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*ln(c*x^n)))*sin(d*(a+
b*ln(c*x^n)))/e/((1+m)^2+4*b^2*d^2*n^2)+(1+m)*(e*x)^(1+m)*sin(d*(a+b*ln(c*x^n)))^2/e/((1+m)^2+4*b^2*d^2*n^2)
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Rubi [A] time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00,
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 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]
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]
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*}
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Mathematica [C] time = 0.30, 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]
-1/2*(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])]))/((1 + m)*(1 + m - (2*I)*b*d*n)*(1 + m + (2*I)*b*d*n))
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fricas [A] time = 0.65, size = 155, normalized size = 1.01 \[ -\frac {2 \, {\left (b d m + b d\right )} n x \cos \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) e^{\left (m \log \relax (e) + m \log \relax (x)\right )} \sin \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) + {\left ({\left (m^{2} + 2 \, m + 1\right )} x \cos \left (b d n \log \relax (x) + b d \log \relax (c) + a d\right )^{2} - {\left (2 \, b^{2} d^{2} n^{2} + m^{2} + 2 \, m + 1\right )} x\right )} e^{\left (m \log \relax (e) + m \log \relax (x)\right )}}{m^{3} + 4 \, {\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1} \]
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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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\sin ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]
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 = 0.49, size = 2551, normalized size = 16.56 \[ \text {result too large to display} \]
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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mupad [B] time = 3.05, size = 95, normalized size = 0.62 \[ \frac {x\,{\left (e\,x\right )}^m}{2\,m+2}-\frac {x\,{\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}\,{\left (e\,x\right )}^m}{4\,m+4+b\,d\,n\,8{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{-a\,d\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}}\,{\left (e\,x\right )}^m\,1{}\mathrm {i}}{m\,4{}\mathrm {i}+8\,b\,d\,n+4{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(d*(a + b*log(c*x^n)))^2*(e*x)^m,x)
[Out]
(x*(e*x)^m)/(2*m + 2) - (x*exp(a*d*2i)*(c*x^n)^(b*d*2i)*(e*x)^m)/(4*m + b*d*n*8i + 4) - (x*exp(-a*d*2i)/(c*x^n
)^(b*d*2i)*(e*x)^m*1i)/(m*4i + 8*b*d*n + 4i)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\begin {cases} \frac {\log {\relax (x )} \cos {\left (2 a d \right )}}{e} & \text {for}\: b = 0 \wedge m = -1 \\\int \left (e x\right )^{m} \cos {\left (- 2 a d + \frac {i m \log {\left (c x^{n} \right )}}{n} + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i \left (m + 1\right )}{2 d n} \\\int \left (e x\right )^{m} \cos {\left (2 a d + \frac {i m \log {\left (c x^{n} \right )}}{n} + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i \left (m + 1\right )}{2 d n} \\\frac {2 b d e^{m} n x x^{m} \sin {\left (2 a d + 2 b d n \log {\relax (x )} + 2 b d \log {\relax (c )} \right )}}{4 b^{2} d^{2} n^{2} + m^{2} + 2 m + 1} + \frac {e^{m} m x x^{m} \cos {\left (2 a d + 2 b d n \log {\relax (x )} + 2 b d \log {\relax (c )} \right )}}{4 b^{2} d^{2} n^{2} + m^{2} + 2 m + 1} + \frac {e^{m} x x^{m} \cos {\left (2 a d + 2 b d n \log {\relax (x )} + 2 b d \log {\relax (c )} \right )}}{4 b^{2} d^{2} n^{2} + m^{2} + 2 m + 1} & \text {otherwise} \end {cases}}{2} + \frac {\begin {cases} \frac {\left (e x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (e x \right )} & \text {otherwise} \end {cases}}{2 e} \]
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]
-Piecewise((log(x)*cos(2*a*d)/e, Eq(b, 0) & Eq(m, -1)), (Integral((e*x)**m*cos(-2*a*d + I*m*log(c*x**n)/n + I*
log(c*x**n)/n), x), Eq(b, -I*(m + 1)/(2*d*n))), (Integral((e*x)**m*cos(2*a*d + I*m*log(c*x**n)/n + I*log(c*x**
n)/n), x), Eq(b, I*(m + 1)/(2*d*n))), (2*b*d*e**m*n*x*x**m*sin(2*a*d + 2*b*d*n*log(x) + 2*b*d*log(c))/(4*b**2*
d**2*n**2 + m**2 + 2*m + 1) + e**m*m*x*x**m*cos(2*a*d + 2*b*d*n*log(x) + 2*b*d*log(c))/(4*b**2*d**2*n**2 + m**
2 + 2*m + 1) + e**m*x*x**m*cos(2*a*d + 2*b*d*n*log(x) + 2*b*d*log(c))/(4*b**2*d**2*n**2 + m**2 + 2*m + 1), Tru
e))/2 + Piecewise(((e*x)**(m + 1)/(m + 1), Ne(m, -1)), (log(e*x), True))/(2*e)
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