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

Optimal. Leaf size=88 \[ \frac{2 b c \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e \sqrt{c d-e} \sqrt{c d+e}}-\frac{a+b \cosh ^{-1}(c x)}{e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.0564455, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5802, 93, 208} \[ \frac{2 b c \tanh ^{-1}\left (\frac{\sqrt{c x+1} \sqrt{c d+e}}{\sqrt{c x-1} \sqrt{c d-e}}\right )}{e \sqrt{c d-e} \sqrt{c d+e}}-\frac{a+b \cosh ^{-1}(c x)}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5802

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.191636, size = 121, normalized size = 1.38 \[ -\frac{\frac{a}{d+e x}-\frac{b c \log (d+e x)}{\sqrt{c^2 d^2-e^2}}+\frac{b c \log \left (-\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d^2-e^2}+c^2 d x+e\right )}{\sqrt{c^2 d^2-e^2}}+\frac{b \cosh ^{-1}(c x)}{d+e x}}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 145, normalized size = 1.7 \begin{align*} -{\frac{ca}{ \left ( cxe+cd \right ) e}}-{\frac{bc{\rm arccosh} \left (cx\right )}{ \left ( cxe+cd \right ) e}}-{\frac{bc}{{e}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( -2\,{\frac{1}{cxe+cd} \left ({c}^{2}dx-\sqrt{{c}^{2}{x}^{2}-1}\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}e+e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{d}^{2}-{e}^{2}}{{e}^{2}}}}}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.50001, size = 1008, normalized size = 11.45 \begin{align*} \left [-\frac{a c^{2} d^{3} - a d e^{2} -{\left (b c^{2} d^{2} e - b e^{3}\right )} x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (b c d e x + b c d^{2}\right )} \sqrt{c^{2} d^{2} - e^{2}} \log \left (\frac{c^{3} d^{2} x + c d e + \sqrt{c^{2} d^{2} - e^{2}}{\left (c^{2} d x + e\right )} +{\left (c^{2} d^{2} + \sqrt{c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{e x + d}\right ) -{\left (b c^{2} d^{3} - b d e^{2} +{\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac{a c^{2} d^{3} - a d e^{2} -{\left (b c^{2} d^{2} e - b e^{3}\right )} x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \,{\left (b c d e x + b c d^{2}\right )} \sqrt{-c^{2} d^{2} + e^{2}} \arctan \left (-\frac{\sqrt{-c^{2} d^{2} + e^{2}} \sqrt{c^{2} x^{2} - 1} e - \sqrt{-c^{2} d^{2} + e^{2}}{\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) -{\left (b c^{2} d^{3} - b d e^{2} +{\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} +{\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.35395, size = 308, normalized size = 3.5 \begin{align*}{\left ({\left (\frac{e^{\left (-1\right )} \log \left ({\left | -c^{2} d + \sqrt{c^{2} d^{2} - e^{2}}{\left | c \right |} \right |}\right ) \mathrm{sgn}\left (\frac{1}{x e + d}\right )}{\sqrt{c^{2} d^{2} - e^{2}}} - \frac{e^{\left (-1\right )} \log \left ({\left | -c^{2} d + \sqrt{c^{2} d^{2} - e^{2}}{\left (\sqrt{c^{2} - \frac{2 \, c^{2} d}{x e + d} + \frac{c^{2} d^{2}}{{\left (x e + d\right )}^{2}} - \frac{e^{2}}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c^{2} d^{2} e^{2} - e^{4}} e^{\left (-1\right )}}{x e + d}\right )} \right |}\right )}{\sqrt{c^{2} d^{2} - e^{2}} \mathrm{sgn}\left (\frac{1}{x e + d}\right )}\right )} c - \frac{e^{\left (-1\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right )}{x e + d}\right )} b - \frac{a e^{\left (-1\right )}}{x e + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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