\(\int \frac {a+b \text {arcsinh}(c+d x)}{c e+d e x} \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 81 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {(a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5859, 12, 5775, 3797, 2221, 2317, 2438} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {\log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))}{d e}-\frac {b \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]

[In]

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

[Out]

(a + b*ArcSinh[c + d*x])^2/(2*b*d*e) + ((a + b*ArcSinh[c + d*x])*Log[1 - E^(-2*ArcSinh[c + d*x])])/(d*e) - (b*
PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(2*d*e)

Rule 12

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

Rule 2221

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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 x} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x} \, dx,x,c+d x\right )}{d e} \\ & = -\frac {\text {Subst}\left (\int x \coth \left (\frac {a}{b}-\frac {x}{b}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {2 \text {Subst}\left (\int \frac {e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )} x}{1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{b d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {(a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {\text {Subst}\left (\int \log \left (1-e^{2 \left (\frac {a}{b}-\frac {x}{b}\right )}\right ) \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {(a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ & = \frac {(a+b \text {arcsinh}(c+d x))^2}{2 b d e}+\frac {(a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}\right )}{2 d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{c e+d e x} \, dx=\frac {-\left ((a+b \text {arcsinh}(c+d x)) \left (a+b \text {arcsinh}(c+d x)-2 b \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )\right )\right )+b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )}{2 b d e} \]

[In]

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

[Out]

(-((a + b*ArcSinh[c + d*x])*(a + b*ArcSinh[c + d*x] - 2*b*Log[1 - E^(2*ArcSinh[c + d*x])])) + b^2*PolyLog[2, E
^(2*ArcSinh[c + d*x])])/(2*b*d*e)

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(129\)
default	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(129\)
parts	\(\frac {a \ln \left (d x +c \right )}{e d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}\)	\(131\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arcsinh(d*x + c) + a)/(d*e*x + c*e), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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