3.40 \(\int x^m \sin ^3(a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log (c x^n)) \, dx\)
Optimal. Leaf size=226 \[ -\frac{4 x^{m+1} \sin ^3\left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}+\frac{8 x^{m+1} \sin \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}-\frac{4 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2}+\frac{6 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right ) \cos \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2} \]
[Out]
(-4*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])/(5*(1 + m)^2) + (8*x^(1
+ m)*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])/(5*(1 + m)) + (6*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[
a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2)/(5*(1 + m)^2) - (
4*x^(1 + m)*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3)/(5*(1 + m))
________________________________________________________________________________________
Rubi [A] time = 0.0790366, antiderivative size = 226, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used =
{4487, 4485} \[ -\frac{4 x^{m+1} \sin ^3\left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}+\frac{8 x^{m+1} \sin \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)}-\frac{4 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2}+\frac{6 n \sqrt{-\frac{(m+1)^2}{n^2}} x^{m+1} \sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right ) \cos \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (m+1)^2} \]
Antiderivative was successfully verified.
[In]
Int[x^m*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3,x]
[Out]
(-4*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])/(5*(1 + m)^2) + (8*x^(1
+ m)*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])/(5*(1 + m)) + (6*Sqrt[-((1 + m)^2/n^2)]*n*x^(1 + m)*Cos[
a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^2)/(5*(1 + m)^2) - (
4*x^(1 + m)*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3)/(5*(1 + m))
Rule 4487
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> Simp[((m + 1)*(e*x)
^(m + 1)*Sin[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b
^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^(p - 2), x], x] - Simp[(b*d*n*p*(e*x)^(m +
1)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; Free
Q[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]
Rule 4485
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[((m + 1)*(e*x)^(m +
1)*Sin[d*(a + b*Log[c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] - Simp[(b*d*n*(e*x)^(m + 1)*Cos[d*(a + b*Log[
c*x^n])])/(b^2*d^2*e*n^2 + e*(m + 1)^2), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2,
0]
Rubi steps
\begin{align*} \int x^m \sin ^3\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{6 \sqrt{-\frac{(1+m)^2}{n^2}} n x^{1+m} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}-\frac{4 x^{1+m} \sin ^3\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}+\frac{6}{5} \int x^m \sin \left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \, dx\\ &=-\frac{4 \sqrt{-\frac{(1+m)^2}{n^2}} n x^{1+m} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}+\frac{8 x^{1+m} \sin \left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}+\frac{6 \sqrt{-\frac{(1+m)^2}{n^2}} n x^{1+m} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right ) \sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)^2}-\frac{4 x^{1+m} \sin ^3\left (a+\frac{1}{2} \sqrt{-\frac{(1+m)^2}{n^2}} \log \left (c x^n\right )\right )}{5 (1+m)}\\ \end{align*}
Mathematica [A] time = 1.1883, size = 169, normalized size = 0.75 \[ \frac{x^{m+1} \left (2 (m+1) \left (5 \sin \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )+\sin \left (3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )\right )-5 n \sqrt{-\frac{(m+1)^2}{n^2}} \cos \left (a+\frac{1}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )-3 n \sqrt{-\frac{(m+1)^2}{n^2}} \cos \left (3 a+\frac{3}{2} \sqrt{-\frac{(m+1)^2}{n^2}} \log \left (c x^n\right )\right )\right )}{10 (m+1)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2]^3,x]
