3.103 \(\int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx\)

Optimal. Leaf size=137 \[ -\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{3 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{40 d e^6 (c+d x)^2}+\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{20 d e^6 (c+d x)^4}+\frac{3 b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{40 d e^6} \]

[Out]

(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(20*d*e^6*(c + d*x)^4) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(
40*d*e^6*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])/(5*d*e^6*(c + d*x)^5) + (3*b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1
+ c + d*x]])/(40*d*e^6)

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Rubi [A]  time = 0.0866639, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 103, 92, 203} \[ -\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{3 b \sqrt{c+d x-1} \sqrt{c+d x+1}}{40 d e^6 (c+d x)^2}+\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{20 d e^6 (c+d x)^4}+\frac{3 b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{40 d e^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^6,x]

[Out]

(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(20*d*e^6*(c + d*x)^4) + (3*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])/(
40*d*e^6*(c + d*x)^2) - (a + b*ArcCosh[c + d*x])/(5*d*e^6*(c + d*x)^5) + (3*b*ArcTan[Sqrt[-1 + c + d*x]*Sqrt[1
+ c + d*x]])/(40*d*e^6)

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{a+b \cosh ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x^5 \sqrt{1+x}} \, dx,x,c+d x\right )}{5 d e^6}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{20 d e^6 (c+d x)^4}-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{b \operatorname{Subst}\left (\int \frac{3}{\sqrt{-1+x} x^3 \sqrt{1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{20 d e^6 (c+d x)^4}-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x^3 \sqrt{1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{20 d e^6 (c+d x)^4}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{40 d e^6 (c+d x)^2}-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx,x,c+d x\right )}{40 d e^6}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{20 d e^6 (c+d x)^4}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{40 d e^6 (c+d x)^2}-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{40 d e^6}\\ &=\frac{b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{20 d e^6 (c+d x)^4}+\frac{3 b \sqrt{-1+c+d x} \sqrt{1+c+d x}}{40 d e^6 (c+d x)^2}-\frac{a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{3 b \tan ^{-1}\left (\sqrt{-1+c+d x} \sqrt{1+c+d x}\right )}{40 d e^6}\\ \end{align*}

Mathematica [A]  time = 0.265301, size = 136, normalized size = 0.99 \[ \frac{-\frac{a+b \cosh ^{-1}(c+d x)}{(c+d x)^5}+\frac{b \sqrt{c+d x-1} \sqrt{c+d x+1}}{4 (c+d x)^4}+\frac{3 b \left (\frac{(c+d x-1) (c+d x+1)}{(c+d x)^2}+\sqrt{(c+d x)^2-1} \tan ^{-1}\left (\sqrt{(c+d x)^2-1}\right )\right )}{8 \sqrt{c+d x-1} \sqrt{c+d x+1}}}{5 d e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^6,x]

[Out]

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

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Maple [A]  time = 0.005, size = 152, normalized size = 1.1 \begin{align*} -{\frac{a}{5\,d{e}^{6} \left ( dx+c \right ) ^{5}}}-{\frac{b{\rm arccosh} \left (dx+c\right )}{5\,d{e}^{6} \left ( dx+c \right ) ^{5}}}-{\frac{3\,b}{40\,d{e}^{6}}\sqrt{dx+c-1}\sqrt{dx+c+1}\arctan \left ({\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}} \right ){\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}}}+{\frac{3\,b}{40\,d{e}^{6} \left ( dx+c \right ) ^{2}}\sqrt{dx+c-1}\sqrt{dx+c+1}}+{\frac{b}{20\,d{e}^{6} \left ( dx+c \right ) ^{4}}\sqrt{dx+c-1}\sqrt{dx+c+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x)

[Out]

-1/5/d*a/e^6/(d*x+c)^5-1/5/d*b/e^6/(d*x+c)^5*arccosh(d*x+c)-3/40/d*b/e^6*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/((d*x
+c)^2-1)^(1/2)*arctan(1/((d*x+c)^2-1)^(1/2))+3/40*b*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^6/(d*x+c)^2+1/20*b*(d*
x+c-1)^(1/2)*(d*x+c+1)^(1/2)/d/e^6/(d*x+c)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="maxima")

[Out]

