3.4.71 \(\int \frac {x (d-c^2 d x^2)}{(a+b \cosh ^{-1}(c x))^{3/2}} \, dx\) [371]

Optimal. Leaf size=241 \[ \frac {2 d x (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2} \]

[Out]

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

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Rubi [A]
time = 0.69, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5942, 5907, 3393, 3388, 2211, 2236, 2235, 5953, 5556} \begin {gather*} -\frac {\sqrt {\pi } d e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {\sqrt {\pi } d e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {2 d x (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) - (d*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[
a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2) + (d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x
]])/Sqrt[b]])/(2*b^(3/2)*c^2) - (d*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*b^(3/2)*c^2*E^((4*a
)/b)) + (d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(2*b^(3/2)*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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5907

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(1/(b*c))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subst[Int[x^n*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[2*p, 0]

Rule 5942

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (
Dist[f*(m/(b*c*(n + 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] - Dist[c*((m + 2*p + 1)/(b*f*(n + 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, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0
] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

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 \left (d-c^2 d x^2\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {(2 d) \int \frac {\sqrt {-1+c x} \sqrt {1+c x}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac {(8 c d) \int \frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {(2 d) \text {Subst}\left (\int \frac {\sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}-\frac {(8 d) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {(2 d) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}-\frac {(8 d) \text {Subst}\left (\int \left (-\frac {1}{8 \sqrt {a+b x}}+\frac {\cosh (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {d \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}-\frac {d \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {d \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}+\frac {d \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}-\frac {d \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^2}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}-\frac {d \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 d x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}\\ \end {align*}

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Mathematica [A]
time = 2.85, size = 331, normalized size = 1.37 \begin {gather*} \frac {d e^{-\frac {4 a}{b}} \left (2 e^{\frac {6 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )+\frac {\sqrt {b} \left (-\sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 a}{b}} \left (8 c x \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3+\sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )\right )}{\sqrt {a+b \cosh ^{-1}(c x)}}\right )}{4 b^{3/2} c^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(2*E^((6*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + 2*E^((2*a)/b)*Sqrt[2*Pi]*Erfi[(
Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + (Sqrt[b]*(-(Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b
*ArcCosh[c*x]))/b]) - Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))
/b] + E^((4*a)/b)*(8*c*x*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3 + Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x
]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[
c*x]))/b])))/Sqrt[a + b*ArcCosh[c*x]]))/(4*b^(3/2)*c^2*E^((4*a)/b))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x \left (-c^{2} d \,x^{2}+d \right )}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - d \left (\int \left (- \frac {x}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{2} x^{3}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(Integral(-x/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**2*x**3/(a*s
qrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst 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\,\left (d-c^2\,d\,x^2\right )}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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