[Out]
(x^(1 + m)*(-5*Sqrt[-((1 + m)^2/n^2)]*n*Cos[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] - 3*Sqrt[-((1 + m)^2/n^
2)]*n*Cos[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2] + 2*(1 + m)*(5*Sin[a + (Sqrt[-((1 + m)^2/n^2)]*Log[c*
x^n])/2] + Sin[3*a + (3*Sqrt[-((1 + m)^2/n^2)]*Log[c*x^n])/2])))/(10*(1 + m)^2)
________________________________________________________________________________________
Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{2}\sqrt{-{\frac{ \left ( 1+m \right ) ^{2}}{{n}^{2}}}}} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*sin(a+1/2*ln(c*x^n)*(-(1+m)^2/n^2)^(1/2))^3,x)
[Out]
int(x^m*sin(a+1/2*ln(c*x^n)*(-(1+m)^2/n^2)^(1/2))^3,x)
________________________________________________________________________________________
Maxima [A] time = 1.29846, size = 263, normalized size = 1.16 \begin{align*} -\frac{{\left (c^{\frac{3 \, m}{n} + \frac{3}{n}} x e^{\left (m \log \left (x\right ) + \frac{3 \, m \log \left (x^{n}\right )}{n} + \frac{3 \, \log \left (x^{n}\right )}{n}\right )} \sin \left (3 \, a\right ) - 5 \, c^{\frac{2 \, m}{n} + \frac{2}{n}} x e^{\left (m \log \left (x\right ) + \frac{2 \, m \log \left (x^{n}\right )}{n} + \frac{2 \, \log \left (x^{n}\right )}{n}\right )} \sin \left (a\right ) - 15 \, c^{\frac{m}{n} + \frac{1}{n}} x e^{\left (m \log \left (x\right ) + \frac{m \log \left (x^{n}\right )}{n} + \frac{\log \left (x^{n}\right )}{n}\right )} \sin \left (a\right ) - 5 \, x x^{m} \sin \left (3 \, a\right )\right )} e^{\left (-\frac{3 \, m \log \left (x^{n}\right )}{2 \, n} - \frac{3 \, \log \left (x^{n}\right )}{2 \, n}\right )}}{20 \,{\left (c^{\frac{3 \, m}{2 \, n} + \frac{3}{2 \, n}} m + c^{\frac{3 \, m}{2 \, n} + \frac{3}{2 \, n}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sin(a+1/2*log(c*x^n)*(-(1+m)^2/n^2)^(1/2))^3,x, algorithm="maxima")
[Out]
-1/20*(c^(3*m/n + 3/n)*x*e^(m*log(x) + 3*m*log(x^n)/n + 3*log(x^n)/n)*sin(3*a) - 5*c^(2*m/n + 2/n)*x*e^(m*log(
x) + 2*m*log(x^n)/n + 2*log(x^n)/n)*sin(a) - 15*c^(m/n + 1/n)*x*e^(m*log(x) + m*log(x^n)/n + log(x^n)/n)*sin(a
) - 5*x*x^m*sin(3*a))*e^(-3/2*m*log(x^n)/n - 3/2*log(x^n)/n)/(c^(3/2*m/n + 3/2/n)*m + c^(3/2*m/n + 3/2/n))
________________________________________________________________________________________
Fricas [C] time = 0.502763, size = 387, normalized size = 1.71 \begin{align*} \frac{{\left (5 i \, e^{\left (-\frac{{\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )}{n}\right )} - 15 i \, e^{\left (-\frac{2 \,{\left ({\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )\right )}}{n}\right )} - 5 i \, e^{\left (-\frac{3 \,{\left ({\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )\right )}}{n}\right )} - i\right )} e^{\left (\frac{5 \,{\left ({\left (m + 1\right )} n \log \left (x\right ) - 2 i \, a n +{\left (m + 1\right )} \log \left (c\right )\right )}}{2 \, n} + \frac{2 i \, a n -{\left (m + 1\right )} \log \left (c\right )}{n}\right )}}{20 \,{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sin(a+1/2*log(c*x^n)*(-(1+m)^2/n^2)^(1/2))^3,x, algorithm="fricas")
[Out]
1/20*(5*I*e^(-((m + 1)*n*log(x) - 2*I*a*n + (m + 1)*log(c))/n) - 15*I*e^(-2*((m + 1)*n*log(x) - 2*I*a*n + (m +
1)*log(c))/n) - 5*I*e^(-3*((m + 1)*n*log(x) - 2*I*a*n + (m + 1)*log(c))/n) - I)*e^(5/2*((m + 1)*n*log(x) - 2*
I*a*n + (m + 1)*log(c))/n + (2*I*a*n - (m + 1)*log(c))/n)/(m + 1)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sin ^{3}{\left (a + \frac{\sqrt{- \frac{m^{2}}{n^{2}} - \frac{2 m}{n^{2}} - \frac{1}{n^{2}}} \log{\left (c x^{n} \right )}}{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*sin(a+1/2*ln(c*x**n)*(-(1+m)**2/n**2)**(1/2))**3,x)
[Out]
Integral(x**m*sin(a + sqrt(-m**2/n**2 - 2*m/n**2 - 1/n**2)*log(c*x**n)/2)**3, x)
