Integrand size = 18, antiderivative size = 245 \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=\frac {d n \sqrt {1-a^2 x^2}}{a}+\frac {\left (3 a^2 d+e\right ) n \sqrt {1-a^2 x^2}}{3 a^3}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2}}{27 a^3}-d n x \arccos (a x)-\frac {1}{9} e n x^3 \arccos (a x)+\frac {e n \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}-\frac {\left (3 a^2 d+e\right ) n \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{3 a^3}-\frac {\left (3 a^2 d+e\right ) \sqrt {1-a^2 x^2} \log \left (c x^n\right )}{3 a^3}+\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x) \log \left (c x^n\right ) \]
-2/27*e*n*(-a^2*x^2+1)^(3/2)/a^3-d*n*x*arccos(a*x)-1/9*e*n*x^3*arccos(a*x) +1/9*e*n*arctanh((-a^2*x^2+1)^(1/2))/a^3-1/3*(3*a^2*d+e)*n*arctanh((-a^2*x ^2+1)^(1/2))/a^3+1/9*e*(-a^2*x^2+1)^(3/2)*ln(c*x^n)/a^3+d*x*arccos(a*x)*ln (c*x^n)+1/3*e*x^3*arccos(a*x)*ln(c*x^n)+d*n*(-a^2*x^2+1)^(1/2)/a+1/3*(3*a^ 2*d+e)*n*(-a^2*x^2+1)^(1/2)/a^3-1/3*(3*a^2*d+e)*ln(c*x^n)*(-a^2*x^2+1)^(1/ 2)/a^3
Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.01 \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=-\frac {-54 a^2 d n \sqrt {1-a^2 x^2}-7 e n \sqrt {1-a^2 x^2}-2 a^2 e n x^2 \sqrt {1-a^2 x^2}-3 \left (9 a^2 d+2 e\right ) n \log (x)+27 a^2 d \sqrt {1-a^2 x^2} \log \left (c x^n\right )+6 e \sqrt {1-a^2 x^2} \log \left (c x^n\right )+3 a^2 e x^2 \sqrt {1-a^2 x^2} \log \left (c x^n\right )+3 a^3 x \arccos (a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+27 a^2 d n \log \left (1+\sqrt {1-a^2 x^2}\right )+6 e n \log \left (1+\sqrt {1-a^2 x^2}\right )}{27 a^3} \]
-1/27*(-54*a^2*d*n*Sqrt[1 - a^2*x^2] - 7*e*n*Sqrt[1 - a^2*x^2] - 2*a^2*e*n *x^2*Sqrt[1 - a^2*x^2] - 3*(9*a^2*d + 2*e)*n*Log[x] + 27*a^2*d*Sqrt[1 - a^ 2*x^2]*Log[c*x^n] + 6*e*Sqrt[1 - a^2*x^2]*Log[c*x^n] + 3*a^2*e*x^2*Sqrt[1 - a^2*x^2]*Log[c*x^n] + 3*a^3*x*ArcCos[a*x]*(n*(9*d + e*x^2) - 3*(3*d + e* x^2)*Log[c*x^n]) + 27*a^2*d*n*Log[1 + Sqrt[1 - a^2*x^2]] + 6*e*n*Log[1 + S qrt[1 - a^2*x^2]])/a^3
Time = 0.47 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2834, 2009}
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.
| \(\displaystyle \int \arccos (a x) \left (d+e x^2\right ) \log \left (c x^n\right ) \, dx\) |
| \(\Big \downarrow \) 2834 |
| \(\displaystyle -n \int \left (\frac {1}{3} e \arccos (a x) x^2+d \arccos (a x)+\frac {e \left (1-a^2 x^2\right )^{3/2}}{9 a^3 x}-\frac {\left (3 d a^2+e\right ) \sqrt {1-a^2 x^2}}{3 a^3 x}\right )dx-\frac {\sqrt {1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}+\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x) \log \left (c x^n\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -n \left (-\frac {d \sqrt {1-a^2 x^2}}{a}+\frac {\text {arctanh}\left (\sqrt {1-a^2 x^2}\right ) \left (3 a^2 d+e\right )}{3 a^3}-\frac {e \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{9 a^3}-\frac {\sqrt {1-a^2 x^2} \left (3 a^2 d+e\right )}{3 a^3}+\frac {2 e \left (1-a^2 x^2\right )^{3/2}}{27 a^3}+d x \arccos (a x)+\frac {1}{9} e x^3 \arccos (a x)\right )-\frac {\sqrt {1-a^2 x^2} \left (3 a^2 d+e\right ) \log \left (c x^n\right )}{3 a^3}+\frac {e \left (1-a^2 x^2\right )^{3/2} \log \left (c x^n\right )}{9 a^3}+d x \arccos (a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \arccos (a x) \log \left (c x^n\right )\) |
-(n*(-((d*Sqrt[1 - a^2*x^2])/a) - ((3*a^2*d + e)*Sqrt[1 - a^2*x^2])/(3*a^3 ) + (2*e*(1 - a^2*x^2)^(3/2))/(27*a^3) + d*x*ArcCos[a*x] + (e*x^3*ArcCos[a *x])/9 - (e*ArcTanh[Sqrt[1 - a^2*x^2]])/(9*a^3) + ((3*a^2*d + e)*ArcTanh[S qrt[1 - a^2*x^2]])/(3*a^3))) - ((3*a^2*d + e)*Sqrt[1 - a^2*x^2]*Log[c*x^n] )/(3*a^3) + (e*(1 - a^2*x^2)^(3/2)*Log[c*x^n])/(9*a^3) + d*x*ArcCos[a*x]*L og[c*x^n] + (e*x^3*ArcCos[a*x]*Log[c*x^n])/3
3.2.87.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)* (x_))]^(m_.), x_Symbol] :> With[{u = IntHide[Px*F[d*(e + f*x)]^m, x]}, Simp [(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && IGtQ[m, 0] && MemberQ[{ArcSi n, ArcCos, ArcSinh, ArcCosh}, F]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.11 (sec) , antiderivative size = 5619, normalized size of antiderivative = 22.93
Time = 0.48 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.26 \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=\frac {18 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \arccos \left (a x\right ) \log \left (c\right ) + 18 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \arccos \left (a x\right ) \log \left (x\right ) - 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) - 6 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x - {\left (9 \, a^{3} d + a^{3} e\right )} n\right )} \arccos \left (a x\right ) - 6 \, {\left ({\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (3 \, a^{3} d + a^{3} e\right )} \log \left (c\right )\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 2 \, {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \left (c\right ) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \left (x\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{54 \, a^{3}} \]
1/54*(18*(a^3*e*x^3 + 3*a^3*d*x - 3*a^3*d - a^3*e)*arccos(a*x)*log(c) + 18 *(a^3*e*n*x^3 + 3*a^3*d*n*x)*arccos(a*x)*log(x) - 3*(9*a^2*d + 2*e)*n*log( sqrt(-a^2*x^2 + 1) + 1) + 3*(9*a^2*d + 2*e)*n*log(sqrt(-a^2*x^2 + 1) - 1) - 6*(a^3*e*n*x^3 + 9*a^3*d*n*x - (9*a^3*d + a^3*e)*n)*arccos(a*x) - 6*((9* a^3*d + a^3*e)*n - 3*(3*a^3*d + a^3*e)*log(c))*arctan(sqrt(-a^2*x^2 + 1)*a *x/(a^2*x^2 - 1)) + 2*(2*a^2*e*n*x^2 + (54*a^2*d + 7*e)*n - 3*(a^2*e*x^2 + 9*a^2*d + 2*e)*log(c) - 3*(a^2*e*n*x^2 + (9*a^2*d + 2*e)*n)*log(x))*sqrt( -a^2*x^2 + 1))/a^3
Time = 50.33 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.82 \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=- \frac {a e n \left (\begin {cases} - \frac {\begin {cases} \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3} - \frac {\sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases}}{3 a^{2}} - \frac {2 \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{3 a^{4}} & \text {for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac {x^{4}}{16} & \text {otherwise} \end {cases}\right )}{3} - \frac {a e n \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{9} + \frac {a e \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a^{2} \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{3} - d n \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\\begin {cases} x \operatorname {acos}{\left (a x \right )} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} - \frac {\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}}{a} & \text {otherwise} \end {cases}\right ) + d \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\x \operatorname {acos}{\left (a x \right )} - \frac {\sqrt {- a^{2} x^{2} + 1}}{a} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - \frac {e n x^{3} \operatorname {acos}{\left (a x \right )}}{9} + \frac {e x^{3} \log {\left (c x^{n} \right )} \operatorname {acos}{\left (a x \right )}}{3} \]
-a*e*n*Piecewise((-Piecewise((x**2*sqrt(-a**2*x**2 + 1)/3 - sqrt(-a**2*x** 2 + 1)/(3*a**2), Ne(a, 0)), (x**2/2, True))/(3*a**2) - 2*Piecewise((I*sqrt (a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin(1/(a*x)), Abs(a**2* x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/(3*a**4), (a > -oo) & (a < oo) & Ne(a, 0)), (x**4/16, T rue))/3 - a*e*n*Piecewise((-x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a **2*x**2 + 1)/(3*a**4), Ne(a**2, 0)), (x**4/4, True))/9 + a*e*Piecewise((- x**2*sqrt(-a**2*x**2 + 1)/(3*a**2) - 2*sqrt(-a**2*x**2 + 1)/(3*a**4), Ne(a **2, 0)), (x**4/4, True))*log(c*x**n)/3 - d*n*Piecewise((pi*x/2, Eq(a, 0)) , (Piecewise((x*acos(a*x) - sqrt(-a**2*x**2 + 1)/a, Ne(a, 0)), (pi*x/2, Tr ue)) - Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I* asin(1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2) /2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a, True)) + d*Piecewise((pi*x/2 , Eq(a, 0)), (x*acos(a*x) - sqrt(-a**2*x**2 + 1)/a, True))*log(c*x**n) - e *n*x**3*acos(a*x)/9 + e*x**3*log(c*x**n)*acos(a*x)/3
\[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=\int { {\left (e x^{2} + d\right )} \arccos \left (a x\right ) \log \left (c x^{n}\right ) \,d x } \]
-1/54*(-I*(27*a^2*d*n*(2*x/a^2 - log(a*x + 1)/a^3 + log(a*x - 1)/a^3) + a^ 2*e*n*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5) - 162*a^2*e*n*integrate(1/9*x^4*log(x)/(a^2*x^2 - 1), x) - 486*a^2*d*n*integ rate(1/9*x^2*log(x)/(a^2*x^2 - 1), x) - 27*a^2*d*(2*x/a^2 - log(a*x + 1)/a ^3 + log(a*x - 1)/a^3)*log(c) - 3*a^2*e*(2*(a^2*x^3 + 3*x)/a^4 - 3*log(a*x + 1)/a^5 + 3*log(a*x - 1)/a^5)*log(c))*a^3 - 2*(-2*I*a^3*e*n + 3*I*a^3*e* log(c))*x^3 + 54*a^3*integrate(-1/9*((a*e*n - 3*a*e*log(c))*x^3 + 9*(a*d*n - a*d*log(c))*x - 3*(a*e*x^3 + 3*a*d*x)*log(x^n))*sqrt(a*x + 1)*sqrt(-a*x + 1)/(a^2*x^2 - 1), x) - 9*(3*I*a^2*d + I*e)*n*dilog(a*x) - 9*(-3*I*a^2*d - I*e)*n*dilog(-a*x) - 6*(9*I*a^3*d*log(c) + 3*I*a*e*log(c) + 2*(-9*I*a^3 *d - 2*I*a*e)*n)*x + 6*((a^3*e*n - 3*a^3*e*log(c))*x^3 + 9*(a^3*d*n - a^3* d*log(c))*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) - 3*(-9*I*a^2*d*lo g(c) + (9*I*a^2*d + I*e)*n - 3*I*e*log(c))*log(a*x + 1) - 3*(9*I*a^2*d*log (c) + (-9*I*a^2*d - I*e)*n + 3*I*e*log(c))*log(a*x - 1) - 3*(2*I*a^3*e*x^3 + 6*(3*I*a^3*d + I*a*e)*x + 6*(a^3*e*x^3 + 3*a^3*d*x)*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) + 3*(-3*I*a^2*d - I*e)*log(a*x + 1) + 3*(3*I*a^2*d + I*e)*log(-a*x + 1))*log(x^n))/a^3
Leaf count of result is larger than twice the leaf count of optimal. 11159 vs. \(2 (215) = 430\).
