3.72 \(\int (e x)^m \sin ^2(d (a+b \log (c x^n))) \, dx\)
Optimal. Leaf size=154 \[ \frac{(m+1) (e x)^{m+1} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (4 b^2 d^2 n^2+(m+1)^2\right )}-\frac{2 b d n (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (4 b^2 d^2 n^2+(m+1)^2\right )}+\frac{2 b^2 d^2 n^2 (e x)^{m+1}}{e (m+1) \left (4 b^2 d^2 n^2+(m+1)^2\right )} \]
[Out]
(2*b^2*d^2*n^2*(e*x)^(1 + m))/(e*(1 + m)*((1 + m)^2 + 4*b^2*d^2*n^2)) - (2*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Lo
g[c*x^n])]*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + 4*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Lo
g[c*x^n])]^2)/(e*((1 + m)^2 + 4*b^2*d^2*n^2))
________________________________________________________________________________________
Rubi [A] time = 0.055039, antiderivative size = 154, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used =
{4487, 32} \[ \frac{(m+1) (e x)^{m+1} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (4 b^2 d^2 n^2+(m+1)^2\right )}-\frac{2 b d n (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (4 b^2 d^2 n^2+(m+1)^2\right )}+\frac{2 b^2 d^2 n^2 (e x)^{m+1}}{e (m+1) \left (4 b^2 d^2 n^2+(m+1)^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^2,x]
[Out]
(2*b^2*d^2*n^2*(e*x)^(1 + m))/(e*(1 + m)*((1 + m)^2 + 4*b^2*d^2*n^2)) - (2*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Lo
g[c*x^n])]*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + 4*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Lo
g[c*x^n])]^2)/(e*((1 + m)^2 + 4*b^2*d^2*n^2))
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 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rubi steps
\begin{align*} \int (e x)^m \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=-\frac{2 b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )}+\frac{(1+m) (e x)^{1+m} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )}+\frac{\left (2 b^2 d^2 n^2\right ) \int (e x)^m \, dx}{(1+m)^2+4 b^2 d^2 n^2}\\ &=\frac{2 b^2 d^2 n^2 (e x)^{1+m}}{e (1+m) \left ((1+m)^2+4 b^2 d^2 n^2\right )}-\frac{2 b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )}+\frac{(1+m) (e x)^{1+m} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )}\\ \end{align*}
Mathematica [C] time = 0.287066, size = 102, normalized size = 0.66 \[ -\frac{x (e x)^m \left (2 b d (m+1) n \sin \left (2 d \left (a+b \log \left (c x^n\right )\right )\right )+(m+1)^2 \cos \left (2 d \left (a+b \log \left (c x^n\right )\right )\right )-4 b^2 d^2 n^2-m^2-2 m-1\right )}{2 (m+1) (-2 i b d n+m+1) (2 i b d n+m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^2,x]
[Out]
-(x*(e*x)^m*(-1 - 2*m - m^2 - 4*b^2*d^2*n^2 + (1 + m)^2*Cos[2*d*(a + b*Log[c*x^n])] + 2*b*d*(1 + m)*n*Sin[2*d*
(a + b*Log[c*x^n])]))/(2*(1 + m)*(1 + m - (2*I)*b*d*n)*(1 + m + (2*I)*b*d*n))
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^m*sin(d*(a+b*ln(c*x^n)))^2,x)
[Out]
int((e*x)^m*sin(d*(a+b*ln(c*x^n)))^2,x)
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Maxima [B] time = 1.53686, size = 3444, normalized size = 22.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^2,x, algorithm="maxima")
[Out]
-1/4*(((((cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) + (cos(2*a*d)*sin(4*a*d) - cos(4*a*
d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + cos(2*b*d*log(c))*cos(2*a*d) - ((cos(2*a*d)*sin(4*a*d) -
cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))
*sin(4*b*d*log(c)) - sin(2*b*d*log(c))*sin(2*a*d))*e^m*m^2 + 2*(((cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d
))*cos(2*b*d*log(c)) + (cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) +
cos(2*b*d*log(c))*cos(2*a*d) - ((cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (cos(4*a*d
)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*sin(4*b*d*log(c)) - sin(2*b*d*log(c))*sin(2*a*d))*e^m
*m + (((cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) + (cos(2*a*d)*sin(4*a*d) - cos(4*a*d)
*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + cos(2*b*d*log(c))*cos(2*a*d) - ((cos(2*a*d)*sin(4*a*d) - c
os(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*s
in(4*b*d*log(c)) - sin(2*b*d*log(c))*sin(2*a*d))*e^m + 2*((b*d*cos(2*a*d)*sin(2*b*d*log(c)) + b*d*cos(2*b*d*lo
g(c))*sin(2*a*d) + ((b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (b*d*cos(4*a*d
)*cos(2*a*d) + b*d*sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + ((b*d*cos(4*a*d)*cos(2*a*d) +
b*d*sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) + (b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*sin(2*
b*d*log(c)))*sin(4*b*d*log(c)))*e^m*m + (b*d*cos(2*a*d)*sin(2*b*d*log(c)) + b*d*cos(2*b*d*log(c))*sin(2*a*d) +
