\(\int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx\) [161]

Optimal result

Integrand size = 14, antiderivative size = 229 \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=-\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {8 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2}+\frac {8 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2} \]

[Out]

4/15/b^2/c^2/(a+b*arccosh(c*x))^(3/2)-8/15*x^2/b^2/(a+b*arccosh(c*x))^(3/2)+8/15*exp(2*a/b)*erf(2^(1/2)*(a+b*a
rccosh(c*x))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/c^2+8/15*erfi(2^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*2
^(1/2)*Pi^(1/2)/b^(7/2)/c^2/exp(2*a/b)-2/5*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(5/2)-32/15*x*
(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^3/c/(a+b*arccosh(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5886, 5951, 5885, 3388, 2211, 2236, 2235, 5893} \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\frac {8 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2}+\frac {8 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2}-\frac {32 x \sqrt {c x-1} \sqrt {c x+1}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {2 x \sqrt {c x-1} \sqrt {c x+1}}{5 b c (a+b \text {arccosh}(c x))^{5/2}} \]

[In]

Int[x/(a + b*ArcCosh[c*x])^(7/2),x]

[Out]

(-2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(5*b*c*(a + b*ArcCosh[c*x])^(5/2)) + 4/(15*b^2*c^2*(a + b*ArcCosh[c*x])^(3
/2)) - (8*x^2)/(15*b^2*(a + b*ArcCosh[c*x])^(3/2)) - (32*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*b^3*c*Sqrt[a + b*
ArcCosh[c*x]]) + (8*E^((2*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2) +
(8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(15*b^(7/2)*c^2*E^((2*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 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 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}-\frac {2 \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{5/2}} \, dx}{5 b c}+\frac {(4 c) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{5/2}} \, dx}{5 b} \\ & = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}+\frac {16 \int \frac {x}{(a+b \text {arccosh}(c x))^{3/2}} \, dx}{15 b^2} \\ & = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {32 \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{15 b^4 c^2} \\ & = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {16 \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{15 b^4 c^2}+\frac {16 \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{15 b^4 c^2} \\ & = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {32 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{15 b^4 c^2}+\frac {32 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{15 b^4 c^2} \\ & = -\frac {2 x \sqrt {-1+c x} \sqrt {1+c x}}{5 b c (a+b \text {arccosh}(c x))^{5/2}}+\frac {4}{15 b^2 c^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {8 x^2}{15 b^2 (a+b \text {arccosh}(c x))^{3/2}}-\frac {32 x \sqrt {-1+c x} \sqrt {1+c x}}{15 b^3 c \sqrt {a+b \text {arccosh}(c x)}}+\frac {8 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2}+\frac {8 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^2} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {7}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/(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)

Sympy [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \text {arccosh}(c x))^{7/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{7/2}} \,d x \]

[In]

int(x/(a + b*acosh(c*x))^(7/2),x)

[Out]

int(x/(a + b*acosh(c*x))^(7/2), x)