3.146 \(\int \frac {1}{(a+b \cosh ^{-1}(c+d x))^3} \, dx\)
Optimal. Leaf size=132 \[ -\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]
[Out]
1/2*(-d*x-c)/b^2/d/(a+b*arccosh(d*x+c))+1/2*cosh(a/b)*Shi((a+b*arccosh(d*x+c))/b)/b^3/d-1/2*Chi((a+b*arccosh(d
*x+c))/b)*sinh(a/b)/b^3/d-1/2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^2
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Rubi [A] time = 0.26, antiderivative size = 132, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used =
{5864, 5656, 5775, 5658, 3303, 3298, 3301} \[ -\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\sqrt {c+d x-1} \sqrt {c+d x+1}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcCosh[c + d*x])^(-3),x]
[Out]
-(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(2*b*d*(a + b*ArcCosh[c + d*x])^2) - (c + d*x)/(2*b^2*d*(a + b*ArcCosh
[c + d*x])) - (CoshIntegral[(a + b*ArcCosh[c + d*x])/b]*Sinh[a/b])/(2*b^3*d) + (Cosh[a/b]*SinhIntegral[(a + b*
ArcCosh[c + d*x])/b])/(2*b^3*d)
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 5656
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c
*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcCosh[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqr
t[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]
Rule 5658
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Dist[(b*c)^(-1), Subst[Int[x^n*Sinh[a/b - x/b], x]
, x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Rule 5775
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*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), 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, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]
Rule 5864
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCosh[x])^n, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\cosh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac {\sinh \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac {\sqrt {-1+c+d x} \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {c+d x}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{2 b^3 d}+\frac {\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{2 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 109, normalized size = 0.83 \[ -\frac {\sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\frac {b \left (a c+a d x+b \sqrt {c+d x-1} \sqrt {c+d x+1}+b (c+d x) \cosh ^{-1}(c+d x)\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}}{2 b^3 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcCosh[c + d*x])^(-3),x]
[Out]
-1/2*((b*(a*c + a*d*x + b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] + b*(c + d*x)*ArcCosh[c + d*x]))/(a + b*ArcCosh
[c + d*x])^2 + CoshIntegral[a/b + ArcCosh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c + d*x]]
)/(b^3*d)
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fricas [F] time = 1.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (d x + c\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*arccosh(d*x+c))^3,x, algorithm="fricas")
[Out]
integral(1/(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), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*arccosh(d*x+c))^3,x, algorithm="giac")
[Out]
integrate((b*arccosh(d*x + c) + a)^(-3), x)
