3.111 \(\int \frac{(a+b \cosh ^{-1}(c+d x))^2}{(c e+d e x)^3} \, dx\)

Optimal. Leaf size=92 \[ \frac{b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}-\frac{b^2 \log (c+d x)}{d e^3} \]

[Out]

(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])
^2/(2*d*e^3*(c + d*x)^2) - (b^2*Log[c + d*x])/(d*e^3)

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Rubi [A]  time = 0.213952, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {5866, 12, 5662, 5724, 29} \[ \frac{b \sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d e^3 (c+d x)}-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)^2}-\frac{b^2 \log (c+d x)}{d e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/(d*e^3*(c + d*x)) - (a + b*ArcCosh[c + d*x])
^2/(2*d*e^3*(c + d*x)^2) - (b^2*Log[c + d*x])/(d*e^3)

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 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-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, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.204559, size = 81, normalized size = 0.88 \[ \frac{b \left (\frac{\sqrt{c+d x-1} \sqrt{c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{c+d x}-b \log (c+d x)\right )-\frac{\left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 (c+d x)^2}}{d e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a + b*ArcCosh[c + d*x])^2/(2*(c + d*x)^2) + b*((Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x
]))/(c + d*x) - b*Log[c + d*x]))/(d*e^3)

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Maple [B]  time = 0.063, size = 194, normalized size = 2.1 \begin{align*} -{\frac{{a}^{2}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{3}}}+{\frac{{b}^{2}{\rm arccosh} \left (dx+c\right )}{d{e}^{3} \left ( dx+c \right ) }\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{2\,d{e}^{3} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{2}}{d{e}^{3}}\ln \left ( \left ( dx+c+\sqrt{dx+c-1}\sqrt{dx+c+1} \right ) ^{2}+1 \right ) }-{\frac{ab{\rm arccosh} \left (dx+c\right )}{d{e}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{ab}{d{e}^{3} \left ( dx+c \right ) }\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))^2/(d*e*x+c*e)^3,x)

[Out]

-1/2/d*a^2/e^3/(d*x+c)^2+1/d*b^2/e^3*arccosh(d*x+c)+1/d*b^2/e^3*arccosh(d*x+c)/(d*x+c)*(d*x+c+1)^(1/2)*(d*x+c-
1)^(1/2)-1/2/d*b^2/e^3*arccosh(d*x+c)^2/(d*x+c)^2-1/d*b^2/e^3*ln((d*x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2+1)-
1/d*a*b/e^3/(d*x+c)^2*arccosh(d*x+c)+1/d*a*b/e^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/(d*x+c)

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Maxima [B]  time = 1.7738, size = 309, normalized size = 3.36 \begin{align*}{\left (\frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d \operatorname{arcosh}\left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac{\log \left (d x + c\right )}{d e^{3}}\right )} b^{2} + a b{\left (\frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac{\operatorname{arcosh}\left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac{b^{2} \operatorname{arcosh}\left (d x + c\right )^{2}}{2 \,{\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac{a^{2}}{2 \,{\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d*arccosh(d*x + c)/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c)/(d*e^3))*b^2 + a*
b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d/(d^3*e^3*x + c*d^2*e^3) - arccosh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x
+ c^2*d*e^3)) - 1/2*b^2*arccosh(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^2/(d^3*e^3*x^2 +
2*c*d^2*e^3*x + c^2*d*e^3)

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Fricas [B]  time = 2.76066, size = 713, normalized size = 7.75 \begin{align*} \frac{2 \, a b c^{2} d^{2} x^{2} + 4 \, a b c^{3} d x + 2 \, a b c^{4} - b^{2} c^{2} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} - a^{2} c^{2} + 2 \,{\left (a b d^{2} x^{2} + 2 \, a b c d x +{\left (b^{2} c^{2} d x + b^{2} c^{3}\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \,{\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right ) + 2 \,{\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} \log \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 2 \,{\left (a b c^{2} d x + a b c^{3}\right )} \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{2 \,{\left (c^{2} d^{3} e^{3} x^{2} + 2 \, c^{3} d^{2} e^{3} x + c^{4} d e^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*c^2*d^2*x^2 + 4*a*b*c^3*d*x + 2*a*b*c^4 - b^2*c^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^
2 - a^2*c^2 + 2*(a*b*d^2*x^2 + 2*a*b*c*d*x + (b^2*c^2*d*x + b^2*c^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*
x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - 2*(b^2*c^2*d^2*x^2 + 2*b^2*c^3*d*x + b^2*c^4)*log(d*x + c) + 2*(a
*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*log(-d*x - c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) + 2*(a*b*c^2*d*x + a*b*c
^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))/(c^2*d^3*e^3*x^2 + 2*c^3*d^2*e^3*x + c^4*d*e^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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