\(\int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx\) [32]

Optimal result

Integrand size = 21, antiderivative size = 70 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx=\frac {2 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c d}-\frac {i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c d}+\frac {i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c d} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.93 \[ \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx=-\frac {c \left (a \sqrt {-c^2} \arctan (c x)-b c \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+b c \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+b c \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-b c \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )\right )}{\left (-c^2\right )^{3/2} d} \] Input:

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

Output:

-((c*(a*Sqrt[-c^2]*ArcTan[c*x] - b*c*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x 
])/Sqrt[-c^2]] + b*c*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 
 b*c*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - b*c*PolyLog[2, (Sqrt[-c^2 
]*E^ArcSinh[c*x])/c]))/((-c^2)^(3/2)*d))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6204, 3042, 4668, 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.

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{c^2 d x^2+d} \, dx\)

\(\Big \downarrow \) 6204

\(\displaystyle \frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 4668

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

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.04

method	result	size
derivativedivides	\(\frac {\frac {a \arctan \left (x c \right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )+\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c}\)	\(143\)
default	\(\frac {\frac {a \arctan \left (x c \right )}{d}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )+\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d}}{c}\)	\(143\)
parts	\(\frac {a \arctan \left (x c \right )}{d c}+\frac {b \left (\operatorname {arcsinh}\left (x c \right ) \arctan \left (x c \right )+\arctan \left (x c \right ) \ln \left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-\arctan \left (x c \right ) \ln \left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i x c +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )\right )}{d c}\)	\(145\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*arcsinh(c*x) + a)/(c^2*d*x^2 + d), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx=\frac {\int \frac {a}{c^{2} x^{2} + 1}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \] Input:

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

Output:

(Integral(a/(c**2*x**2 + 1), x) + Integral(b*asinh(c*x)/(c**2*x**2 + 1), x 
))/d
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

integrate((b*arcsinh(c*x) + a)/(c^2*d*x^2 + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{d+c^2 d x^2} \, dx=\frac {\mathit {atan} \left (c x \right ) a +\left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{2} x^{2}+1}d x \right ) b c}{c d} \] Input:

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

Output:

(atan(c*x)*a + int(asinh(c*x)/(c**2*x**2 + 1),x)*b*c)/(c*d)