Integrand size = 23, antiderivative size = 87 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {b \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{d e^3 (c+d x)}-\frac {(a+b \arcsin (c+d x))^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3} \]
-1/2*(a+b*arcsin(d*x+c))^2/d/e^3/(d*x+c)^2+b^2*ln(d*x+c)/d/e^3-b*(a+b*arcs in(d*x+c))*(1-(d*x+c)^2)^(1/2)/d/e^3/(d*x+c)
Time = 0.46 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {a \left (a+2 b (c+d x) \sqrt {1-c^2-2 c d x-d^2 x^2}\right )+2 b \left (a+b (c+d x) \sqrt {1-c^2-2 c d x-d^2 x^2}\right ) \arcsin (c+d x)+b^2 \arcsin (c+d x)^2-2 b^2 (c+d x)^2 \log (c+d x)}{2 d e^3 (c+d x)^2} \]
-1/2*(a*(a + 2*b*(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2]) + 2*b*(a + b *(c + d*x)*Sqrt[1 - c^2 - 2*c*d*x - d^2*x^2])*ArcSin[c + d*x] + b^2*ArcSin [c + d*x]^2 - 2*b^2*(c + d*x)^2*Log[c + d*x])/(d*e^3*(c + d*x)^2)
Time = 0.37 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5304, 27, 5138, 5186, 14}
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.
| \(\displaystyle \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arcsin (c+d x))^2}{(c+d x)^3}d(c+d x)}{d e^3}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {b \int \frac {a+b \arcsin (c+d x)}{(c+d x)^2 \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {(a+b \arcsin (c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {b \left (b \int \frac {1}{c+d x}d(c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{c+d x}\right )-\frac {(a+b \arcsin (c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {b \left (b \log (c+d x)-\frac {\sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))}{c+d x}\right )-\frac {(a+b \arcsin (c+d x))^2}{2 (c+d x)^2}}{d e^3}\) |
(-1/2*(a + b*ArcSin[c + d*x])^2/(c + d*x)^2 + b*(-((Sqrt[1 - (c + d*x)^2]* (a + b*ArcSin[c + d*x]))/(c + d*x)) + b*Log[c + d*x]))/(d*e^3)
3.2.95.3.1 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[((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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x ^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
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 = 0.66 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{d x +c}+\ln \left (d x +c \right )\right )}{e^{3}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(125\) |
| default | \(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{d x +c}+\ln \left (d x +c \right )\right )}{e^{3}}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(125\) |
| parts | \(-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{2} \left (-\frac {\arcsin \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}}{d x +c}+\ln \left (d x +c \right )\right )}{e^{3} d}+\frac {2 a b \left (-\frac {\arcsin \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3} d}\) | \(130\) |
1/d*(-1/2*a^2/e^3/(d*x+c)^2+b^2/e^3*(-1/2*arcsin(d*x+c)^2/(d*x+c)^2-arcsin (d*x+c)*(1-(d*x+c)^2)^(1/2)/(d*x+c)+ln(d*x+c))+2*a*b/e^3*(-1/2/(d*x+c)^2*a rcsin(d*x+c)-1/2/(d*x+c)*(1-(d*x+c)^2)^(1/2)))
Time = 0.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right ) + 2 \, {\left (a b d x + a b c + {\left (b^{2} d x + b^{2} c\right )} \arcsin \left (d x + c\right )\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
-1/2*(b^2*arcsin(d*x + c)^2 + 2*a*b*arcsin(d*x + c) + a^2 - 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(d*x + c) + 2*(a*b*d*x + a*b*c + (b^2*d*x + b ^2*c)*arcsin(d*x + c))*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1))/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
\[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
(Integral(a**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integ ral(b**2*asin(c + d*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*a*b*asin(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d* *3*x**3), x))/e**3
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=-{\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d \arcsin \left (d x + c\right )}{d^{3} e^{3} x + c d^{2} e^{3}} - \frac {\log \left (d x + c\right )}{d e^{3}}\right )} b^{2} - a b {\left (\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} d}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\arcsin \left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} - \frac {b^{2} \arcsin \left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \]
-(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1)*d*arcsin(d*x + c)/(d^3*e^3*x + c*d^2* e^3) - log(d*x + c)/(d*e^3))*b^2 - a*b*(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) *d/(d^3*e^3*x + c*d^2*e^3) + arcsin(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 1/2*b^2*arcsin(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c ^2*d*e^3) - 1/2*a^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
Leaf count of result is larger than twice the leaf count of optimal. 510 vs. \(2 (83) = 166\).
