\(\int \frac {a+b \arcsin (c x)}{d+e x} \, dx\) [5]

Optimal result

Integrand size = 16, antiderivative size = 229 \[ \int \frac {a+b \arcsin (c x)}{d+e x} \, dx=-\frac {i (a+b \arcsin (c x))^2}{2 b e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \arcsin (c x)}{d+e x} \, dx=-\frac {i \left ((a+b \arcsin (c x)) \left (a+b \arcsin (c x)+2 i b \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 i b \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+2 b^2 \operatorname {PolyLog}\left (2,-\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 b^2 \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{2 b e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5240, 5030, 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.

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

\(\Big \downarrow \) 5240

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

\(\Big \downarrow \) 5030

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

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 5030

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] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( 
I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] 
 - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] 
 && PosQ[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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 757 vs. \(2 (244 ) = 488\).

Time = 1.56 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.31

method	result	size
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \left (-\frac {i \arcsin \left (c x \right )^{2} c}{2 e}+\frac {c^{3} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{3} \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {e c \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {e c \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i c^{3} \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i c^{3} \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {i e c \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e c \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}\right )}{c}\)	\(758\)
derivativedivides	\(\frac {\frac {a c \ln \left (c x e +c d \right )}{e}+b c \left (-\frac {i \arcsin \left (c x \right )^{2}}{2 e}+\frac {i e \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {e \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {e \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}\right )}{c}\)	\(759\)
default	\(\frac {\frac {a c \ln \left (c x e +c d \right )}{e}+b c \left (-\frac {i \arcsin \left (c x \right )^{2}}{2 e}+\frac {i e \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {e \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {e \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}\right )}{c}\)	\(759\)

Input:

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

Output:

a/e*ln(e*x+d)+b/c*(-1/2*I*arcsin(c*x)^2*c/e+1/e*c^3*arcsin(c*x)/(c^2*d^2-e 
^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(- 
c^2*d^2+e^2)^(1/2)))*d^2+1/e*c^3*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c* 
x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)) 
)*d^2-e*c*arcsin(c*x)/(c^2*d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e 
+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-e*c*arcsin(c*x)/(c^2* 
d^2-e^2)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d 
*c-(-c^2*d^2+e^2)^(1/2)))-I/e*c^3/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2* 
x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))*d^2-I/ 
e*c^3/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^ 
2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))*d^2+I*e*c/(c^2*d^2-e^2)*dilog((I*d 
*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2 
)^(1/2)))+I*e*c/(c^2*d^2-e^2)*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(- 
c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \arcsin (c x)}{d+e x} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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