\(\int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx\) [99]

Optimal result

Integrand size = 21, antiderivative size = 56 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=-\frac {a+b \text {arccosh}(c+d x)}{d e^2 (c+d x)}+\frac {b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{d e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5996, 12, 5883, 94, 209} \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\frac {b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{d e^2}-\frac {a+b \text {arccosh}(c+d x)}{d e^2 (c+d x)} \]

[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 12

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

Rule 94

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 209

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

Rule 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5996

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
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {a+b \text {arccosh}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {a+b \text {arccosh}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {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 \text {arccosh}(c+d x)}{d e^2 (c+d x)}+\frac {b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{d e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\frac {\frac {-a-b \text {arccosh}(c+d x)}{c+d x}+\frac {b \sqrt {-1+(c+d x)^2} \arctan \left (\sqrt {-1+(c+d x)^2}\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}}{d e^2} \]

[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)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2}}}{d}\)	\(81\)
default	\(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2}}}{d}\)	\(81\)
parts	\(-\frac {a}{e^{2} \left (d x +c \right ) d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{d x +c}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{2} d}\)	\(83\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (52) = 104\).

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.38 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\frac {b d x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - a c + 2 \, {\left (b c d x + b c^{2}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (b d x + b c\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \]

[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)

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\frac {\int \frac {a}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]

[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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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