Integrand size = 33, antiderivative size = 390 \[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\frac {i m (a+b \arcsin (c x))^3}{6 b^2 c}-\frac {m (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c}-\frac {m (a+b \arcsin (c x))^2 \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c}+\frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}+\frac {i m (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}+\frac {i m (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {b m \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c}-\frac {b m \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c} \]
1/6*I*m*(a+b*arcsin(c*x))^3/b^2/c+1/2*(a+b*arcsin(c*x))^2*ln(h*(g*x+f)^m)/ b/c-1/2*m*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^ 2*f^2-g^2)^(1/2)))/b/c-1/2*m*(a+b*arcsin(c*x))^2*ln(1-I*(I*c*x+(-c^2*x^2+1 )^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/b/c+I*m*(a+b*arcsin(c*x))*polylog(2, I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/c+I*m*(a+b*arcsi n(c*x))*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)) )/c-b*m*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)) )/c-b*m*polylog(3,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)) )/c
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2724\) vs. \(2(390)=780\).
Time = 10.46 (sec) , antiderivative size = 2724, normalized size of antiderivative = 6.98 \[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\text {Result too large to show} \]
(m*ArcSin[c*x]*(2*a + b*ArcSin[c*x])*Log[f + g*x])/(2*c) + (a*ArcSin[c*x]* (-(m*Log[f + g*x]) + Log[h*(f + g*x)^m]))/c + (b*f*(-(m*Log[f + g*x]) + Lo g[h*(f + g*x)^m])*((-I)*ArcSin[c*x]*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c* f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c ^2*f^2 - g^2])]) - PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^ 2*f^2 - g^2])] + PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2] + (a*g*m*(-1/2*(((3*I)/2)*Pi*ArcSin[c*x] - (I /2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - Pi*Log[1 + I*E^(I *ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] - 2*Pi*Log[Cos [ArcSin[c*x]/2]] + Pi*Log[-Cos[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2, (-I)*E^(I*ArcSin[c*x])])/(c*(-c^(-1) - f/g)*g) + ((I/2)*Pi*ArcSin[c*x] - (I/2)*ArcSin[c*x]^2 + 2*Pi*Log[1 + E^((-I)*ArcSin[c*x])] + Pi*Log[1 - I*E^ (I*ArcSin[c*x])] + 2*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] - 2*Pi*Log[C os[ArcSin[c*x]/2]] - Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] - (2*I)*PolyLog[2 , I*E^(I*ArcSin[c*x])])/(2*c*(c^(-1) - f/g)*g) + (((-1/2*I)*ArcSin[c*x]^2) /g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^ 2])])/g + (ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (I*PolyLog[2, ((-I)*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2 *f^2 - g^2])])/g - (I*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f ^2 - g^2])])/g)/(c^2*(-c^(-1) - f/g)*(c^(-1) - f/g))))/c - a*c*g*m*(-1/...
Time = 1.26 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {5278, 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.
\(\displaystyle \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5278 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \int \frac {(a+b \arcsin (c x))^2}{f+g x}dx}{2 b c}\) |
\(\Big \downarrow \) 5240 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c f+c g x}d\arcsin (c x)}{2 b c}\) |
\(\Big \downarrow \) 5030 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \left (\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c f-i e^{i \arcsin (c x)} g-\sqrt {c^2 f^2-g^2}}d\arcsin (c x)+\int \frac {e^{i \arcsin (c x)} (a+b \arcsin (c x))^2}{c f-i e^{i \arcsin (c x)} g+\sqrt {c^2 f^2-g^2}}d\arcsin (c x)-\frac {i (a+b \arcsin (c x))^3}{3 b g}\right )}{2 b c}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \left (-\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}-\frac {2 b \int (a+b \arcsin (c x)) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^3}{3 b g}\right )}{2 b c}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \left (-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-i b \int \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )d\arcsin (c x)\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^3}{3 b g}\right )}{2 b c}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \left (-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \int e^{-i \arcsin (c x)} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )de^{i \arcsin (c x)}\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^3}{3 b g}\right )}{2 b c}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {(a+b \arcsin (c x))^2 \log \left (h (f+g x)^m\right )}{2 b c}-\frac {g m \left (-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )\right )}{g}-\frac {2 b \left (i (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (3,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}+\frac {(a+b \arcsin (c x))^2 \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {i (a+b \arcsin (c x))^3}{3 b g}\right )}{2 b c}\) |
((a + b*ArcSin[c*x])^2*Log[h*(f + g*x)^m])/(2*b*c) - (g*m*(((-1/3*I)*(a + b*ArcSin[c*x])^3)/(b*g) + ((a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c* x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + ((a + b*ArcSin[c*x])^2*Log[1 - (I *E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (2*b*(I*(a + b*Arc Sin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])]))/g - (2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt [c^2*f^2 - g^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f ^2 - g^2])]))/g))/(2*b*c)
3.1.85.3.1 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[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]]]
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]
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]
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]
Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcSin[(c_.)*(x_)]*(b_. ))^(n_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[Log[h*(f + g*x)^m]*( (a + b*ArcSin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[g*(m/(b*c*Sqr t[d]*(n + 1))) Int[(a + b*ArcSin[c*x])^(n + 1)/(f + g*x), x], x] /; FreeQ [{a, b, c, d, e, f, g, h, m}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && IGtQ[ n, 0]
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]
\[\int \frac {\left (a +b \arcsin \left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}d x\]
\[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
\[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
(b*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(sqrt(c*x + 1)*s qrt(-c*x + 1)), x)*log(h) + b*c*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt( -c*x + 1))*log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a*c*integ rate(log((g*x + f)^m)/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a*arctan2(c*x, sqrt(-c^2*x^2 + 1))*log(h))/c
\[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arcsin (c x)) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{\sqrt {1-c^2\,x^2}} \,d x \]