3.3.14 \(\int \frac {x^5 (a+b \cosh ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [214]

Optimal. Leaf size=568 \[ -\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

1/3*x^4*(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*x^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-7/3*b^2*(-c*x
+1)*(c*x+1)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-4/3*x^2*(a+b*arccosh(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*a*b*x*
(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*b^2*x*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
^5/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*b*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/
2)+1/3*b*x^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)-22/3*b*(
a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^
(1/2)-11/3*b^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^
(1/2)+11/3*b^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(
1/2)-8/3*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {5934, 5914, 5879, 75, 5912, 5938, 5903, 4267, 2317, 2438, 100, 21} \begin {gather*} \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {c x-1} \sqrt {c x+1}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {c x-1} \sqrt {c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

-1/3*(b^2*x^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2
*d*x^2]) - (7*b^2*(1 - c*x)*(1 + c*x))/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (16*b^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]*ArcCosh[c*x])/(3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (11*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(
3*c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2
*x^2)*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcC
osh[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^6*d^3) - (2
2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])
- (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*b^2
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 75

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

Rule 100

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5879

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

Rule 5903

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-(c*d)^(-1), Subst[Int[
(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 5934

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

Rule 5938

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

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (8 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x (-2-2 c x)}{\sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {3 b^2 (1-c x) (1+c x)}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 3.62, size = 437, normalized size = 0.77 \begin {gather*} -\frac {8 a^2 \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 a b \left (25 \cosh ^{-1}(c x)-36 \cosh ^{-1}(c x) \cosh \left (2 \cosh ^{-1}(c x)\right )+3 \cosh ^{-1}(c x) \cosh \left (4 \cosh ^{-1}(c x)\right )-33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )+4 \sinh \left (2 \cosh ^{-1}(c x)\right )+11 \log \left (\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+b^2 \left (22+25 \cosh ^{-1}(c x)^2-4 \left (7+9 \cosh ^{-1}(c x)^2\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )+3 \left (2+\cosh ^{-1}(c x)^2\right ) \cosh \left (4 \cosh ^{-1}(c x)\right )-66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )+66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1+e^{-\cosh ^{-1}(c x)}\right )+88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )+8 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )+22 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-22 \cosh ^{-1}(c x) \log \left (1+e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-6 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

-1/24*(8*a^2*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*a*b*(25*ArcCosh[c*x] - 36*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + 3*
ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 33*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + 4*Sinh
[2*ArcCosh[c*x]] + 11*Log[Tanh[ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] - 3*Sinh[4*ArcCosh[c*x]]) + b^2*(22 + 25*
ArcCosh[c*x]^2 - 4*(7 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] + 3*(2 + ArcCosh[c*x]^2)*Cosh[4*ArcCosh[c*x]] -
66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 66*Sqrt[(-1 + c*x)/(1 + c*x
)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2,
-E^(-ArcCosh[c*x])] - 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 8*ArcCosh[c*
x]*Sinh[2*ArcCosh[c*x]] + 22*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 22*ArcCosh[c*x]*Lo
g[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(c^6*d*(d - c^2*d*x^2)^(
3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1211\) vs. \(2(531)=1062\).
time = 5.54, size = 1212, normalized size = 2.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+2*
b^2*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)-b^2*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arccosh(c*x)^2*x
^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^4*arccosh(c*x)^2*x^2-11/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-
1)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+11/3*b^2*(-d*(c^
2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+11
/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x
+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^5*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x-11/3*b^2*(-d*
(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-
2*b^2*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*x^2+b^2*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*
x)^2-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^4
*x^2-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6*arccosh(c*x)^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/
(c^2*x^2-1)*arccosh(c*x)*x^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+2*
a*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)+4*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^4
*arccosh(c*x)*x^2+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x-10/3*a*b*
(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6*arccosh(c*x)+11/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-11/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2
)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)

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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(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*a^2*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^(3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(
3/2)*c^6*d)) - 1/3*(3*b^2*c^4*sqrt(d)*x^4 - 12*b^2*c^2*sqrt(d)*x^2 + 8*b^2*sqrt(d))*sqrt(c*x + 1)*sqrt(-c*x +
1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) - integrate(2/3*((12*b^2*
c^3*x^3 + 3*(a*b*c^5 - b^2*c^5)*x^5 - 8*b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (15*b^2*c^4*x^4 + 3*(a*b*c^6 - b^2*
c^6)*x^6 - 20*b^2*c^2*x^2 + 8*b^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^12*
d^(5/2)*x^7 - 3*c^10*d^(5/2)*x^5 + 3*c^8*d^(5/2)*x^3 - c^6*d^(5/2)*x + (c^11*d^(5/2)*x^6 - 3*c^9*d^(5/2)*x^4 +
3*c^7*d^(5/2)*x^2 - c^5*d^(5/2))*sqrt(c*x + 1)*sqrt(c*x - 1)), 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(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(b^2*x^5*arccosh(c*x)^2 + 2*a*b*x^5*arccosh(c*x) + a^2*x^5)*sqrt(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^
4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**5*(a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)

[Out]

int((x^5*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)

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