3.73 \(\int (e x)^m \sin (d (a+b \log (c x^n))) \, dx\)
Optimal. Leaf size=92 \[ \frac{(m+1) (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )}-\frac{b d n (e x)^{m+1} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )} \]
[Out]
-((b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))) + ((1 + m)*(e*x)^(1 + m)*Sin[
d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))
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Rubi [A] time = 0.0245137, antiderivative size = 92, normalized size of antiderivative = 1.,
number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used =
{4485} \[ \frac{(m+1) (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )}-\frac{b d n (e x)^{m+1} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])],x]
[Out]
-((b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))) + ((1 + m)*(e*x)^(1 + m)*Sin[
d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2))
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 (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=-\frac{b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+b^2 d^2 n^2\right )}+\frac{(1+m) (e x)^{1+m} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+b^2 d^2 n^2\right )}\\ \end{align*}
Mathematica [A] time = 0.13483, size = 63, normalized size = 0.68 \[ \frac{x (e x)^m \left ((m+1) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )-b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b^2 d^2 n^2+m^2+2 m+1} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])],x]
[Out]
(x*(e*x)^m*(-(b*d*n*Cos[d*(a + b*Log[c*x^n])]) + (1 + m)*Sin[d*(a + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*
n^2)
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Maple [F] time = 0.05, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m}\sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, 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))),x)
[Out]
int((e*x)^m*sin(d*(a+b*ln(c*x^n))),x)
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Maxima [B] time = 1.26394, size = 1705, normalized size = 18.53 \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))),x, algorithm="maxima")
[Out]
1/2*(((((cos(a*d)*sin(2*a*d) - cos(2*a*d)*sin(a*d))*cos(b*d*log(c)) - (cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*
d))*sin(b*d*log(c)))*cos(2*b*d*log(c)) + ((cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*cos(b*d*log(c)) + (cos(a
*d)*sin(2*a*d) - cos(2*a*d)*sin(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(c)) + cos(a*d)*sin(b*d*log(c)) + cos(b*d*
log(c))*sin(a*d))*e^m*m - (b*d*cos(b*d*log(c))*cos(a*d) - b*d*sin(b*d*log(c))*sin(a*d) + ((b*d*cos(2*a*d)*cos(
a*d) + b*d*sin(2*a*d)*sin(a*d))*cos(b*d*log(c)) + (b*d*cos(a*d)*sin(2*a*d) - b*d*cos(2*a*d)*sin(a*d))*sin(b*d*
log(c)))*cos(2*b*d*log(c)) - ((b*d*cos(a*d)*sin(2*a*d) - b*d*cos(2*a*d)*sin(a*d))*cos(b*d*log(c)) - (b*d*cos(2
*a*d)*cos(a*d) + b*d*sin(2*a*d)*sin(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(c)))*e^m*n + (((cos(a*d)*sin(2*a*d) -
cos(2*a*d)*sin(a*d))*cos(b*d*log(c)) - (cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*sin(b*d*log(c)))*cos(2*b*d
*log(c)) + ((cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*cos(b*d*log(c)) + (cos(a*d)*sin(2*a*d) - cos(2*a*d)*si
n(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(c)) + cos(a*d)*sin(b*d*log(c)) + cos(b*d*log(c))*sin(a*d))*e^m)*x*x^m*c
os(b*d*log(x^n)) + ((((cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*cos(b*d*log(c)) + (cos(a*d)*sin(2*a*d) - cos
(2*a*d)*sin(a*d))*sin(b*d*log(c)))*cos(2*b*d*log(c)) + cos(b*d*log(c))*cos(a*d) - ((cos(a*d)*sin(2*a*d) - cos(
2*a*d)*sin(a*d))*cos(b*d*log(c)) - (cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(
c)) - sin(b*d*log(c))*sin(a*d))*e^m*m + (b*d*cos(a*d)*sin(b*d*log(c)) + b*d*cos(b*d*log(c))*sin(a*d) + ((b*d*c
os(a*d)*sin(2*a*d) - b*d*cos(2*a*d)*sin(a*d))*cos(b*d*log(c)) - (b*d*cos(2*a*d)*cos(a*d) + b*d*sin(2*a*d)*sin(
a*d))*sin(b*d*log(c)))*cos(2*b*d*log(c)) + ((b*d*cos(2*a*d)*cos(a*d) + b*d*sin(2*a*d)*sin(a*d))*cos(b*d*log(c)
) + (b*d*cos(a*d)*sin(2*a*d) - b*d*cos(2*a*d)*sin(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(c)))*e^m*n + (((cos(2*a
*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*cos(b*d*log(c)) + (cos(a*d)*sin(2*a*d) - cos(2*a*d)*sin(a*d))*sin(b*d*log(
c)))*cos(2*b*d*log(c)) + cos(b*d*log(c))*cos(a*d) - ((cos(a*d)*sin(2*a*d) - cos(2*a*d)*sin(a*d))*cos(b*d*log(c
)) - (cos(2*a*d)*cos(a*d) + sin(2*a*d)*sin(a*d))*sin(b*d*log(c)))*sin(2*b*d*log(c)) - sin(b*d*log(c))*sin(a*d)
)*e^m)*x*x^m*sin(b*d*log(x^n)))/(((cos(a*d)^2 + sin(a*d)^2)*cos(b*d*log(c))^2 + (cos(a*d)^2 + sin(a*d)^2)*sin(
b*d*log(c))^2)*m^2 + ((b^2*d^2*cos(a*d)^2 + b^2*d^2*sin(a*d)^2)*cos(b*d*log(c))^2 + (b^2*d^2*cos(a*d)^2 + b^2*
d^2*sin(a*d)^2)*sin(b*d*log(c))^2)*n^2 + (cos(a*d)^2 + sin(a*d)^2)*cos(b*d*log(c))^2 + (cos(a*d)^2 + sin(a*d)^
2)*sin(b*d*log(c))^2 + 2*((cos(a*d)^2 + sin(a*d)^2)*cos(b*d*log(c))^2 + (cos(a*d)^2 + sin(a*d)^2)*sin(b*d*log(
c))^2)*m)
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Fricas [A] time = 0.496398, size = 238, normalized size = 2.59 \begin{align*} -\frac{b d 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 )} -{\left (m + 1\right )} x 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 )}{b^{2} d^{2} n^{2} + m^{2} + 2 \, 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))),x, algorithm="fricas")
[Out]
-(b*d*n*x*cos(b*d*n*log(x) + b*d*log(c) + a*d)*e^(m*log(e) + m*log(x)) - (m + 1)*x*e^(m*log(e) + m*log(x))*sin
