\(\int \frac {a+b \text {arcsinh}(c+d x)}{c e+d e x} \, dx\) [62]

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {6274, 27, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c+d x)}{e (c+d x)}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c+d x)}{c+d x}d(c+d x)}{d e}\)

\(\Big \downarrow \) 6190

\(\displaystyle \frac {\int -\left ((a+b \text {arcsinh}(c+d x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a+b \text {arcsinh}(c+d x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -i (a+b \text {arcsinh}(c+d x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {i \int (a+b \text {arcsinh}(c+d x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 4201

\(\displaystyle \frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi } (a+b \text {arcsinh}(c+d x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i (a+b \text {arcsinh}(c+d x))^2\right )}{b d e}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c+d x))^2\right )}{b d e}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {i \left (2 i \left (-\frac {1}{4} b^2 \int \exp \left (-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+i \pi \right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c+d x))^2\right )}{b d e}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-c-d x)-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c+d x))^2\right )}{b d e}\)

Input:

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

Output:

(I*((-1/2*I)*(a + b*ArcSinh[c + d*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c 
+ d*x])*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c + d*x]))/b)]) + (b 
^2*PolyLog[2, -c - d*x])/4)))/(b*d*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

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

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[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 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 0.49 (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	\(\frac {a \ln \left (d x +c \right )}{e d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}\)	\(131\)

Input:

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

Output:

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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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