3.2.82 \(\int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx\) [182]

3.2.82.1 Optimal result

Integrand size = 21, antiderivative size = 89 \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=-\frac {i (a+b \arcsin (c+d x))^2}{2 b d e}+\frac {(a+b \arcsin (c+d x)) \log \left (1-e^{2 i \arcsin (c+d x)}\right )}{d e}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )}{2 d e} \]

output

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

3.2.82.2 Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \arcsin (c+d x)}{c e+d e x} \, dx=\frac {b \arcsin (c+d x) \log \left (1-e^{2 i \arcsin (c+d x)}\right )+a \log (c+d x)-\frac {1}{2} i b \left (\arcsin (c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c+d x)}\right )\right )}{d e} \]

input

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

output

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

3.2.82.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5304, 27, 5136, 3042, 25, 4200, 25, 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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5136

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.2.82.3.1 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 3042

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

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5136

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( 
a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, 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]
 

3.2.82.4 Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(154\)
default	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\)	\(154\)
parts	\(\frac {a \ln \left (d x +c \right )}{e d}+\frac {b \left (-\frac {i \arcsin \left (d x +c \right )^{2}}{2}+\arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, -i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )+\arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-i \operatorname {polylog}\left (2, i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e d}\)	\(156\)

input

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

output

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

3.2.82.5 Fricas [F]

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

input

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

output

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

3.2.82.6 Sympy [F]

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

input

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

output

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

3.2.82.7 Maxima [F]

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

input

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

output

b*integrate(arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1))/(d*e*x 
+ c*e), x) + a*log(d*e*x + c*e)/(d*e)
 

3.2.82.8 Giac [F]

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

input

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

output

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

3.2.82.9 Mupad [F(-1)]

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

input

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

output

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