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

Optimal. Leaf size=243 \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d^3}-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \cosh ^{-1}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {11 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^6 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )} \]

[Out]

1/3*(a+b*arccosh(c*x))/c^6/d/(-c^2*d*x^2+d)^(3/2)-2*(a+b*arccosh(c*x))/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-(a+b*arcco
sh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3+b*x*(-c^2*d*x^2+d)^(1/2)/c^5/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/6*b*x*(-c
^2*d*x^2+d)^(1/2)/c^5/d^3/(-c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11/6*b*arctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^
6/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 0.44, antiderivative size = 280, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 98, 21, 74, 5733, 12, 1157, 388, 206} \[ \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{6 c^5 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {11 b \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c^5*d^2*Sqrt[d - c^2*d*x^2])) + (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*c^
5*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (4*x^2*(a + b*ArcCosh[c*x]))/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (x^4
*(a + b*ArcCosh[c*x]))/(3*c^2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (8*(1 - c*x)*(1 + c*x)*(a + b*Arc
Cosh[c*x]))/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(6*c^6*d^2*Sqrt
[d - c^2*d*x^2])

Rule 12

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

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 74

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 98

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 388

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

Rule 1157

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, -Simp[(R*x*(d + e*x^2)^(q + 1))/(2*d*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*
ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && N
eQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5798

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

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\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 )}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {8-12 c^2 x^2+3 c^4 x^4}{3 c^6 \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {4 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{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 {8-12 c^2 x^2+3 c^4 x^4}{\left (1-c^2 x^2\right )^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c^5 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 )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{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 {-17+6 c^2 x^2}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c^5 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 )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (11 b \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{6 c^5 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 )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{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 )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b \sqrt {-1+c x} \sqrt {1+c x} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 167, normalized size = 0.69 \[ \frac {6 a c^4 x^4-24 a c^2 x^2+16 a-6 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}-11 b \sqrt {c x-1} \sqrt {c x+1} \left (c^2 x^2-1\right ) \tanh ^{-1}(c x)+2 b \left (3 c^4 x^4-12 c^2 x^2+8\right ) \cosh ^{-1}(c x)+5 b c x \sqrt {c x-1} \sqrt {c x+1}}{6 c^6 d^2 \left (c^2 x^2-1\right ) \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a - 24*a*c^2*x^2 + 6*a*c^4*x^4 + 5*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1
+ c*x] + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] - 11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*ArcT
anh[c*x])/(6*c^6*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])

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fricas [A]  time = 0.83, size = 529, normalized size = 2.18 \[ \left [-\frac {8 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} + 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} \sqrt {-d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) - 4 \, {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(8*(3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 11*(b*c^4*x^4
- 2*b*c^2*x^2 + b)*sqrt(-d)*log(-(c^6*d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x^2 + 4*(c^3*x^3 + c*x)*sqrt(-c^2*d*x^2 +
d)*sqrt(c^2*x^2 - 1)*sqrt(-d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) - 4*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c
^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 8*(3*a*c^4*x^4 - 12*a*c^2*x^2 + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x^4 - 2
*c^8*d^3*x^2 + c^6*d^3), 1/12*(11*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(c^2
*x^2 - 1)*c*sqrt(d)*x/(c^4*d*x^4 - d)) - 4*(3*b*c^4*x^4 - 12*b*c^2*x^2 + 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + s
qrt(c^2*x^2 - 1)) + 2*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*(3*a*c^4*x^4 - 12*a*c
^2*x^2 + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3)]

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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(x^5*(a+b*arccosh(c*x))/(-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,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 0.65, size = 466, normalized size = 1.92 \[ -\frac {a \,x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {4 a \,x^{2}}{c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8 a}{3 c^{6} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2}}{c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x}{c^{5} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right ) x^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x +1}\, \sqrt {c x -1}\, x}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{5}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{6}}+\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}-1\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4*a/c^4*x^2/d/(-c^2*d*x^2+d)^(3/2)-8/3*a/c^6/d/(-c^2*d*x^2+d)^(3/2)-b*(-d*(c
^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arccosh(c*x)*x^2+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+b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)+2*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c
^2*x^2-1)^2/c^4*arccosh(c*x)*x^2+1/6*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-5/3*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6*arccosh(c*x)+11/6*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))-11/6*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 \[ -\frac {1}{9} \, b {\left (\frac {\frac {{\left (9 \, c^{4} \sqrt {d} x^{4} - 8 \, \sqrt {d}\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{\sqrt {-c x + 1}} - \frac {3 \, {\left (3 \, c^{5} \sqrt {d} x^{5} - 12 \, c^{3} \sqrt {d} x^{3} + 8 \, c \sqrt {d} x + {\left (3 \, c^{4} \sqrt {d} x^{4} - 12 \, c^{2} \sqrt {d} x^{2} + 8 \, \sqrt {d}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {-c x + 1}}}{{\left (c^{8} d^{3} x^{2} - c^{6} d^{3}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} + {\left (c^{9} d^{3} x^{3} - c^{7} d^{3} x\right )} \sqrt {c x + 1}} + 9 \, \int \frac {9 \, c^{7} \sqrt {d} x^{7} - 45 \, c^{5} \sqrt {d} x^{5} + 60 \, c^{3} \sqrt {d} x^{3} - 24 \, c \sqrt {d} x + {\left (9 \, c^{6} \sqrt {d} x^{6} - 54 \, c^{4} \sqrt {d} x^{4} + 60 \, c^{2} \sqrt {d} x^{2} - 16 \, \sqrt {d}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{9 \, \sqrt {-c x + 1} {\left ({\left (c^{9} d^{3} x^{4} - 2 \, c^{7} d^{3} x^{2} + c^{5} d^{3}\right )} e^{\left (\frac {3}{2} \, \log \left (c x + 1\right ) + \log \left (c x - 1\right )\right )} + 2 \, {\left (c^{10} d^{3} x^{5} - 2 \, c^{8} d^{3} x^{3} + c^{6} d^{3} x\right )} e^{\left (\log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )} + {\left (c^{11} d^{3} x^{6} - 2 \, c^{9} d^{3} x^{4} + c^{7} d^{3} x^{2}\right )} \sqrt {c x + 1}\right )}}\,{d x}\right )} - \frac {1}{3} \, a {\left (\frac {3 \, x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {12 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d} + \frac {8}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{6} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*b*(((9*c^4*sqrt(d)*x^4 - 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*x - 1)/sqrt(-c*x + 1) - 3*(3*c^5*sqrt(d)*x^5 - 1
2*c^3*sqrt(d)*x^3 + 8*c*sqrt(d)*x + (3*c^4*sqrt(d)*x^4 - 12*c^2*sqrt(d)*x^2 + 8*sqrt(d))*sqrt(c*x + 1)*sqrt(c*
x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c*x + 1))/((c^8*d^3*x^2 - c^6*d^3)*(c*x + 1)*sqrt(c*x - 1
) + (c^9*d^3*x^3 - c^7*d^3*x)*sqrt(c*x + 1)) + 9*integrate(1/9*(9*c^7*sqrt(d)*x^7 - 45*c^5*sqrt(d)*x^5 + 60*c^
3*sqrt(d)*x^3 - 24*c*sqrt(d)*x + (9*c^6*sqrt(d)*x^6 - 54*c^4*sqrt(d)*x^4 + 60*c^2*sqrt(d)*x^2 - 16*sqrt(d))*e^
(1/2*log(c*x + 1) + 1/2*log(c*x - 1)))/(sqrt(-c*x + 1)*((c^9*d^3*x^4 - 2*c^7*d^3*x^2 + c^5*d^3)*e^(3/2*log(c*x
+ 1) + log(c*x - 1)) + 2*(c^10*d^3*x^5 - 2*c^8*d^3*x^3 + c^6*d^3*x)*e^(log(c*x + 1) + 1/2*log(c*x - 1)) + (c^
11*d^3*x^6 - 2*c^9*d^3*x^4 + c^7*d^3*x^2)*sqrt(c*x + 1))), x)) - 1/3*a*(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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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