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maple [A] time = 0.06, size = 207, normalized size = 1.57 \[ \frac {-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) \left (b \,\mathrm {arccosh}\left (d x +c \right )+a -b \right )}{4 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{4 b^{3}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{4 b^{3}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*arccosh(d*x+c))^3,x)
[Out]
1/d*(-1/4*(-(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)+d*x+c)*(b*arccosh(d*x+c)+a-b)/b^2/(b^2*arccosh(d*x+c)^2+2*a*b*arcc
osh(d*x+c)+a^2)+1/4/b^3*exp(a/b)*Ei(1,arccosh(d*x+c)+a/b)-1/4/b*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*a
rccosh(d*x+c))^2-1/4/b^2*(d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))/(a+b*arccosh(d*x+c))-1/4/b^3*exp(-a/b)*Ei(1,-
arccosh(d*x+c)-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(1/(a+b*arccosh(d*x+c))^3,x, algorithm="maxima")
[Out]
-1/2*((a*d^7 + b*d^7)*x^7 + 7*(a*c*d^6 + b*c*d^6)*x^6 + 3*((7*c^2*d^5 - d^5)*a + (7*c^2*d^5 - d^5)*b)*x^5 + 5*
((7*c^3*d^4 - 3*c*d^4)*a + (7*c^3*d^4 - 3*c*d^4)*b)*x^4 + ((a*d^4 + b*d^4)*x^4 + 4*(a*c*d^3 + b*c*d^3)*x^3 + (
6*a*c^2*d^2 + (6*c^2*d^2 - d^2)*b)*x^2 + (c^4 - 1)*a + (c^4 - c^2)*b + 2*(2*a*c^3*d + (2*c^3*d - c*d)*b)*x)*(d
*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + ((35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*a + (35*c^4*d^3 - 30*c^2*d^3 + 3*d^
3)*b)*x^3 + (3*(a*d^5 + b*d^5)*x^5 + 15*(a*c*d^4 + b*c*d^4)*x^4 + (3*(10*c^2*d^3 - d^3)*a + 5*(6*c^2*d^3 - d^3
)*b)*x^3 + 3*((10*c^3*d^2 - 3*c*d^2)*a + 5*(2*c^3*d^2 - c*d^2)*b)*x^2 + 3*(c^5 - c^3)*a + (3*c^5 - 5*c^3 + 2*c
)*b + (3*(5*c^4*d - 3*c^2*d)*a + (15*c^4*d - 15*c^2*d + 2*d)*b)*x)*(d*x + c + 1)*(d*x + c - 1) + 3*((7*c^5*d^2
- 10*c^3*d^2 + 3*c*d^2)*a + (7*c^5*d^2 - 10*c^3*d^2 + 3*c*d^2)*b)*x^2 + (3*(a*d^6 + b*d^6)*x^6 + 18*(a*c*d^5
+ b*c*d^5)*x^5 + (3*(15*c^2*d^4 - 2*d^4)*a + (45*c^2*d^4 - 7*d^4)*b)*x^4 + 4*(3*(5*c^3*d^3 - 2*c*d^3)*a + (15*
c^3*d^3 - 7*c*d^3)*b)*x^3 + ((45*c^4*d^2 - 36*c^2*d^2 + 4*d^2)*a + (45*c^4*d^2 - 42*c^2*d^2 + 5*d^2)*b)*x^2 +
(3*c^6 - 6*c^4 + 4*c^2 - 1)*a + (3*c^6 - 7*c^4 + 5*c^2 - 1)*b + 2*((9*c^5*d - 12*c^3*d + 4*c*d)*a + (9*c^5*d -
14*c^3*d + 5*c*d)*b)*x)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^7 - 3*c^5 + 3*c^3 - c)*a + (c^7 - 3*c^5 + 3*
c^3 - c)*b + ((7*c^6*d - 15*c^4*d + 9*c^2*d - d)*a + (7*c^6*d - 15*c^4*d + 9*c^2*d - d)*b)*x + (b*d^7*x^7 + 7*
b*c*d^6*x^6 + 3*(7*c^2*d^5 - d^5)*b*x^5 + 5*(7*c^3*d^4 - 3*c*d^4)*b*x^4 + (35*c^4*d^3 - 30*c^2*d^3 + 3*d^3)*b*
x^3 + (b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + (c^4 - 1)*b)*(d*x + c + 1)^(3/2)*(d*x + c -
1)^(3/2) + 3*(7*c^5*d^2 - 10*c^3*d^2 + 3*c*d^2)*b*x^2 + 3*(b*d^5*x^5 + 5*b*c*d^4*x^4 + (10*c^2*d^3 - d^3)*b*x
^3 + (10*c^3*d^2 - 3*c*d^2)*b*x^2 + (5*c^4*d - 3*c^2*d)*b*x + (c^5 - c^3)*b)*(d*x + c + 1)*(d*x + c - 1) + (7*
c^6*d - 15*c^4*d + 9*c^2*d - d)*b*x + (3*b*d^6*x^6 + 18*b*c*d^5*x^5 + 3*(15*c^2*d^4 - 2*d^4)*b*x^4 + 12*(5*c^3
*d^3 - 2*c*d^3)*b*x^3 + (45*c^4*d^2 - 36*c^2*d^2 + 4*d^2)*b*x^2 + 2*(9*c^5*d - 12*c^3*d + 4*c*d)*b*x + (3*c^6
- 6*c^4 + 4*c^2 - 1)*b)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + (c^7 - 3*c^5 + 3*c^3 - c)*b)*log(d*x + sqrt(d*x