________________________________________________________________________________________
Giac [C] time = 3.68101, size = 2525, normalized size = 11.17 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*sin(a+1/2*log(c*x^n)*(-(1+m)^2/n^2)^(1/2))^3,x, algorithm="giac")
[Out]
1/4*(8*I*m^3*n^4*x*x^m*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 24*I*m^3*n^4*x*x^m*
e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 24*I*m^3*n^4*x*x^m*e^(-I*a + 1/2*(n*abs(m*n
+ n)*log(x) + abs(m*n + n)*log(c))/n^2) - 8*I*m^3*n^4*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n +
n)*log(c))/n^2) + 24*I*m^2*n^4*x*x^m*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 12*I
*m^2*n^3*x*x^m*abs(m*n + n)*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 72*I*m^2*n^4*x
*x^m*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 12*I*m^2*n^3*x*x^m*abs(m*n + n)*e^(I*a
- 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 72*I*m^2*n^4*x*x^m*e^(-I*a + 1/2*(n*abs(m*n + n)*lo
g(x) + abs(m*n + n)*log(c))/n^2) - 12*I*m^2*n^3*x*x^m*abs(m*n + n)*e^(-I*a + 1/2*(n*abs(m*n + n)*log(x) + abs(
m*n + n)*log(c))/n^2) - 24*I*m^2*n^4*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2)
+ 12*I*m^2*n^3*x*x^m*abs(m*n + n)*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 2*I*(m*
n + n)^2*m*n^2*x*x^m*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 24*I*m*n^4*x*x^m*e^(3
*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 24*I*m*n^3*x*x^m*abs(m*n + n)*e^(3*I*a - 3/2*(
n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 54*I*(m*n + n)^2*m*n^2*x*x^m*e^(I*a - 1/2*(n*abs(m*n + n)*
log(x) + abs(m*n + n)*log(c))/n^2) - 72*I*m*n^4*x*x^m*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c
))/n^2) - 24*I*m*n^3*x*x^m*abs(m*n + n)*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 54*I
*(m*n + n)^2*m*n^2*x*x^m*e^(-I*a + 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 72*I*m*n^4*x*x^m*e
^(-I*a + 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 24*I*m*n^3*x*x^m*abs(m*n + n)*e^(-I*a + 1/2*
(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 2*I*(m*n + n)^2*m*n^2*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n +
n)*log(x) + abs(m*n + n)*log(c))/n^2) - 24*I*m*n^4*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)
*log(c))/n^2) + 24*I*m*n^3*x*x^m*abs(m*n + n)*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^
2) - 2*I*(m*n + n)^2*n^2*x*x^m*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 8*I*n^4*x*x
^m*e^(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 3*I*(m*n + n)^2*n*x*x^m*abs(m*n + n)*e^
(3*I*a - 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 12*I*n^3*x*x^m*abs(m*n + n)*e^(3*I*a - 3/2*(
n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 54*I*(m*n + n)^2*n^2*x*x^m*e^(I*a - 1/2*(n*abs(m*n + n)*lo
g(x) + abs(m*n + n)*log(c))/n^2) - 24*I*n^4*x*x^m*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n
^2) + 27*I*(m*n + n)^2*n*x*x^m*abs(m*n + n)*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) -
12*I*n^3*x*x^m*abs(m*n + n)*e^(I*a - 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 54*I*(m*n + n)^2
*n^2*x*x^m*e^(-I*a + 1/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 24*I*n^4*x*x^m*e^(-I*a + 1/2*(n*
abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 27*I*(m*n + n)^2*n*x*x^m*abs(m*n + n)*e^(-I*a + 1/2*(n*abs(m
*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 12*I*n^3*x*x^m*abs(m*n + n)*e^(-I*a + 1/2*(n*abs(m*n + n)*log(x)
+ abs(m*n + n)*log(c))/n^2) + 2*I*(m*n + n)^2*n^2*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*
log(c))/n^2) - 8*I*n^4*x*x^m*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) - 3*I*(m*n + n
)^2*n*x*x^m*abs(m*n + n)*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2) + 12*I*n^3*x*x^m*a
bs(m*n + n)*e^(-3*I*a + 3/2*(n*abs(m*n + n)*log(x) + abs(m*n + n)*log(c))/n^2))/(16*m^4*n^4 + 64*m^3*n^4 - 40*
(m*n + n)^2*m^2*n^2 + 96*m^2*n^4 - 80*(m*n + n)^2*m*n^2 + 64*m*n^4 + 9*(m*n + n)^4 - 40*(m*n + n)^2*n^2 + 16*n
^4)