\(\int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx\) [13]

Optimal result

Integrand size = 18, antiderivative size = 347 \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx=-\frac {i (a+b \arcsin (c x))^3}{3 b e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \] Output:

-1/3*I*(a+b*arcsin(c*x))^3/b/e+(a+b*arcsin(c*x))^2*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))^2*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-2*I*b*(a+b*arcsin(c*x 
))*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-2 
*I*b*(a+b*arcsin(c*x))*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+2*b^2*polylog(3,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^ 
2*d^2-e^2)^(1/2)))/e+2*b^2*polylog(3,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 = 0.21 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx=\frac {-\frac {i (a+b \arcsin (c x))^3}{b}+3 (a+b \arcsin (c x))^2 \log \left (1+\frac {i e e^{i \arcsin (c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+3 (a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+6 b \left (-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )\right )+6 b \left (-i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{3 e} \] Input:

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

Output:

(((-I)*(a + b*ArcSin[c*x])^3)/b + 3*(a + b*ArcSin[c*x])^2*Log[1 + (I*e*E^( 
I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 3*(a + b*ArcSin[c*x])^2* 
Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + 6*b*((-I)*( 
a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 
- e^2])] + b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2] 
)]) + 6*b*((-I)*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c* 
d + Sqrt[c^2*d^2 - e^2])] + b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sq 
rt[c^2*d^2 - e^2])]))/(3*e)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5240, 5030, 2620, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5240

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

\(\Big \downarrow \) 5030

\(\displaystyle \int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c d-i e e^{i \arcsin (c x)}-\sqrt {c^2 d^2-e^2}}d\arcsin (c x)+\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c d-i e e^{i \arcsin (c x)}+\sqrt {c^2 d^2-e^2}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b e}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}-\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i (a+b \arcsin (c x))^3}{3 b e}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)\right )}{e}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )d\arcsin (c x)\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i (a+b \arcsin (c x))^3}{3 b e}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}\right )}{e}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )de^{i \arcsin (c x)}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i (a+b \arcsin (c x))^3}{3 b e}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )\right )}{e}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e e^{i \arcsin (c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i e e^{i \arcsin (c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i (a+b \arcsin (c x))^3}{3 b e}\)

Input:

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

Output:

((-1/3*I)*(a + b*ArcSin[c*x])^3)/(b*e) + ((a + b*ArcSin[c*x])^2*Log[1 - (I 
*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 - e^2])])/e + ((a + b*ArcSin[c*x 
])^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])/e - (2* 
b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^ 
2*d^2 - e^2])] - b*PolyLog[3, (I*e*E^(I*ArcSin[c*x]))/(c*d - Sqrt[c^2*d^2 
- e^2])]))/e - (2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*e*E^(I*ArcSin[c*x 
]))/(c*d + Sqrt[c^2*d^2 - e^2])] - b*PolyLog[3, (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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{e x +d}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a + b*asin(c*x))**2/(d + e*x), x)
 

Maxima [F]

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

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

Output:

a^2*log(e*x + d)/e + integrate((b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 
 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(e*x + d), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arcsin(c*x))^2/(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. \[ \int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b \arcsin (c x))^2}{d+e x} \, dx=\frac {2 \left (\int \frac {\mathit {asin} \left (c x \right )}{e x +d}d x \right ) a b e +\left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{e x +d}d x \right ) b^{2} e +\mathrm {log}\left (e x +d \right ) a^{2}}{e} \] Input:

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

Output:

(2*int(asin(c*x)/(d + e*x),x)*a*b*e + int(asin(c*x)**2/(d + e*x),x)*b**2*e 
 + log(d + e*x)*a**2)/e