(b*d*n*log(x) + b*d*log(c) + a*d))/(b^2*d^2*n^2 + m^2 + 2*m + 1)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{m} \sin{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \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))),x)
[Out]
Integral((e*x)**m*sin(a*d + b*d*log(c*x**n)), x)
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Giac [B] time = 1.75122, size = 7776, normalized size = 84.52 \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))),x, algorithm="giac")
[Out]
1/2*(b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*
n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + b*d*n*x*abs(x)^m*e^(
-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*
log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 - b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/
2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi
*m*sgn(x) - 1/4*pi*m)^2 - b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi
*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + 4*b*d*n*x*abs
(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) +
1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) - 4*b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(
x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan
(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) - 4*b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*
b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^
2*tan(1/2*a*d) - 4*b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d +
m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) - b*d*n*x*a
bs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x))
+ 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 - b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d
*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 + 4*b*d*n*x*abs(x)
^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2
*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 - 4*b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x)
+ 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*
pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 - b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*s
gn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 - b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*s
gn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 +
2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(a
bs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d
*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))
)^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) - 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/
2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4
*pi*m)*tan(1/2*a*d)^2 + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d
+ m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 + 2*m*
x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x
)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*s
gn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*ta
n(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*
pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 + b*d*n*x*abs(x)^m*e^(-1/2*
pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(a
bs(c)))^2 - 4*b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan
(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + 4*b*d*n*x*abs(x)^m*e^(-1/2*pi*
b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(
c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn
(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*
d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + 4*b*d*n*x*abs(x)^m*e^(1/2*pi*b*d
*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))
)*tan(1/2*a*d) + 4*b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d +
m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/2*a*d) - 4*b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) -
1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) + 4*b*d*n*x*a
bs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4
*pi*m)*tan(1/2*a*d) + 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)
*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) + 2*x*abs(x
)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1
/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) + b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x)
- 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*a*d)^2 + b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x
) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*a*d)^2 - 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) -
1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*
pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 + 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(
c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/
2*a*d)^2 + 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*
d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*x*abs(x)^m*e^(-1/2
*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(
abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d
*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x
) - 1/4*pi*m) - 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*ta
n(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - 2*m*x*abs(x)^m*e^(1/2*pi*b*
d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)
))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(
c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*m*
x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x
)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d) - 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*
sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d) + 8*m*x*abs(x)^m*e^(1
/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*lo
g(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) - 8*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*
n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) -
1/4*pi*m)*tan(1/2*a*d) - 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*
d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) - 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n
- 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) - 2*m*x*abs(x)^m*e^(1/2*p
i*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(ab
s(c)))*tan(1/2*a*d)^2 - 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d
+ m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/2*a*d)^2 + 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x)
- 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 - 2*m*x*ab
s(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*
pi*m)*tan(1/2*a*d)^2 - b*d*n*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d
+ m) - b*d*n*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m) + 2*x*ab
s(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) +
1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) - 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n
- 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) -
1/4*pi*m) - 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*
b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn
(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(
1/4*pi*m*sgn(x) - 1/4*pi*m)^2 - 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*p
i*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d) - 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn
(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*ta
n(1/2*a*d) + 8*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*
b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) - 8*x*abs(x)^m*e^(-1/2*p
i*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(ab
s(c)))*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d) - 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2
*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) - 2*x*abs(x)^m*e^(-1/2*pi*b*d*
n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d) -
2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(ab
s(x)) + 1/2*b*d*log(abs(c)))*tan(1/2*a*d)^2 - 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d
*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/2*a*d)^2 + 2*x*abs(x)^m*e^(1/
2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)*tan(1/2
*a*d)^2 - 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi
*m*sgn(x) - 1/4*pi*m)*tan(1/2*a*d)^2 + 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c
) - 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x)
+ 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) - 2*m*x*
abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4
*pi*m) + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*p
i*m*sgn(x) - 1/4*pi*m) + 2*m*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d
+ m)*tan(1/2*a*d) + 2*m*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d +
m)*tan(1/2*a*d) + 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan
(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) + 2*x*abs(x)^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b
*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) - 2*x*abs(x)^m*e^(1/2*pi*b*d*n*sg
n(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + 2*x*abs(x)^m*e^(-1
/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/4*pi*m*sgn(x) - 1/4*pi*m) + 2*x*
abs(x)^m*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*pi*b*d*sgn(c) - 1/2*pi*b*d + m)*tan(1/2*a*d) + 2*x*abs(x)
^m*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m)*tan(1/2*a*d))/(b^2*d^2*n^2*tan
(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + b^2*d^2*n^2
*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 + b^2*d^2*n^2*tan(1/2*b*
d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 + b^2*d^2*n^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/
2*a*d)^2 + b^2*d^2*n^2*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 + b^2*d^2*n^2*tan(1/4*pi*m*sgn(x) -
1/4*pi*m)^2 + b^2*d^2*n^2*tan(1/2*a*d)^2 + m^2*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m
*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*m*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn
(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + b^2*d^2*n^2 + m^2*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/
4*pi*m*sgn(x) - 1/4*pi*m)^2 + m^2*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 + m^2*tan(
1/4*pi*m*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m
*sgn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*m*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn
(x) - 1/4*pi*m)^2 + 2*m*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 + 2*m*tan(1/4*pi*m*s
gn(x) - 1/4*pi*m)^2*tan(1/2*a*d)^2 + m^2*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 + m^2*tan(1/4*pi*m
*sgn(x) - 1/4*pi*m)^2 + tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(x) - 1/4*pi*m)^2 +
m^2*tan(1/2*a*d)^2 + tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d)^2 + tan(1/4*pi*m*sgn(x)
- 1/4*pi*m)^2*tan(1/2*a*d)^2 + 2*m*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 + 2*m*tan(1/4*pi*m*sgn(x
) - 1/4*pi*m)^2 + 2*m*tan(1/2*a*d)^2 + m^2 + tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 + tan(1/4*pi*m
*sgn(x) - 1/4*pi*m)^2 + tan(1/2*a*d)^2 + 2*m + 1)