3.104 \(\int \frac{x^5 (a+b \cosh ^{-1}(c x))}{\sqrt{d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=236 \[ -\frac{x^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4 d}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 c^6 d}-\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{25 c \sqrt{d-c^2 d x^2}}-\frac{4 b x^3 \sqrt{c x-1} \sqrt{c x+1}}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{8 b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}} \]

[Out]

(-8*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (4*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4
5*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(25*c*Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x]))/(15*c^6*d) - (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*c^4*d) - (x^4*
Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*c^2*d)

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Rubi [A]  time = 0.708788, antiderivative size = 260, normalized size of antiderivative = 1.1, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {5798, 5759, 5718, 8, 30} \[ -\frac{x^4 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{25 c \sqrt{d-c^2 d x^2}}-\frac{4 b x^3 \sqrt{c x-1} \sqrt{c x+1}}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{8 b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]

[Out]

(-8*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^5*Sqrt[d - c^2*d*x^2]) - (4*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4
5*c^3*Sqrt[d - c^2*d*x^2]) - (b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(25*c*Sqrt[d - c^2*d*x^2]) - (8*(1 - c*x)*(1
+ c*x)*(a + b*ArcCosh[c*x]))/(15*c^6*Sqrt[d - c^2*d*x^2]) - (4*x^2*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/
(15*c^4*Sqrt[d - c^2*d*x^2]) - (x^4*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]))/(5*c^2*Sqrt[d - c^2*d*x^2])

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^4 \, dx}{5 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{25 c \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (4 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \, dx}{15 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{4 b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (8 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int 1 \, dx}{15 c^5 \sqrt{d-c^2 d x^2}}\\ &=-\frac{8 b x \sqrt{-1+c x} \sqrt{1+c x}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{4 b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^6 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{15 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^4 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{5 c^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.253573, size = 140, normalized size = 0.59 \[ \frac{\sqrt{d-c^2 d x^2} \left (-15 a \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right )+b c x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^4 x^4+20 c^2 x^2+120\right )-15 b \left (3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right ) \cosh ^{-1}(c x)\right )}{225 c^6 d (c x-1) (c x+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]

[Out]

(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2
+ c^4*x^4 + 3*c^6*x^6) - 15*b*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcCosh[c*x]))/(225*c^6*d*(-1 + c*x)*(1
+ c*x))

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Maple [B]  time = 0.314, size = 670, normalized size = 2.8 \begin{align*} a \left ( -{\frac{{x}^{4}}{5\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{4}{5\,{c}^{2}} \left ( -{\frac{{x}^{2}}{3\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{2}{3\,d{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}} \right ) } \right ) +b \left ( -{\frac{-1+5\,{\rm arccosh} \left (cx\right )}{800\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 16\,{c}^{6}{x}^{6}-28\,{c}^{4}{x}^{4}+16\,\sqrt{cx+1}\sqrt{cx-1}{x}^{5}{c}^{5}+13\,{c}^{2}{x}^{2}-20\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}+5\,\sqrt{cx+1}\sqrt{cx-1}xc-1 \right ) }-{\frac{-5+15\,{\rm arccosh} \left (cx\right )}{288\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-5\,{c}^{2}{x}^{2}+4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}-3\,\sqrt{cx+1}\sqrt{cx-1}xc+1 \right ) }-{\frac{-5+5\,{\rm arccosh} \left (cx\right )}{16\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( \sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{5+5\,{\rm arccosh} \left (cx\right )}{16\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -\sqrt{cx+1}\sqrt{cx-1}xc+{c}^{2}{x}^{2}-1 \right ) }-{\frac{5+15\,{\rm arccosh} \left (cx\right )}{288\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -4\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}+4\,{c}^{4}{x}^{4}+3\,\sqrt{cx+1}\sqrt{cx-1}xc-5\,{c}^{2}{x}^{2}+1 \right ) }-{\frac{1+5\,{\rm arccosh} \left (cx\right )}{800\,d{c}^{6} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( -16\,\sqrt{cx+1}\sqrt{cx-1}{x}^{5}{c}^{5}+16\,{c}^{6}{x}^{6}+20\,\sqrt{cx+1}\sqrt{cx-1}{x}^{3}{c}^{3}-28\,{c}^{4}{x}^{4}-5\,\sqrt{cx+1}\sqrt{cx-1}xc+13\,{c}^{2}{x}^{2}-1 \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)

[Out]

a*(-1/5*x^4/c^2/d*(-c^2*d*x^2+d)^(1/2)+4/5/c^2*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(
1/2)))+b*(-1/800*(-d*(c^2*x^2-1))^(1/2)*(16*c^6*x^6-28*c^4*x^4+16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+13*c^2*x
^2-20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-1)*(-1+5*arccosh(c*x))/c^6/d/(c^2*
x^2-1)-5/288*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2+4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3-3*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x*c+1)*(-1+3*arccosh(c*x))/c^6/d/(c^2*x^2-1)-5/16*(-d*(c^2*x^2-1))^(1/2)*((c*x+1)^(1/2)*(c*x-1)
^(1/2)*x*c+c^2*x^2-1)*(-1+arccosh(c*x))/c^6/d/(c^2*x^2-1)-5/16*(-d*(c^2*x^2-1))^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^
(1/2)*x*c+c^2*x^2-1)*(1+arccosh(c*x))/c^6/d/(c^2*x^2-1)-5/288*(-d*(c^2*x^2-1))^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)
^(1/2)*x^3*c^3+4*c^4*x^4+3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)*(1+3*arccosh(c*x))/c^6/d/(c^2*x^2-1)-1
/800*(-d*(c^2*x^2-1))^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)
*x^3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)*(1+5*arccosh(c*x))/c^6/d/(c^2*x^2-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.13713, size = 382, normalized size = 1.62 \begin{align*} -\frac{15 \,{\left (3 \, b c^{6} x^{6} + b c^{4} x^{4} + 4 \, b c^{2} x^{2} - 8 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (9 \, b c^{5} x^{5} + 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 15 \,{\left (3 \, a c^{6} x^{6} + a c^{4} x^{4} + 4 \, a c^{2} x^{2} - 8 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{225 \,{\left (c^{8} d x^{2} - c^{6} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

-1/225*(15*(3*b*c^6*x^6 + b*c^4*x^4 + 4*b*c^2*x^2 - 8*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (
9*b*c^5*x^5 + 20*b*c^3*x^3 + 120*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 15*(3*a*c^6*x^6 + a*c^4*x^4 +
4*a*c^2*x^2 - 8*a)*sqrt(-c^2*d*x^2 + d))/(c^8*d*x^2 - c^6*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**5*(a + b*acosh(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )} x^{5}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x^5/sqrt(-c^2*d*x^2 + d), x)