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

Optimal. Leaf size=56 \[ \frac{b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^2}-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0516872, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {5866, 12, 5662, 92, 203} \[ \frac{b \tan ^{-1}\left (\sqrt{c+d x-1} \sqrt{c+d x+1}\right )}{d e^2}-\frac{a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.108186, size = 76, normalized size = 1.36 \[ \frac{\frac{b \sqrt{(c+d x)^2-1} \tan ^{-1}\left (\sqrt{(c+d x)^2-1}\right )}{\sqrt{c+d x-1} \sqrt{c+d x+1}}-\frac{a+b \cosh ^{-1}(c+d x)}{c+d x}}{d e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 88, normalized size = 1.6 \begin{align*} -{\frac{a}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{b{\rm arccosh} \left (dx+c\right )}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{b}{d{e}^{2}}\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}}}} \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)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.53022, size = 312, normalized size = 5.57 \begin{align*} \frac{b d x \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - a c + 2 \,{\left (b c d x + b c^{2}\right )} \arctan \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) +{\left (b d x + b c\right )} \log \left (-d x - c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \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)^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError