Integrand size = 38, antiderivative size = 287 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\frac {i b p q \arcsin \left (\frac {h x}{2}\right )^2}{2 h}-\frac {b p q \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}-\frac {b p q \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}+\frac {i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{h}+\frac {i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{h} \]
1/2*I*b*p*q*arcsin(1/2*h*x)^2/h+arcsin(1/2*h*x)*(a+b*ln(c*(d*(f*x+e)^p)^q) )/h-b*p*q*arcsin(1/2*h*x)*ln(1+2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e *h-(-e^2*h^2+4*f^2)^(1/2)))/h-b*p*q*arcsin(1/2*h*x)*ln(1+2*(1/2*I*h*x+1/2* (-h^2*x^2+4)^(1/2))*f/(I*e*h+(-e^2*h^2+4*f^2)^(1/2)))/h+I*b*p*q*polylog(2, -2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e*h-(-e^2*h^2+4*f^2)^(1/2)))/h+ I*b*p*q*polylog(2,-2*(1/2*I*h*x+1/2*(-h^2*x^2+4)^(1/2))*f/(I*e*h+(-e^2*h^2 +4*f^2)^(1/2)))/h
Time = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\frac {\arcsin \left (\frac {h x}{2}\right ) \left (2 a+i b p q \arcsin \left (\frac {h x}{2}\right )-2 b p q \log \left (1-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{-i e h+\sqrt {4 f^2-e^2 h^2}}\right )-2 b p q \log \left (1+\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )+2 b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+2 i b p q \operatorname {PolyLog}\left (2,\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{-i e h+\sqrt {4 f^2-e^2 h^2}}\right )+2 i b p q \operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{2 h} \]
(ArcSin[(h*x)/2]*(2*a + I*b*p*q*ArcSin[(h*x)/2] - 2*b*p*q*Log[1 - (2*E^(I* ArcSin[(h*x)/2])*f)/((-I)*e*h + Sqrt[4*f^2 - e^2*h^2])] - 2*b*p*q*Log[1 + (2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + Sqrt[4*f^2 - e^2*h^2])] + 2*b*Log[c*( d*(e + f*x)^p)^q]) + (2*I)*b*p*q*PolyLog[2, (2*E^(I*ArcSin[(h*x)/2])*f)/(( -I)*e*h + Sqrt[4*f^2 - e^2*h^2])] + (2*I)*b*p*q*PolyLog[2, (-2*E^(I*ArcSin [(h*x)/2])*f)/(I*e*h + Sqrt[4*f^2 - e^2*h^2])])/(2*h)
Time = 1.48 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {2895, 2852, 27, 5240, 5032, 27, 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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {h x+2}} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {h x+2}}dx\) |
\(\Big \downarrow \) 2852 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-b f p q \int \frac {\arcsin \left (\frac {h x}{2}\right )}{h (e+f x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \int \frac {\arcsin \left (\frac {h x}{2}\right )}{e+f x}dx}{h}\) |
\(\Big \downarrow \) 5240 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \int \frac {\sqrt {1-\frac {h^2 x^2}{4}} \arcsin \left (\frac {h x}{2}\right )}{\frac {e h}{2}+\frac {f x h}{2}}d\arcsin \left (\frac {h x}{2}\right )}{h}\) |
\(\Big \downarrow \) 5032 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \left (i \int \frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} \arcsin \left (\frac {h x}{2}\right )}{2 e^{i \arcsin \left (\frac {h x}{2}\right )} f+i e h-\sqrt {4 f^2-e^2 h^2}}d\arcsin \left (\frac {h x}{2}\right )+i \int \frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} \arcsin \left (\frac {h x}{2}\right )}{2 e^{i \arcsin \left (\frac {h x}{2}\right )} f+i e h+\sqrt {4 f^2-e^2 h^2}}d\arcsin \left (\frac {h x}{2}\right )-\frac {i \arcsin \left (\frac {h x}{2}\right )^2}{2 f}\right )}{h}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \left (2 i \int \frac {e^{i \arcsin \left (\frac {h x}{2}\right )} \arcsin \left (\frac {h x}{2}\right )}{2 e^{i \arcsin \left (\frac {h x}{2}\right )} f+i e h-\sqrt {4 f^2-e^2 h^2}}d\arcsin \left (\frac {h x}{2}\right )+2 i \int \frac {e^{i \arcsin \left (\frac {h x}{2}\right )} \arcsin \left (\frac {h x}{2}\right )}{2 e^{i \arcsin \left (\frac {h x}{2}\right )} f+i e h+\sqrt {4 f^2-e^2 h^2}}d\arcsin \left (\frac {h x}{2}\right )-\frac {i \arcsin \left (\frac {h x}{2}\right )^2}{2 f}\right )}{h}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \left (2 i \left (\frac {i \int \log \left (\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}+1\right )d\arcsin \left (\frac {h x}{2}\right )}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )+2 i \left (\frac {i \int \log \left (\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}+1\right )d\arcsin \left (\frac {h x}{2}\right )}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )-\frac {i \arcsin \left (\frac {h x}{2}\right )^2}{2 f}\right )}{h}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \left (2 i \left (\frac {\int e^{-i \arcsin \left (\frac {h x}{2}\right )} \log \left (\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}+1\right )de^{i \arcsin \left (\frac {h x}{2}\right )}}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )+2 i \left (\frac {\int e^{-i \arcsin \left (\frac {h x}{2}\right )} \log \left (\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}+1\right )de^{i \arcsin \left (\frac {h x}{2}\right )}}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )-\frac {i \arcsin \left (\frac {h x}{2}\right )^2}{2 f}\right )}{h}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\arcsin \left (\frac {h x}{2}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h}-\frac {b f p q \left (2 i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h-\sqrt {4 f^2-e^2 h^2}}\right )}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{-\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )+2 i \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {2 e^{i \arcsin \left (\frac {h x}{2}\right )} f}{i e h+\sqrt {4 f^2-e^2 h^2}}\right )}{2 f}-\frac {i \arcsin \left (\frac {h x}{2}\right ) \log \left (1+\frac {2 f e^{i \arcsin \left (\frac {h x}{2}\right )}}{\sqrt {4 f^2-e^2 h^2}+i e h}\right )}{2 f}\right )-\frac {i \arcsin \left (\frac {h x}{2}\right )^2}{2 f}\right )}{h}\) |
(ArcSin[(h*x)/2]*(a + b*Log[c*(d*(e + f*x)^p)^q]))/h - (b*f*p*q*(((-1/2*I) *ArcSin[(h*x)/2]^2)/f + (2*I)*(((-1/2*I)*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*A rcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2])])/f - PolyLog[2, (-2*E^ (I*ArcSin[(h*x)/2])*f)/(I*e*h - Sqrt[4*f^2 - e^2*h^2])]/(2*f)) + (2*I)*((( -1/2*I)*ArcSin[(h*x)/2]*Log[1 + (2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + Sqrt[ 4*f^2 - e^2*h^2])])/f - PolyLog[2, (-2*E^(I*ArcSin[(h*x)/2])*f)/(I*e*h + S qrt[4*f^2 - e^2*h^2])]/(2*f))))/h
3.6.21.3.1 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[(((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[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/(Sqrt[(f1_) + (g1_ .)*(x_)]*Sqrt[(f2_) + (g2_.)*(x_)]), x_Symbol] :> With[{u = IntHide[1/Sqrt[ f1*f2 + g1*g2*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Simp[b*e*n Int[SimplifyIntegrand[u/(d + e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f1, g1, f2, g2, n}, x] && EqQ[f2*g1 + f1*g2, 0] && GtQ[f1, 0] && GtQ[f2, 0]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
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] + (Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x] + Simp[I Int[(e + f*x)^m*(E^(I*(c + d*x) )/(I*a + Rt[-a^2 + b^2, 2] + b*E^(I*(c + d*x)))), x], x]) /; FreeQ[{a, b, c , d, e, f}, x] && IGtQ[m, 0] && NegQ[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]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {-h x +2}\, \sqrt {h x +2}}d x\]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + 2} \sqrt {-h x + 2}} \,d x } \]
integral(-(sqrt(h*x + 2)*sqrt(-h*x + 2)*b*log(((f*x + e)^p*d)^q*c) + sqrt( h*x + 2)*sqrt(-h*x + 2)*a)/(h^2*x^2 - 4), x)
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {- h x + 2} \sqrt {h x + 2}}\, dx \]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + 2} \sqrt {-h x + 2}} \,d x } \]
b*integrate((q*log(d) + log(((f*x + e)^p)^q) + log(c))/(sqrt(h*x + 2)*sqrt (-h*x + 2)), x) + a*arcsin(1/2*h*x)/h
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x + 2} \sqrt {-h x + 2}} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2-h x} \sqrt {2+h x}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {2-h\,x}\,\sqrt {h\,x+2}} \,d x \]