1/30*b*((6*d^4*x^4 + 24*c*d^3*x^3 + 6*c^4 + 2*(18*c^2*d^2 + d^2)*x^2 + 2*c^2 + 4*(6*c^3*d + c*d)*x - 3*(d^5*x^
5 + 5*c*d^4*x^4 + 10*c^2*d^3*x^3 + 10*c^3*d^2*x^2 + 5*c^4*d*x + c^5)*log(d*x + c + 1) + 3*(d^5*x^5 + 5*c*d^4*x
^4 + 10*c^2*d^3*x^3 + 10*c^3*d^2*x^2 + 5*c^4*d*x + c^5)*log(d*x + c - 1) - 6*log(d*x + sqrt(d*x + c + 1)*sqrt(
d*x + c - 1) + c))/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x
+ c^5*d*e^6) - 30*integrate(1/5/(d^8*e^6*x^8 + 8*c*d^7*e^6*x^7 + c^8*e^6 - c^6*e^6 + (28*c^2*d^6*e^6 - d^6*e^6
)*x^6 + 2*(28*c^3*d^5*e^6 - 3*c*d^5*e^6)*x^5 + 5*(14*c^4*d^4*e^6 - 3*c^2*d^4*e^6)*x^4 + 4*(14*c^5*d^3*e^6 - 5*
c^3*d^3*e^6)*x^3 + (28*c^6*d^2*e^6 - 15*c^4*d^2*e^6)*x^2 + 2*(4*c^7*d*e^6 - 3*c^5*d*e^6)*x + (d^7*e^6*x^7 + 7*
c*d^6*e^6*x^6 + c^7*e^6 - c^5*e^6 + (21*c^2*d^5*e^6 - d^5*e^6)*x^5 + 5*(7*c^3*d^4*e^6 - c*d^4*e^6)*x^4 + 5*(7*
c^4*d^3*e^6 - 2*c^2*d^3*e^6)*x^3 + (21*c^5*d^2*e^6 - 10*c^3*d^2*e^6)*x^2 + (7*c^6*d*e^6 - 5*c^4*d*e^6)*x)*e^(1
/2*log(d*x + c + 1) + 1/2*log(d*x + c - 1))), x)) - 1/5*a/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3
+ 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)

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Fricas [B]  time = 3.22602, size = 909, normalized size = 6.64 \begin{align*} -\frac{8 \, a c^{5} - 6 \,{\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \arctan \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \,{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \,{\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) -{\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} + 2 \, b c^{6} +{\left (9 \, b c^{7} + 2 \, b c^{5}\right )} d x\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{40 \,{\left (c^{5} d^{6} e^{6} x^{5} + 5 \, c^{6} d^{5} e^{6} x^{4} + 10 \, c^{7} d^{4} e^{6} x^{3} + 10 \, c^{8} d^{3} e^{6} x^{2} + 5 \, c^{9} d^{2} e^{6} x + c^{10} d e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="fricas")

[Out]

-1/40*(8*a*c^5 - 6*(b*c^5*d^5*x^5 + 5*b*c^6*d^4*x^4 + 10*b*c^7*d^3*x^3 + 10*b*c^8*d^2*x^2 + 5*b*c^9*d*x + b*c^
10)*arctan(-d*x - c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 8*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 1
0*b*c^3*d^2*x^2 + 5*b*c^4*d*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 8*(b*d^5*x^5 + 5*b*c*d^4*x^4
+ 10*b*c^2*d^3*x^3 + 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*log(-d*x - c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)
) - (3*b*c^5*d^3*x^3 + 9*b*c^6*d^2*x^2 + 3*b*c^8 + 2*b*c^6 + (9*b*c^7 + 2*b*c^5)*d*x)*sqrt(d^2*x^2 + 2*c*d*x +
c^2 - 1))/(c^5*d^6*e^6*x^5 + 5*c^6*d^5*e^6*x^4 + 10*c^7*d^4*e^6*x^3 + 10*c^8*d^3*e^6*x^2 + 5*c^9*d^2*e^6*x +
c^10*d*e^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac{b \operatorname{acosh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**6,x)

[Out]

(Integral(a/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x**5 + d
**6*x**6), x) + Integral(b*acosh(c + d*x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2
*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x))/e**6

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^6,x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^6, x)