\(\int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx\) [121]

Optimal result

Integrand size = 21, antiderivative size = 49 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{d e^2 (c+d x)}-\frac {b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5859, 12, 5776, 272, 65, 213} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{d e^2 (c+d x)}-\frac {b \text {arctanh}\left (\sqrt {(c+d x)^2+1}\right )}{d e^2} \]

[In]

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

[Out]

-((a + b*ArcSinh[c + d*x])/(d*e^2*(c + d*x))) - (b*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5859

Int[((a_.) + ArcSinh[(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*ArcSinh[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 {arcsinh}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^2} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{d e^2 (c+d x)}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{2 d e^2} \\ & = -\frac {a+b \text {arcsinh}(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)^2}\right )}{d e^2} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{d e^2 (c+d x)}-\frac {b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx=-\frac {\frac {a+b \text {arcsinh}(c+d x)}{c+d x}+b \text {arctanh}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\)	\(54\)
default	\(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\)	\(54\)
parts	\(-\frac {a}{e^{2} \left (d x +c \right ) d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\)	\(56\)

[In]

int((a+b*arcsinh(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)*arcsinh(d*x+c)-arctanh(1/(1+(d*x+c)^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (47) = 94\).

Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.57 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx=\frac {b d x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - a c - {\left (b c d x + b c^{2}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) + {\left (b d x + b c\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (b c d x + b c^{2}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \]

[In]

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

Sympy [F]

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

[In]

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

Maxima [F(-2)]

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

[In]

integrate((a+b*arcsinh(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (47) = 94\).

Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.73 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^2} \, dx=-b {\left (\frac {\log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )}{{\left (d e x + c e\right )} d e} + \frac {d \log \left (\sqrt {\frac {e^{2}}{{\left (d e x + c e\right )}^{2}} + 1} + \frac {\sqrt {d^{2} e^{4}}}{{\left (d e x + c e\right )} d e}\right )}{e^{2} {\left | d \right |}^{2} \mathrm {sgn}\left (\frac {1}{d e x + c e}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}\right )} - \frac {a}{{\left (d e x + c e\right )} d e} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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