Time = 0.37 (sec) , antiderivative size = 510, normalized size of antiderivative = 5.86 \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {b^{2} \arcsin \left (d x + c\right )^{2}}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} b^{2} \arcsin \left (d x + c\right )^{2}}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )^{2}}{8 \, {\left (d x + c\right )}^{2} d e^{3}} - \frac {a b \arcsin \left (d x + c\right )}{2 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} a b \arcsin \left (d x + c\right )}{4 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )}{2 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right )}{2 \, {\left (d x + c\right )} d e^{3}} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2} \arcsin \left (d x + c\right )}{4 \, {\left (d x + c\right )}^{2} d e^{3}} + \frac {2 \, b^{2} \log \left (2\right )}{d e^{3}} - \frac {b^{2} \log \left (2 \, \sqrt {-{\left (d x + c\right )}^{2} + 1} + 2\right )}{d e^{3}} + \frac {b^{2} \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{d e^{3}} + \frac {b^{2} \log \left ({\left | d x + c \right |}\right )}{d e^{3}} - \frac {a^{2}}{4 \, d e^{3}} - \frac {{\left (d x + c\right )}^{2} a^{2}}{8 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} + \frac {{\left (d x + c\right )} a b}{2 \, d e^{3} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}}{2 \, {\left (d x + c\right )} d e^{3}} - \frac {a^{2} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}}{8 \, {\left (d x + c\right )}^{2} d e^{3}} \]
-1/4*b^2*arcsin(d*x + c)^2/(d*e^3) - 1/8*(d*x + c)^2*b^2*arcsin(d*x + c)^2 /(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)^2) - 1/8*b^2*(sqrt(-(d*x + c)^2 + 1) + 1)^2*arcsin(d*x + c)^2/((d*x + c)^2*d*e^3) - 1/2*a*b*arcsin(d*x + c)/(d* e^3) - 1/4*(d*x + c)^2*a*b*arcsin(d*x + c)/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)^2) + 1/2*(d*x + c)*b^2*arcsin(d*x + c)/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/2*b^2*(sqrt(-(d*x + c)^2 + 1) + 1)*arcsin(d*x + c)/((d*x + c)*d *e^3) - 1/4*a*b*(sqrt(-(d*x + c)^2 + 1) + 1)^2*arcsin(d*x + c)/((d*x + c)^ 2*d*e^3) + 2*b^2*log(2)/(d*e^3) - b^2*log(2*sqrt(-(d*x + c)^2 + 1) + 2)/(d *e^3) + b^2*log(sqrt(-(d*x + c)^2 + 1) + 1)/(d*e^3) + b^2*log(abs(d*x + c) )/(d*e^3) - 1/4*a^2/(d*e^3) - 1/8*(d*x + c)^2*a^2/(d*e^3*(sqrt(-(d*x + c)^ 2 + 1) + 1)^2) + 1/2*(d*x + c)*a*b/(d*e^3*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/2*a*b*(sqrt(-(d*x + c)^2 + 1) + 1)/((d*x + c)*d*e^3) - 1/8*a^2*(sqrt(-(d *x + c)^2 + 1) + 1)^2/((d*x + c)^2*d*e^3)
Timed out. \[ \int \frac {(a+b \arcsin (c+d x))^2}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]