3.4.69 \(\int \frac {x^3 (d-c^2 d x^2)}{(a+b \cosh ^{-1}(c x))^{3/2}} \, dx\) [369]

Optimal. Leaf size=259 \[ \frac {2 d x^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {3 d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}-\frac {d e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {3 d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}-\frac {d e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4} \]

[Out]

3/32*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^4+3/32*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^4/exp(2*a/b)-1/32*d*exp(6*a/b)*erf(6^(1/2)*(a+
b*arccosh(c*x))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4-1/32*d*erfi(6^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/
2))*6^(1/2)*Pi^(1/2)/b^(3/2)/c^4/exp(6*a/b)+2*d*x^3*(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 = 1.11, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5942, 5953, 5556, 3388, 2211, 2236, 2235} \begin {gather*} \frac {3 \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 )}{16 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d e^{\frac {6 a}{b}} \text {Erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {3 \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 )}{16 b^{3/2} c^4}-\frac {\sqrt {\frac {3 \pi }{2}} d e^{-\frac {6 a}{b}} \text {Erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {2 d x^3 (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^3*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]

[Out]

(2*d*x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (3*d*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(S
qrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) - (d*E^((6*a)/b)*Sqrt[(3*Pi)/2]*Erf[(Sqrt[6]*Sqrt[
a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b^(3/2)*c^4) + (3*d*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt
[b]])/(16*b^(3/2)*c^4*E^((2*a)/b)) - (d*Sqrt[(3*Pi)/2]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(16*b
^(3/2)*c^4*E^((6*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 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 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^3 \left (d-c^2 d x^2\right )}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d x^3 \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 {(6 d) \int \frac {x^2 \sqrt {-1+c x} \sqrt {1+c x}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b c}-\frac {(12 c d) \int \frac {x^4 \sqrt {-1+c x} \sqrt {1+c x}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d x^3 \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 {(6 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^4}-\frac {(12 d) \text {Subst}\left (\int \frac {\cosh ^4(x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {2 d x^3 \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 {(6 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^4}-\frac {(12 d) \text {Subst}\left (\int \left (-\frac {1}{16 \sqrt {a+b x}}-\frac {\cosh (2 x)}{32 \sqrt {a+b x}}+\frac {\cosh (4 x)}{16 \sqrt {a+b x}}+\frac {\cosh (6 x)}{32 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {2 d x^3 \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 {(3 d) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}-\frac {(3 d) \text {Subst}\left (\int \frac {\cosh (6 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac {2 d x^3 \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 {(3 d) \text {Subst}\left (\int \frac {e^{-6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}-\frac {(3 d) \text {Subst}\left (\int \frac {e^{6 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^4}\\ &=-\frac {2 d x^3 \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 {(3 d) \text {Subst}\left (\int e^{\frac {6 a}{b}-\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}+\frac {(3 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 )}{8 b^2 c^4}+\frac {(3 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 )}{8 b^2 c^4}-\frac {(3 d) \text {Subst}\left (\int e^{-\frac {6 a}{b}+\frac {6 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{8 b^2 c^4}\\ &=-\frac {2 d x^3 \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 {3 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 )}{16 b^{3/2} c^4}-\frac {d e^{\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}+\frac {3 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 )}{16 b^{3/2} c^4}-\frac {d e^{-\frac {6 a}{b}} \sqrt {\frac {3 \pi }{2}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^4}\\ \end {align*}

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Mathematica [A]
time = 1.86, size = 300, normalized size = 1.16 \begin {gather*} \frac {d e^{-\frac {6 a}{b}} \left (-\sqrt {6} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt {2} e^{\frac {4 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 {6 a}{b}} \left (-64 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}}-64 c^4 x^4 \sqrt {\frac {-1+c x}{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 )+\sqrt {6} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+10 \sinh \left (2 \cosh ^{-1}(c x)\right )+8 \sinh \left (4 \cosh ^{-1}(c x)\right )+2 \sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{32 b c^4 \sqrt {a+b \cosh ^{-1}(c x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d*(-(Sqrt[6]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b]) + 3*Sqrt[2]*E^((4*a)/b)
*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + E^((6*a)/b)*(-64*c^3*x^3*Sqrt[(-1 +
c*x)/(1 + c*x)] - 64*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamm
a[1/2, (2*(a + b*ArcCosh[c*x]))/b] + Sqrt[6]*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh
[c*x]))/b] + 10*Sinh[2*ArcCosh[c*x]] + 8*Sinh[4*ArcCosh[c*x]] + 2*Sinh[6*ArcCosh[c*x]])))/(32*b*c^4*E^((6*a)/b
)*Sqrt[a + b*ArcCosh[c*x]])

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*(-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^3*(-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^3/(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^3*(-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^{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 )}}\right )\, dx + \int \frac {c^{2} x^{5}}{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**3*(-c**2*d*x**2+d)/(a+b*acosh(c*x))**(3/2),x)

[Out]

-d*(Integral(-x**3/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**2*x**5/(
a*sqrt(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^3*(-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^3\,\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^3*(d - c^2*d*x^2))/(a + b*acosh(c*x))^(3/2),x)

[Out]

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

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