Integrand size = 23, antiderivative size = 116 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=-\frac {(a+b \arcsin (c+d x))^2}{d e^2 (c+d x)}-\frac {4 b (a+b \arcsin (c+d x)) \text {arctanh}\left (e^{i \arcsin (c+d x)}\right )}{d e^2}+\frac {2 i b^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )}{d e^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )}{d e^2} \]
-(a+b*arcsin(d*x+c))^2/d/e^2/(d*x+c)-4*b*(a+b*arcsin(d*x+c))*arctanh(I*(d* x+c)+(1-(d*x+c)^2)^(1/2))/d/e^2+2*I*b^2*polylog(2,-I*(d*x+c)-(1-(d*x+c)^2) ^(1/2))/d/e^2-2*I*b^2*polylog(2,I*(d*x+c)+(1-(d*x+c)^2)^(1/2))/d/e^2
Time = 0.83 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\frac {-\frac {a^2}{c+d x}-2 a b \left (\frac {\arcsin (c+d x)}{c+d x}+\log \left (\frac {1}{2} (c+d x) \csc \left (\frac {1}{2} \arcsin (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} \arcsin (c+d x)\right )\right )\right )+b^2 \left (\arcsin (c+d x) \left (-\frac {\arcsin (c+d x)}{c+d x}+2 \log \left (1-e^{i \arcsin (c+d x)}\right )-2 \log \left (1+e^{i \arcsin (c+d x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c+d x)}\right )-2 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )\right )}{d e^2} \]
(-(a^2/(c + d*x)) - 2*a*b*(ArcSin[c + d*x]/(c + d*x) + Log[((c + d*x)*Csc[ ArcSin[c + d*x]/2])/2] - Log[Sin[ArcSin[c + d*x]/2]]) + b^2*(ArcSin[c + d* x]*(-(ArcSin[c + d*x]/(c + d*x)) + 2*Log[1 - E^(I*ArcSin[c + d*x])] - 2*Lo g[1 + E^(I*ArcSin[c + d*x])]) + (2*I)*PolyLog[2, -E^(I*ArcSin[c + d*x])] - (2*I)*PolyLog[2, E^(I*ArcSin[c + d*x])]))/(d*e^2)
Time = 0.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5304, 27, 5138, 5218, 3042, 4671, 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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2 b \int \frac {a+b \arcsin (c+d x)}{(c+d x) \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 5218 |
| \(\displaystyle \frac {2 b \int \frac {a+b \arcsin (c+d x)}{c+d x}d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b \int (a+b \arcsin (c+d x)) \csc (\arcsin (c+d x))d\arcsin (c+d x)-\frac {(a+b \arcsin (c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-b \int \log \left (1-e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)+b \int \log \left (1+e^{i \arcsin (c+d x)}\right )d\arcsin (c+d x)-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )}{d e^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-i b \int e^{-i \arcsin (c+d x)} \log \left (1+e^{i \arcsin (c+d x)}\right )de^{i \arcsin (c+d x)}+i b \int e^{-i \arcsin (c+d x)} \log (-c-d x+1)de^{i \arcsin (c+d x)}-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))\right )}{d e^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \arcsin (c+d x))^2}{c+d x}+2 b \left (-2 \text {arctanh}\left (e^{i \arcsin (c+d x)}\right ) (a+b \arcsin (c+d x))-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )}{d e^2}\) |
(-((a + b*ArcSin[c + d*x])^2/(c + d*x)) + 2*b*(-2*(a + b*ArcSin[c + d*x])* ArcTanh[E^(I*ArcSin[c + d*x])] - I*b*PolyLog[2, E^(I*ArcSin[c + d*x])] + I *b*PolyLog[2, -c - d*x]))/(d*e^2)
3.2.94.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
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]
Time = 0.46 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.78
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(207\) |
| default | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(207\) |
| parts | \(-\frac {a^{2}}{e^{2} \left (d x +c \right ) d}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{d x +c}+2 \arcsin \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )-2 \arcsin \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )+2 i \operatorname {dilog}\left (1+i \left (d x +c \right )+\sqrt {1-\left (d x +c \right )^{2}}\right )-2 i \operatorname {dilog}\left (1-i \left (d x +c \right )-\sqrt {1-\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(212\) |
1/d*(-a^2/e^2/(d*x+c)+b^2/e^2*(-arcsin(d*x+c)^2/(d*x+c)+2*arcsin(d*x+c)*ln (1-I*(d*x+c)-(1-(d*x+c)^2)^(1/2))-2*arcsin(d*x+c)*ln(1+I*(d*x+c)+(1-(d*x+c )^2)^(1/2))+2*I*dilog(1+I*(d*x+c)+(1-(d*x+c)^2)^(1/2))-2*I*dilog(1-I*(d*x+ c)-(1-(d*x+c)^2)^(1/2)))+2*a*b/e^2*(-1/(d*x+c)*arcsin(d*x+c)-arctanh(1/(1- (d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
integral((b^2*arcsin(d*x + c)^2 + 2*a*b*arcsin(d*x + c) + a^2)/(d^2*e^2*x^ 2 + 2*c*d*e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**2*asin(c + d *x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(2*a*b*asin(c + d*x)/(c* *2 + 2*c*d*x + d**2*x**2), x))/e**2
Exception generated. \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \arcsin \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]