3.23 \(\int (d-c^2 d x^2)^3 (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=191 \[ -\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c}-\frac {6 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c}+\frac {8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{105 c}-\frac {16 b d^3 \sqrt {c x-1} \sqrt {c x+1}}{35 c} \]

[Out]

8/105*b*d^3*(c*x-1)^(3/2)*(c*x+1)^(3/2)/c-6/175*b*d^3*(c*x-1)^(5/2)*(c*x+1)^(5/2)/c+1/49*b*d^3*(c*x-1)^(7/2)*(
c*x+1)^(7/2)/c+d^3*x*(a+b*arccosh(c*x))-c^2*d^3*x^3*(a+b*arccosh(c*x))+3/5*c^4*d^3*x^5*(a+b*arccosh(c*x))-1/7*
c^6*d^3*x^7*(a+b*arccosh(c*x))-16/35*b*d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

________________________________________________________________________________________

Rubi [A]  time = 0.26, antiderivative size = 237, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {194, 5680, 12, 1610, 1799, 1850} \[ -\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+\frac {b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

(16*b*d^3*(1 - c^2*x^2))/(35*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(105*c*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]) + (6*b*d^3*(1 - c^2*x^2)^3)/(175*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)^4)/(49
*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + d^3*x*(a + b*ArcCosh[c*x]) - c^2*d^3*x^3*(a + b*ArcCosh[c*x]) + (3*c^4*d^3*
x^5*(a + b*ArcCosh[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcCosh[c*x]))/7

Rule 12

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 5680

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{35} \left (b c d^3\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c d^3 \sqrt {-1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {16}{\sqrt {-1+c^2 x}}-8 \sqrt {-1+c^2 x}+6 \left (-1+c^2 x\right )^{3/2}-5 \left (-1+c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {16 b d^3 \left (1-c^2 x^2\right )}{35 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {8 b d^3 \left (1-c^2 x^2\right )^2}{105 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {6 b d^3 \left (1-c^2 x^2\right )^3}{175 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^3 \left (1-c^2 x^2\right )^4}{49 c \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} c^4 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{7} c^6 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.27, size = 123, normalized size = 0.64 \[ -\frac {d^3 \left (105 a c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+b \sqrt {c x-1} \sqrt {c x+1} \left (-75 c^6 x^6+351 c^4 x^4-757 c^2 x^2+2161\right )+105 b c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right ) \cosh ^{-1}(c x)\right )}{3675 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]

[Out]

-1/3675*(d^3*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2161 - 757*c^2*x^2 + 351*c^4*x^4 - 75*c^6*x^6) + 105*a*c*x*(-35
+ 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCosh[c*x]))/
c

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 169, normalized size = 0.88 \[ -\frac {525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \, {\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 3675*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 -
21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 35*b*c*d^3*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (75*b*c^6*d^3*x^6 - 351*b*
c^4*d^3*x^4 + 757*b*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(c^2*x^2 - 1))/c

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.01, size = 132, normalized size = 0.69 \[ \frac {-d^{3} a \left (\frac {1}{7} c^{7} x^{7}-\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}-c x \right )-d^{3} b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{7} x^{7}}{7}-\frac {3 \,\mathrm {arccosh}\left (c x \right ) c^{5} x^{5}}{5}+c^{3} x^{3} \mathrm {arccosh}\left (c x \right )-c x \,\mathrm {arccosh}\left (c x \right )-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} x^{6}-351 c^{4} x^{4}+757 c^{2} x^{2}-2161\right )}{3675}\right )}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x)

[Out]

1/c*(-d^3*a*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b*(1/7*arccosh(c*x)*c^7*x^7-3/5*arccosh(c*x)*c^5*x^5+c^3
*x^3*arccosh(c*x)-c*x*arccosh(c*x)-1/3675*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*x^6-351*c^4*x^4+757*c^2*x^2-2161
)))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 302, normalized size = 1.58 \[ -\frac {1}{7} \, a c^{6} d^{3} x^{7} + \frac {3}{5} \, a c^{4} d^{3} x^{5} - \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b c^{2} d^{3} + a d^{3} x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^3*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^
2*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^6*d^3 + 1/25*(15*x^5*arcco
sh(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^4*d^3 -
a*c^2*d^3*x^3 - 1/3*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*c^2*d^3
+ a*d^3*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3,x)

[Out]

int((a + b*acosh(c*x))*(d - c^2*d*x^2)^3, x)

________________________________________________________________________________________

sympy [A]  time = 6.32, size = 228, normalized size = 1.19 \[ \begin {cases} - \frac {a c^{6} d^{3} x^{7}}{7} + \frac {3 a c^{4} d^{3} x^{5}}{5} - a c^{2} d^{3} x^{3} + a d^{3} x - \frac {b c^{6} d^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} + \frac {b c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{49} + \frac {3 b c^{4} d^{3} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {117 b c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{1225} - b c^{2} d^{3} x^{3} \operatorname {acosh}{\left (c x \right )} + \frac {757 b c d^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{3675} + b d^{3} x \operatorname {acosh}{\left (c x \right )} - \frac {2161 b d^{3} \sqrt {c^{2} x^{2} - 1}}{3675 c} & \text {for}\: c \neq 0 \\d^{3} x \left (a + \frac {i \pi b}{2}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**3*(a+b*acosh(c*x)),x)

[Out]

Piecewise((-a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 - a*c**2*d**3*x**3 + a*d**3*x - b*c**6*d**3*x**7*acosh(c
*x)/7 + b*c**5*d**3*x**6*sqrt(c**2*x**2 - 1)/49 + 3*b*c**4*d**3*x**5*acosh(c*x)/5 - 117*b*c**3*d**3*x**4*sqrt(
c**2*x**2 - 1)/1225 - b*c**2*d**3*x**3*acosh(c*x) + 757*b*c*d**3*x**2*sqrt(c**2*x**2 - 1)/3675 + b*d**3*x*acos
h(c*x) - 2161*b*d**3*sqrt(c**2*x**2 - 1)/(3675*c), Ne(c, 0)), (d**3*x*(a + I*pi*b/2), True))

________________________________________________________________________________________