((b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (b*d*cos(4*a*d)*cos(2*a*d) + b*d
*sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + ((b*d*cos(4*a*d)*cos(2*a*d) + b*d*sin(4*a*d)*si
n(2*a*d))*cos(2*b*d*log(c)) + (b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*sin(4
*b*d*log(c)))*e^m)*n)*x*x^m*cos(2*b*d*log(x^n)) - ((((cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*cos(2*b*d
*log(c)) - (cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + ((cos(4*a*d)
*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) + (cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*sin(2
*b*d*log(c)))*sin(4*b*d*log(c)) + cos(2*a*d)*sin(2*b*d*log(c)) + cos(2*b*d*log(c))*sin(2*a*d))*e^m*m^2 + 2*(((
cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a
*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + ((cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c))
+ (cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*sin(4*b*d*log(c)) + cos(2*a*d)*sin(2*b*d
*log(c)) + cos(2*b*d*log(c))*sin(2*a*d))*e^m*m + (((cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*cos(2*b*d*l
og(c)) - (cos(4*a*d)*cos(2*a*d) + sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) + ((cos(4*a*d)*c
os(2*a*d) + sin(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) + (cos(2*a*d)*sin(4*a*d) - cos(4*a*d)*sin(2*a*d))*sin(2*b
*d*log(c)))*sin(4*b*d*log(c)) + cos(2*a*d)*sin(2*b*d*log(c)) + cos(2*b*d*log(c))*sin(2*a*d))*e^m - 2*((b*d*cos
(2*b*d*log(c))*cos(2*a*d) - b*d*sin(2*b*d*log(c))*sin(2*a*d) + ((b*d*cos(4*a*d)*cos(2*a*d) + b*d*sin(4*a*d)*si
n(2*a*d))*cos(2*b*d*log(c)) + (b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4
*b*d*log(c)) - ((b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (b*d*cos(4*a*d)*co
s(2*a*d) + b*d*sin(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*sin(4*b*d*log(c)))*e^m*m + (b*d*cos(2*b*d*log(c))*cos
(2*a*d) - b*d*sin(2*b*d*log(c))*sin(2*a*d) + ((b*d*cos(4*a*d)*cos(2*a*d) + b*d*sin(4*a*d)*sin(2*a*d))*cos(2*b*
d*log(c)) + (b*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*cos(4*b*d*log(c)) - ((b
*d*cos(2*a*d)*sin(4*a*d) - b*d*cos(4*a*d)*sin(2*a*d))*cos(2*b*d*log(c)) - (b*d*cos(4*a*d)*cos(2*a*d) + b*d*sin
(4*a*d)*sin(2*a*d))*sin(2*b*d*log(c)))*sin(4*b*d*log(c)))*e^m)*n)*x*x^m*sin(2*b*d*log(x^n)) - 2*(((cos(2*a*d)^
2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + sin(2*a*d)^2)*sin(2*b*d*log(c))^2)*e^m*m^2 + 4*((b^2*d
^2*cos(2*a*d)^2 + b^2*d^2*sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (b^2*d^2*cos(2*a*d)^2 + b^2*d^2*sin(2*a*d)^2)*si
n(2*b*d*log(c))^2)*e^m*n^2 + 2*((cos(2*a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + sin(2*a*d)
^2)*sin(2*b*d*log(c))^2)*e^m*m + ((cos(2*a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + sin(2*a*
d)^2)*sin(2*b*d*log(c))^2)*e^m)*x*x^m)/(((cos(2*a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + s
in(2*a*d)^2)*sin(2*b*d*log(c))^2)*m^3 + 3*((cos(2*a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 +
sin(2*a*d)^2)*sin(2*b*d*log(c))^2)*m^2 + 4*((b^2*d^2*cos(2*a*d)^2 + b^2*d^2*sin(2*a*d)^2)*cos(2*b*d*log(c))^2
+ (b^2*d^2*cos(2*a*d)^2 + b^2*d^2*sin(2*a*d)^2)*sin(2*b*d*log(c))^2 + ((b^2*d^2*cos(2*a*d)^2 + b^2*d^2*sin(2*
a*d)^2)*cos(2*b*d*log(c))^2 + (b^2*d^2*cos(2*a*d)^2 + b^2*d^2*sin(2*a*d)^2)*sin(2*b*d*log(c))^2)*m)*n^2 + (cos
(2*a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + sin(2*a*d)^2)*sin(2*b*d*log(c))^2 + 3*((cos(2*
a*d)^2 + sin(2*a*d)^2)*cos(2*b*d*log(c))^2 + (cos(2*a*d)^2 + sin(2*a*d)^2)*sin(2*b*d*log(c))^2)*m)
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Fricas [A] time = 0.51077, size = 401, normalized size = 2.6 \begin{align*} -\frac{2 \,{\left (b d m + b d\right )} n x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \sin \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) +{\left ({\left (m^{2} + 2 \, m + 1\right )} x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )^{2} -{\left (2 \, b^{2} d^{2} n^{2} + m^{2} + 2 \, m + 1\right )} x\right )} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{3} + 4 \,{\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^2,x, algorithm="fricas")
[Out]
-(2*(b*d*m + b*d)*n*x*cos(b*d*n*log(x) + b*d*log(c) + a*d)*e^(m*log(e) + m*log(x))*sin(b*d*n*log(x) + b*d*log(
c) + a*d) + ((m^2 + 2*m + 1)*x*cos(b*d*n*log(x) + b*d*log(c) + a*d)^2 - (2*b^2*d^2*n^2 + m^2 + 2*m + 1)*x)*e^(
m*log(e) + m*log(x)))/(m^3 + 4*(b^2*d^2*m + b^2*d^2)*n^2 + 3*m^2 + 3*m + 1)
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**m*sin(d*(a+b*ln(c*x**n)))**2,x)
[Out]
Exception raised: TypeError
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^2,x, algorithm="giac")
[Out]
Timed out