3.2.26 \(\int \frac {\arcsin (a+b x)}{x} \, dx\) [126]

3.2.26.1 Optimal result

Integrand size = 10, antiderivative size = 181 \[ \int \frac {\arcsin (a+b x)}{x} \, dx=-\frac {1}{2} i \arcsin (a+b x)^2+\arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )+\arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right ) \]

output

-1/2*I*arcsin(b*x+a)^2+arcsin(b*x+a)*ln(1-(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/ 
(I*a-(-a^2+1)^(1/2)))+arcsin(b*x+a)*ln(1-(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/( 
I*a+(-a^2+1)^(1/2)))-I*polylog(2,(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(I*a-(-a^ 
2+1)^(1/2)))-I*polylog(2,(I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/ 
2)))
 

3.2.26.2 Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09 \[ \int \frac {\arcsin (a+b x)}{x} \, dx=-\frac {1}{2} i \arcsin (a+b x)^2+\arcsin (a+b x) \log \left (1+\frac {e^{i \arcsin (a+b x)}}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right )+\arcsin (a+b x) \log \left (1+\frac {e^{i \arcsin (a+b x)}}{\left (-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}\right ) b}\right )-i \operatorname {PolyLog}\left (2,-\frac {e^{i \arcsin (a+b x)}}{-i a+\sqrt {1-a^2}}\right )-i \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right ) \]

input

Integrate[ArcSin[a + b*x]/x,x]
 

output

(-1/2*I)*ArcSin[a + b*x]^2 + ArcSin[a + b*x]*Log[1 + E^(I*ArcSin[a + b*x]) 
/((((-I)*a)/b - Sqrt[1 - a^2]/b)*b)] + ArcSin[a + b*x]*Log[1 + E^(I*ArcSin 
[a + b*x])/((((-I)*a)/b + Sqrt[1 - a^2]/b)*b)] - I*PolyLog[2, -(E^(I*ArcSi 
n[a + b*x])/((-I)*a + Sqrt[1 - a^2]))] - I*PolyLog[2, E^(I*ArcSin[a + b*x] 
)/(I*a + Sqrt[1 - a^2])]
 

3.2.26.3 Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5304, 25, 27, 5240, 5032, 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.

\(\displaystyle \int \frac {\arcsin (a+b x)}{x} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int \frac {\arcsin (a+b x)}{x}d(a+b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\frac {\arcsin (a+b x)}{x}d(a+b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\int -\frac {\arcsin (a+b x)}{b x}d(a+b x)\)

\(\Big \downarrow \) 5240

\(\displaystyle -\int -\frac {\sqrt {1-(a+b x)^2} \arcsin (a+b x)}{b x}d\arcsin (a+b x)\)

\(\Big \downarrow \) 5032

\(\displaystyle -i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{i a-e^{i \arcsin (a+b x)}-\sqrt {1-a^2}}d\arcsin (a+b x)-i \int \frac {e^{i \arcsin (a+b x)} \arcsin (a+b x)}{i a-e^{i \arcsin (a+b x)}+\sqrt {1-a^2}}d\arcsin (a+b x)-\frac {1}{2} i \arcsin (a+b x)^2\)

\(\Big \downarrow \) 2620

\(\displaystyle -i \left (i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{-\sqrt {1-a^2}+i a}\right )-i \int \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )d\arcsin (a+b x)\right )-i \left (i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{\sqrt {1-a^2}+i a}\right )-i \int \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )d\arcsin (a+b x)\right )-\frac {1}{2} i \arcsin (a+b x)^2\)

\(\Big \downarrow \) 2715

\(\displaystyle -i \left (i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{-\sqrt {1-a^2}+i a}\right )-\int e^{-i \arcsin (a+b x)} \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )de^{i \arcsin (a+b x)}\right )-i \left (i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{\sqrt {1-a^2}+i a}\right )-\int e^{-i \arcsin (a+b x)} \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )de^{i \arcsin (a+b x)}\right )-\frac {1}{2} i \arcsin (a+b x)^2\)

