c^4*d - 21*c^2*d + 2*d)*x*e^2 + (6*c^5 - 7*c^3 + 2*c)*e^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + 3*(c^6 -
2*c^4 + c^2)*e^2)/(a*b*d^4*x^4 + 4*a*b*c*d^3*x^3 + 2*(3*c^2*d^2 - d^2)*a*b*x^2 + 4*(c^3*d - c*d)*a*b*x + (a*b*
d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*(d*x + c + 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*a*b + 2*(a*b*d^3*x^3 + 3*a*b*
c*d^2*x^2 + (3*c^2*d - d)*a*b*x + (c^3 - c)*a*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (b^2*d^4*x^4 + 4*b^2*c*
d^3*x^3 + 2*(3*c^2*d^2 - d^2)*b^2*x^2 + 4*(c^3*d - c*d)*b^2*x + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*(d*x + c
+ 1)*(d*x + c - 1) + (c^4 - 2*c^2 + 1)*b^2 + 2*(b^2*d^3*x^3 + 3*b^2*c*d^2*x^2 + (3*c^2*d - d)*b^2*x + (c^3 -
c)*b^2)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)), 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((d*e*x+c*e)^2/(a+b*arccosh(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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