+ c + 1)*sqrt(d*x + c - 1) + c))/(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 + (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*x + c + 1)^(3/2)*(d*x + c - 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*x + c + 1)*(d*x
+ c - 1) + (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*x + c + 1)^(3/2)*(d*x + c - 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*x + c + 1)*(d*x + c - 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*x + c + 1)
*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)^2 + 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*x + c + 1)*sqrt(d*x + c - 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*x + c + 1)^(3/2)*(d*x + c - 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*x + c + 1)*(d*x + c - 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*x +
c + 1)*sqrt(d*x + c - 1))*log(d*x + sqrt(d*x + c + 1)*sqrt(d*x + c - 1) + c)) + integrate(1/2*(d^8*x^8 + 8*c*
d^7*x^7 + c^8 + 4*(7*c^2*d^6 - d^6)*x^6 - 4*c^6 + 8*(7*c^3*d^5 - 3*c*d^5)*x^5 + (d^4*x^4 + 4*c*d^3*x^3 + 6*c^2
*d^2*x^2 + 4*c^3*d*x + c^4 + 3)*(d*x + c + 1)^2*(d*x + c - 1)^2 + 2*(35*c^4*d^4 - 30*c^2*d^4 + 3*d^4)*x^4 + (4
*d^5*x^5 + 20*c*d^4*x^4 + 4*c^5 + 4*(10*c^2*d^3 - d^3)*x^3 - 4*c^3 + 4*(10*c^3*d^2 - 3*c*d^2)*x^2 + (20*c^4*d
- 12*c^2*d + 3*d)*x + 3*c)*(d*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) + 6*c^4 + 8*(7*c^5*d^3 - 10*c^3*d^3 + 3*c*d
^3)*x^3 + 3*(2*d^6*x^6 + 12*c*d^5*x^5 + 2*c^6 + 2*(15*c^2*d^4 - 2*d^4)*x^4 - 4*c^4 + 8*(5*c^3*d^3 - 2*c*d^3)*x
^3 + (30*c^4*d^2 - 24*c^2*d^2 + d^2)*x^2 + c^2 + 2*(6*c^5*d - 8*c^3*d + c*d)*x + 1)*(d*x + c + 1)*(d*x + c - 1
) + 4*(7*c^6*d^2 - 15*c^4*d^2 + 9*c^2*d^2 - d^2)*x^2 + (4*d^7*x^7 + 28*c*d^6*x^6 + 4*c^7 + 12*(7*c^2*d^5 - d^5
)*x^5 - 12*c^5 + 20*(7*c^3*d^4 - 3*c*d^4)*x^4 + (140*c^4*d^3 - 120*c^2*d^3 + 9*d^3)*x^3 + 9*c^3 + 3*(28*c^5*d^
2 - 40*c^3*d^2 + 9*c*d^2)*x^2 + (28*c^6*d - 60*c^4*d + 27*c^2*d - d)*x - c)*sqrt(d*x + c + 1)*sqrt(d*x + c - 1
) - 4*c^2 + 8*(c^7*d - 3*c^5*d + 3*c^3*d - c*d)*x + 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 + (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*x + c + 1)^2*(d*x + c - 1)^2 + 8*(c
^7*d - 3*c^5*d + 3*c^3*d - c*d)*a*b^2*x + 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*x + c + 1)^(3/2)*(d*x
+ c - 1)^(3/2) + (c^8 - 4*c^6 + 6*c^4 - 4*c^2 + 1)*a*b^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*x + c + 1)*(d*x + c - 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*x + c + 1)*sqrt(d*x + c - 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 + (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*x + c + 1)^2*(d*x + c - 1)^2 + 8*(c^7*d
- 3*c^5*d + 3*c^3*d - c*d)*b^3*x + 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*x + c + 1)^(3/2)*(d*x + c - 1)^(3/2) +
(c^8 - 4*c^6 + 6*c^4 - 4*c^2 + 1)*b^3 + 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*x + c + 1)*(d*x + c - 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*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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a + b*acosh(c + d*x))^3,x)
[Out]
int(1/(a + b*acosh(c + d*x))^3, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*acosh(d*x+c))**3,x)
[Out]
Integral((a + b*acosh(c + d*x))**(-3), x)
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