Integrand size = 32, antiderivative size = 217 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}+\frac {2 b \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}}-\frac {i b^2 \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c (d+c d x)^{3/2} (e-c e x)^{3/2}} \]
x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-I*(-c^ 2*x^2+1)^(3/2)*(a+b*arcsin(c*x))^2/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)+2*b* (-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c/ (c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)-I*b^2*(-c^2*x^2+1)^(3/2)*polylog(2,-(I*c* x+(-c^2*x^2+1)^(1/2))^2)/c/(c*d*x+d)^(3/2)/(-c*e*x+e)^(3/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(550\) vs. \(2(217)=434\).
Time = 4.47 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.53 \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\frac {a^2 c x+2 a b c x \arcsin (c x)+2 i b^2 \pi \sqrt {1-c^2 x^2} \arcsin (c x)+b^2 c x \arcsin (c x)^2-i b^2 \sqrt {1-c^2 x^2} \arcsin (c x)^2+4 b^2 \pi \sqrt {1-c^2 x^2} \log \left (1+e^{-i \arcsin (c x)}\right )+b^2 \pi \sqrt {1-c^2 x^2} \log \left (1-i e^{i \arcsin (c x)}\right )+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-b^2 \pi \sqrt {1-c^2 x^2} \log \left (1+i e^{i \arcsin (c x)}\right )+2 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-4 b^2 \pi \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+b^2 \pi \sqrt {1-c^2 x^2} \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+2 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-b^2 \pi \sqrt {1-c^2 x^2} \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c d e \sqrt {d+c d x} \sqrt {e-c e x}} \]
(a^2*c*x + 2*a*b*c*x*ArcSin[c*x] + (2*I)*b^2*Pi*Sqrt[1 - c^2*x^2]*ArcSin[c *x] + b^2*c*x*ArcSin[c*x]^2 - I*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2 + 4*b^ 2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + E^((-I)*ArcSin[c*x])] + b^2*Pi*Sqrt[1 - c^2 *x^2]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*L og[1 - I*E^(I*ArcSin[c*x])] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[1 + I*E^(I*ArcS in[c*x])] + 2*b^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x] )] - 4*b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2]] + b^2*Pi*Sqrt[1 - c^2*x^2]*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[C os[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]] + 2*a*b*Sqrt[1 - c^2*x^2]*Log[Cos[ ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]] - b^2*Pi*Sqrt[1 - c^2*x^2]*Log[Sin[(P i + 2*ArcSin[c*x])/4]] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (-I)*E^(I* ArcSin[c*x])] - (2*I)*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*ArcSin[c*x]) ])/(c*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])
Time = 0.80 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5178, 5160, 5180, 3042, 4202, 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.
\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{(c d x+d)^{3/2} (e-c e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 5178 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(a+b \arcsin (c x))^2}{\left (1-c^2 x^2\right )^{3/2}}dx}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 5160 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-2 b c \int \frac {x (a+b \arcsin (c x))}{1-c^2 x^2}dx\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 5180 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \int \frac {c x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}d\arcsin (c x)}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \int (a+b \arcsin (c x)) \tan (\arcsin (c x))d\arcsin (c x)}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \int \frac {e^{2 i \arcsin (c x)} (a+b \arcsin (c x))}{1+e^{2 i \arcsin (c x)}}d\arcsin (c x)\right )}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (\frac {1}{4} b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\left (1-c^2 x^2\right )^{3/2} \left (\frac {x (a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {2 b \left (\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )\right )}{c}\right )}{(c d x+d)^{3/2} (e-c e x)^{3/2}}\) |
((1 - c^2*x^2)^(3/2)*((x*(a + b*ArcSin[c*x])^2)/Sqrt[1 - c^2*x^2] - (2*b*( ((I/2)*(a + b*ArcSin[c*x])^2)/b - (2*I)*((-1/2*I)*(a + b*ArcSin[c*x])*Log[ 1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcSin[c*x])])/4)))/c ))/((d + c*d*x)^(3/2)*(e - c*e*x)^(3/2))
3.6.68.3.1 Defintions of rubi rules used
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]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[x*((a + b*ArcSin[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp[b *c*(n/d)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSin[c*x ])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 2)^q) Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-e^(-1) Subst[Int[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
\[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{\left (c d x +d \right )^{\frac {3}{2}} \left (-c e x +e \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \]
integral((b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)*sqrt(c*d*x + d)*sqr t(-c*e*x + e)/(c^4*d^2*e^2*x^4 - 2*c^2*d^2*e^2*x^2 + d^2*e^2), x)
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (- e \left (c x - 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \]
-b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^2*d*e*x^2 - d*e)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/(sqrt(d)*sqrt(e)) + 2*a*b*x*arcsi n(c*x)/(sqrt(-c^2*d*e*x^2 + d*e)*d*e) + a^2*x/(sqrt(-c^2*d*e*x^2 + d*e)*d* e) - a*b*sqrt(1/(d*e))*log(x^2 - 1/c^2)/(c*d*e)
\[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c d x + d\right )}^{\frac {3}{2}} {\left (-c e x + e\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{(d+c d x)^{3/2} (e-c e x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\right )}^{3/2}\,{\left (e-c\,e\,x\right )}^{3/2}} \,d x \]