\(\int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx\) [9]

Optimal result

Integrand size = 16, antiderivative size = 82 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx=-\frac {a+b \text {arcsinh}(c x)}{e (d+e x)}-\frac {b c \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5828, 739, 212} \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx=-\frac {a+b \text {arcsinh}(c x)}{e (d+e x)}-\frac {b c \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]

[In]

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

[Out]

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

Rule 212

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 5828

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx=-\frac {\frac {a+b \text {arcsinh}(c x)}{d+e x}+\frac {b c \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{\sqrt {c^2 d^2+e^2}}}{e} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(78)=156\).

Time = 0.96 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.12

method	result	size
parts	\(-\frac {a}{\left (e x +d \right ) e}-\frac {b c \,\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {b c \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\)	\(174\)
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(186\)
default	\(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\right )}{c}\)	\(186\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (78) = 156\).

Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.09 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx=-\frac {a c^{2} d^{3} + a d e^{2} - {\left (b c^{2} d^{2} e + b e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c d e x + b c d^{2}\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e + \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) - {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d^{4} e + d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x} \]

[In]

integrate((a+b*arcsinh(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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx=-b {\left (\frac {\operatorname {arsinh}\left (c x\right )}{e^{2} x + d e} - \frac {c \operatorname {arsinh}\left (\frac {c d \sqrt {e^{4}} x}{e {\left | e^{2} x + d e \right |}} - \frac {\sqrt {e^{4}}}{c {\left | e^{2} x + d e \right |}}\right )}{\sqrt {\frac {c^{2} d^{2}}{e^{2}} + 1} e^{2}}\right )} - \frac {a}{e^{2} x + d e} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (78) = 156\).

Time = 0.47 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.83 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^2} \, dx={\left (\frac {c \log \left (-c^{2} d e + \sqrt {c^{2} d^{2} + e^{2}} {\left | c \right |} {\left | e \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c^{2} d^{2} + e^{2}} {\left | e \right |}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (e x + d\right )} e} - \frac {c \log \left (-c^{2} d e + \sqrt {c^{2} d^{2} + e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{e x + d} + \frac {c^{2} d^{2}}{{\left (e x + d\right )}^{2}} + \frac {e^{2}}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} + e^{4}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |}\right )}{\sqrt {c^{2} d^{2} + e^{2}} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}\right )} b - \frac {a}{{\left (e x + d\right )} e} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*asinh(c*x))/(d + e*x)^2,x)

[Out]

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