Time = 0.76 (sec) , antiderivative size = 11159, normalized size of antiderivative = 45.55 \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=\text {Too large to display} \]
1/54*(18*a^3*e*x^3*arccos(a*x)*log(c) + 54*a^3*d*n*x*arccos(a*x)*log(a*x) - 54*a^3*d*n*x*arccos(a*x)*log(a) + 54*a^3*d*x*arccos(a*x)*log(c) - 6*sqrt (-a^2*x^2 + 1)*a^2*e*x^2*log(c) - 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*log(a*x) + 54*sqrt(-a^2*x^2 + 1)*a^2*d*n*log(a) - 54*sqrt(-a^2*x^2 + 1)*a^2*d*log(c) + 54*a^2*d*n*arccos(a*x)/((a^2*x^2 - 1)/(a*x + 1)^2 - 1) + 54*a^2*d*n*log (abs(a*x + sqrt(-a^2*x^2 + 1) + 1))/((a^2*x^2 - 1)/(a*x + 1)^2 - 1) - 54*a ^2*d*n*log(abs(-a*x + sqrt(-a^2*x^2 + 1) - 1))/((a^2*x^2 - 1)/(a*x + 1)^2 - 1) + 216*sqrt(-a^2*x^2 + 1)*a^2*d*n/(a*x - (a^2*x^2 - 1)*a*x/(a*x + 1)^2 - (a^2*x^2 - 1)/(a*x + 1)^2 + 1) + (18*a^3*x^3*arccos(a*x)*log(a*x) - 18* a^3*x^3*arccos(a*x)*log(a) - 48*(a^2*x^2 - 1)*a^4*x^4*arccos(a*x)/((16*(a^ 2*x^2 - 1)*a^4*x^4/(4*a^3*x^3 - 3*a*x + 1)^2 - 16*(a^2*x^2 - 1)^2*a^4*x^4/ ((4*a^3*x^3 - 3*a*x + 1)^2*(a*x + 1)^2) - 8*(a^2*x^2 - 1)*a^2*x^2/(16*a^6* x^6 - 24*a^4*x^4 + 8*a^3*x^3 + 9*a^2*x^2 - 6*a*x + 1) + 8*(a^2*x^2 - 1)^2* a^2*x^2/((16*a^6*x^6 - 24*a^4*x^4 + 8*a^3*x^3 + 9*a^2*x^2 - 6*a*x + 1)*(a* x + 1)^2) + (a^2*x^2 - 1)/(4*a^3*x^3 - 3*a*x + 1)^2 + (a^2*x^2 - 1)/(a*x + 1)^2 - (a^2*x^2 - 1)^2/((4*a^3*x^3 - 3*a*x + 1)^2*(a*x + 1)^2) - 1)*(4*a^ 3*x^3 - 3*a*x + 1)^2) - 192*(a^2*x^2 - 1)*a^4*x^4*log(abs(a*x + sqrt(-a^2* x^2 + 1) + 1))/((16*(a^2*x^2 - 1)*a^4*x^4/(4*a^3*x^3 - 3*a*x + 1)^2 - 16*( a^2*x^2 - 1)^2*a^4*x^4/((4*a^3*x^3 - 3*a*x + 1)^2*(a*x + 1)^2) - 8*(a^2*x^ 2 - 1)*a^2*x^2/(16*a^6*x^6 - 24*a^4*x^4 + 8*a^3*x^3 + 9*a^2*x^2 - 6*a*x...
Timed out. \[ \int \left (d+e x^2\right ) \arccos (a x) \log \left (c x^n\right ) \, dx=\int \ln \left (c\,x^n\right )\,\mathrm {acos}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \]