3.2.60 \(\int \frac {x^2}{(a+b \cosh ^{-1}(c x))^{7/2}} \, dx\) [160]

Optimal. Leaf size=361 \[ -\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \]

[Out]

8/15*x/b^2/c^2/(a+b*arccosh(c*x))^(3/2)-4/5*x^3/b^2/(a+b*arccosh(c*x))^(3/2)+1/15*exp(a/b)*erf((a+b*arccosh(c*
x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c^3+1/15*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c^3/exp(a
/b)+3/5*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/c^3+3/5*erfi(3^(1/2)
*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(7/2)/c^3/exp(3*a/b)-2/5*x^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/b/c/(a+b*arccosh(c*x))^(5/2)+16/15*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^3/c^3/(a+b*arccosh(c*x))^(1/2)-24/5*x^2*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/b^3/c/(a+b*arccosh(c*x))^(1/2)

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Rubi [A]
time = 1.08, antiderivative size = 361, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5886, 5951, 5885, 3388, 2211, 2236, 2235, 5880, 5953} \begin {gather*} \frac {\sqrt {\pi } e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {\sqrt {\pi } e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 \sqrt {3 \pi } e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {16 \sqrt {c x-1} \sqrt {c x+1}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {2 x^2 \sqrt {c x-1} \sqrt {c x+1}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*ArcCosh[c*x])^(7/2),x]

[Out]

(-2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + (8*x)/(15*b^2*c^2*(a + b*ArcCosh[c*
x])^(3/2)) - (4*x^3)/(5*b^2*(a + b*ArcCosh[c*x])^(3/2)) + (16*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c^3*Sqrt[a
+ b*ArcCosh[c*x]]) - (24*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b^3*c*Sqrt[a + b*ArcCosh[c*x]]) + (E^(a/b)*Sqrt
[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c^3) + (3*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a +
b*ArcCosh[c*x]])/Sqrt[b]])/(5*b^(7/2)*c^3) + (Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(15*b^(7/2)*c^
3*E^(a/b)) + (3*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(5*b^(7/2)*c^3*E^((3*a)/b))

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 5880

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]*Sqrt[
-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && LtQ[n, -1]

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 5886

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCosh
[c*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCosh[c
*x])^(n + 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5951

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*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], 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] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{7/2}} \, dx &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}-\frac {4 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b c}+\frac {(6 c) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \, dx}{5 b}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {12 \int \frac {x^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{5 b^2}-\frac {8 \int \frac {1}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx}{15 b^2 c^2}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}-\frac {16 \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{15 b^3 c}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {6 \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {18 \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {8 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}-\frac {8 \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{15 b^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {9 \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}+\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{5 b^3 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {16 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^3}-\frac {16 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{15 b^4 c^3}+\frac {6 \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {6 \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {18 \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}+\frac {18 \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{5 b^4 c^3}\\ &=-\frac {2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b c \left (a+b \cosh ^{-1}(c x)\right )^{5/2}}+\frac {8 x}{15 b^2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}-\frac {4 x^3}{5 b^2 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}+\frac {16 \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c^3 \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {24 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{5 b^3 c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}+\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}\\ \end {align*}

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Mathematica [A]
time = 1.74, size = 394, normalized size = 1.09 \begin {gather*} \frac {-6 b^2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-2 e^{-\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \left (-2 a+b-2 b \cosh ^{-1}(c x)+2 e^{\frac {a}{b}+\cosh ^{-1}(c x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 e^{-\frac {a}{b}} \left (a+b \cosh ^{-1}(c x)\right ) \left (e^{\frac {a}{b}+\cosh ^{-1}(c x)} \left (2 a+b+2 b \cosh ^{-1}(c x)\right )+2 b \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )-3 \left (a+b \cosh ^{-1}(c x)\right ) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+2 e^{-3 \cosh ^{-1}(c x)} \left (b+6 a \left (-1+e^{6 \cosh ^{-1}(c x)}\right )-6 b \cosh ^{-1}(c x)+b e^{6 \cosh ^{-1}(c x)} \left (1+6 \cosh ^{-1}(c x)\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \left (a+b \cosh ^{-1}(c x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )\right )-6 b^2 \sinh \left (3 \cosh ^{-1}(c x)\right )}{60 b^3 c^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/(a + b*ArcCosh[c*x])^(7/2),x]

[Out]

(-6*b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - (2*(a + b*ArcCosh[c*x])*(-2*a + b - 2*b*ArcCosh[c*x] + 2*E^(a/b
+ ArcCosh[c*x])*Sqrt[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, a/b + ArcCosh[c*x]]))/E^ArcCosh[c*x]
- (2*(a + b*ArcCosh[c*x])*(E^(a/b + ArcCosh[c*x])*(2*a + b + 2*b*ArcCosh[c*x]) + 2*b*(-((a + b*ArcCosh[c*x])/
b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)]))/E^(a/b) - 3*(a + b*ArcCosh[c*x])*((12*Sqrt[3]*b*(-((a + b*Ar
cCosh[c*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b])/E^((3*a)/b) + (2*(b + 6*a*(-1 + E^(6*ArcCosh[c*
x])) - 6*b*ArcCosh[c*x] + b*E^(6*ArcCosh[c*x])*(1 + 6*ArcCosh[c*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcCosh[c*x]))*Sqr
t[a/b + ArcCosh[c*x]]*(a + b*ArcCosh[c*x])*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b]))/E^(3*ArcCosh[c*x])) - 6*b^
2*Sinh[3*ArcCosh[c*x]])/(60*b^3*c^3*(a + b*ArcCosh[c*x])^(5/2))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*arccosh(c*x))^(7/2),x)

[Out]

int(x^2/(a+b*arccosh(c*x))^(7/2),x)

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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^2/(a+b*arccosh(c*x))^(7/2),x, algorithm="maxima")

[Out]

integrate(x^2/(b*arccosh(c*x) + a)^(7/2), x)

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Fricas [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^2/(a+b*arccosh(c*x))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*acosh(c*x))**(7/2),x)

[Out]

Integral(x**2/(a + b*acosh(c*x))**(7/2), 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(x^2/(a+b*arccosh(c*x))^(7/2),x, algorithm="giac")

[Out]

integrate(x^2/(b*arccosh(c*x) + a)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*acosh(c*x))^(7/2),x)

[Out]

int(x^2/(a + b*acosh(c*x))^(7/2), x)

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