\(\int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx\) [566]

Optimal result

Integrand size = 20, antiderivative size = 358 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=-\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \]

[Out]

d*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c+1/4*e*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2
)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3+d*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c/exp(a/b)+1/4*e*erf
i((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/exp(a/b)+1/4*e*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^(3/2)/c^3+1/4*e*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*P
i^(1/2)/b^(3/2)/c^3/exp(3*a/b)-2*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)-2*e*x^2*(c*x-1)^(1
/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5909, 5880, 5953, 3388, 2211, 2236, 2235, 5885} \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {\sqrt {\pi } e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {\pi } e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {3 \pi } e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}-\frac {2 d \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {c x-1} \sqrt {c x+1}}{b c \sqrt {a+b \text {arccosh}(c x)}} \]

[In]

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

[Out]

(-2*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (2*e*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b
*c*Sqrt[a + b*ArcCosh[c*x]]) + (d*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(3/2)*c) + (e*E^(
a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3) + (e*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*
Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^3) + (d*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(b^(
3/2)*c*E^(a/b)) + (e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*b^(3/2)*c^3*E^(a/b)) + (e*Sqrt[3*Pi]*
Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*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 5909

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

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*} \text {integral}& = \int \left (\frac {d}{(a+b \text {arccosh}(c x))^{3/2}}+\frac {e x^2}{(a+b \text {arccosh}(c x))^{3/2}}\right ) \, dx \\ & = d \int \frac {1}{(a+b \text {arccosh}(c x))^{3/2}} \, dx+e \int \frac {x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx \\ & = -\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {(2 c d) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}} \, dx}{b}-\frac {(2 e) \text {Subst}\left (\int \left (-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3} \\ & = -\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c}+\frac {e \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{2 b^2 c^3} \\ & = -\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {d \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c}+\frac {d \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c}+\frac {e \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3}+\frac {e \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3} \\ & = -\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {(2 d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c}+\frac {(2 d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{b^2 c}+\frac {e \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c^3}+\frac {e \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c^3}+\frac {(3 e) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arccosh}(c x)}\right )}{2 b^2 c^3} \\ & = -\frac {2 d \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}-\frac {2 e x^2 \sqrt {-1+c x} \sqrt {1+c x}}{b c \sqrt {a+b \text {arccosh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{b^{3/2} c}+\frac {e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3}+\frac {e e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 1.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.75 \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\frac {e^{-\frac {3 a}{b}} \left (-\left (\left (4 c^2 d+e\right ) e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c x)\right )\right )+\sqrt {3} e \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\left (4 c^2 d+e\right ) e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )-e^{\frac {3 a}{b}} \left (2 \left (4 c^2 d+e\right ) \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+\sqrt {3} e e^{\frac {3 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+2 e \sinh (3 \text {arccosh}(c x))\right )\right )}{4 b c^3 \sqrt {a+b \text {arccosh}(c x)}} \]

[In]

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

[Out]

(-((4*c^2*d + e)*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]]) + Sqrt[3]*e*Sqrt[-((a +
b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + (4*c^2*d + e)*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[
c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - E^((3*a)/b)*(2*(4*c^2*d + e)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 +
c*x) + Sqrt[3]*e*E^((3*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 2*e*Sinh[3*Arc
Cosh[c*x]]))/(4*b*c^3*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])

Maple [F]

\[\int \frac {e \,x^{2}+d}{\left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {d + e x^{2}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((d + e*x**2)/(a + b*acosh(c*x))**(3/2), x)

Maxima [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((e*x^2 + d)/(b*arccosh(c*x) + a)^(3/2), x)

Giac [F]

\[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int { \frac {e x^{2} + d}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((e*x^2 + d)/(b*arccosh(c*x) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x^2}{(a+b \text {arccosh}(c x))^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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