\(\Big \downarrow \) 2838

\(\displaystyle -i \left (\operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )+i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{-\sqrt {1-a^2}+i a}\right )\right )-i \left (\operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )+i \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{\sqrt {1-a^2}+i a}\right )\right )-\frac {1}{2} i \arcsin (a+b x)^2\)

input

Int[ArcSin[a + b*x]/x,x]
 

output

(-1/2*I)*ArcSin[a + b*x]^2 - I*(I*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + 
b*x])/(I*a - Sqrt[1 - a^2])] + PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqr 
t[1 - a^2])]) - I*(I*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + 
Sqrt[1 - a^2])] + PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])
 

3.2.26.3.1 Defintions of rubi rules used

rule 25

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

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 5032

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ 
(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 
))), x] + (Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 
2] + b*E^(I*(c + d*x)))), x], x] + Simp[I   Int[(e + f*x)^m*(E^(I*(c + d*x) 
)/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c 
, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]
 

rule 5240

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Subst[Int[(a + b*x)^n*(Cos[x]/(c*d + e*Sin[x])), x], x, ArcSin[c*x]] /; 
 FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
 

rule 5304

Int[((a_.) + ArcSin[(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*A 
rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.2.26.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (204 ) = 408\).

Time = 0.80 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.20

method	result	size
derivativedivides	\(-\frac {i \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}-\frac {i \arcsin \left (b x +a \right )^{2}}{2}-\frac {i \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}+\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}-\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}-\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1}\)	\(579\)
default	\(-\frac {i \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}-\frac {i \arcsin \left (b x +a \right )^{2}}{2}-\frac {i \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {i \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}+\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right ) a^{2}}{a^{2}-1}-\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}-\frac {\arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1}\)	\(579\)

input

int(arcsin(b*x+a)/x,x,method=_RETURNVERBOSE)
 

output

-I/(a^2-1)*dilog((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a+( 
-a^2+1)^(1/2)))*a^2-1/2*I*arcsin(b*x+a)^2-I/(a^2-1)*dilog((I*a-(-a^2+1)^(1 
/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/2)))*a^2+I/(a^2-1)*dil 
og((I*a-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/2)) 
)+I/(a^2-1)*dilog((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a+ 
(-a^2+1)^(1/2)))+arcsin(b*x+a)/(a^2-1)*ln((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1 
-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))*a^2+arcsin(b*x+a)/(a^2-1)*ln((I*a 
-(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/2)))*a^2-a 
rcsin(b*x+a)/(a^2-1)*ln((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2)) 
/(I*a+(-a^2+1)^(1/2)))-arcsin(b*x+a)/(a^2-1)*ln((I*a-(-a^2+1)^(1/2)-I*(b*x 
+a)-(1-(b*x+a)^2)^(1/2))/(I*a-(-a^2+1)^(1/2)))
 

3.2.26.5 Fricas [F]

\[ \int \frac {\arcsin (a+b x)}{x} \, dx=\int { \frac {\arcsin \left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsin(b*x+a)/x,x, algorithm="fricas")
 

output

integral(arcsin(b*x + a)/x, x)
 

3.2.26.6 Sympy [F]

\[ \int \frac {\arcsin (a+b x)}{x} \, dx=\int \frac {\operatorname {asin}{\left (a + b x \right )}}{x}\, dx \]

input

integrate(asin(b*x+a)/x,x)
 

output

Integral(asin(a + b*x)/x, x)
 

3.2.26.7 Maxima [F]

\[ \int \frac {\arcsin (a+b x)}{x} \, dx=\int { \frac {\arcsin \left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsin(b*x+a)/x,x, algorithm="maxima")
 

output

integrate(arcsin(b*x + a)/x, x)
 

3.2.26.8 Giac [F]

\[ \int \frac {\arcsin (a+b x)}{x} \, dx=\int { \frac {\arcsin \left (b x + a\right )}{x} \,d x } \]

input

integrate(arcsin(b*x+a)/x,x, algorithm="giac")
 

output

integrate(arcsin(b*x + a)/x, x)
 

3.2.26.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a+b x)}{x} \, dx=\int \frac {\mathrm {asin}\left (a+b\,x\right )}{x} \,d x \]

input

int(asin(a + b*x)/x,x)
 

output

int(asin(a + b*x)/x, x)