\(\int (e x)^m \sin (d (a+b \log (c x^n))) \, dx\) [73]

Optimal result

Integrand size = 19, antiderivative size = 92 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+b^2 d^2 n^2\right )}+\frac {(1+m) (e x)^{1+m} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+b^2 d^2 n^2\right )} \]

[Out]

-b*d*n*(e*x)^(1+m)*cos(d*(a+b*ln(c*x^n)))/e/((1+m)^2+b^2*d^2*n^2)+(1+m)*(e*x)^(1+m)*sin(d*(a+b*ln(c*x^n)))/e/(
(1+m)^2+b^2*d^2*n^2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {4573} \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {(m+1) (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )}-\frac {b d n (e x)^{m+1} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right )} \]

[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 4573

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*} \text {integral}& = -\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] (verified)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x (e x)^m \left (-b d n \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )+(1+m) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{1+2 m+m^2+b^2 d^2 n^2} \]

[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)

Maple [F]

\[\int \left (e x \right )^{m} \sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.93 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {b d n x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} - {\left (m + 1\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} \sin \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{b^{2} d^{2} n^{2} + m^{2} + 2 \, m + 1} \]

[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)

Sympy [F]

\[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\int \left (e x\right )^{m} \sin {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (92) = 184\).

Time = 0.27 (sec) , antiderivative size = 1263, normalized size of antiderivative = 13.73 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Too large to display} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6580 vs. \(2 (92) = 184\).

Time = 0.52 (sec) , antiderivative size = 6580, normalized size of antiderivative = 71.52 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x)^m*sin(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

1/2*(b*d*n*x*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*log(abs(e)) + m*log(ab
s(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*
tan(1/2*a*d)^2 + b*d*n*x*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*log(abs(e
)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) -
1/2*pi*m)^2*tan(1/2*a*d)^2 - b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*p
i*m*sgn(x) - 1/2*pi*m)^2 - b*d*n*x*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
*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*
m*sgn(x) - 1/2*pi*m)^2 + 4*b*d*n*x*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*
log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m
*sgn(x) - 1/2*pi*m)*tan(1/2*a*d) - 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e)
+ 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d) - 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*
m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*
tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(a
bs(c)))^2*tan(1/2*a*d)^2 - b*d*n*x*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
*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*
b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 - 4*
b*d*n*x*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*log(abs(e)) + m*log(abs(x)
))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*
a*d)^2 - b*d*n*x*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*log(abs(e)) + m*lo
g(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 - b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*p
i*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(
e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) + 2*m*x*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sg
n(c) + 1/2*pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*
pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 2*m*x*e^(1/2*pi*b*d*n*sgn(x) - 1/2*pi*b*d*n + 1/2*p
i*b*d*sgn(c) - 1/2*pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*
tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(ab
s(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*
d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x))
+ 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*l
og(abs(x)) + 1/2*b*d*log(abs(c)))^2 + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2 - 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*
n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) + 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*lo
g(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1
/4*pi*m*sgn(x) - 1/2*pi*m)^2 + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 + 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*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*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*pi*b*d*n - 1/2*pi*b*d*sg
n(c) + 1/2*pi*b*d + m*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(
abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d) + 4*b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*
sgn(x) - 1/2*pi*m)*tan(1/2*a*d) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*p
i*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e)
+ 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*a*d)^2 + b*d*n*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*a*d)^2 - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x))
+ 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(a
bs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n
*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/
2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2
+ 2*m*x*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*log(abs(e)) + m*log(abs(x))
)*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) - 2*m*x
*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*log(abs(e)) + m*log(abs(x)))*tan(
1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) - 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d
*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 - 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log
(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 - 2*m*x*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*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(abs(x)))*t
an(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)
- 8*m*x*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*log(abs(e)) + m*log(abs(x
)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2
*a*d) - 2*m*x*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*log(abs(e)) + m*log(a
bs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn
(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 2*m*x*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*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(abs(x)))*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*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(
1/2*a*d)^2 - 2*m*x*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*log(abs(e)) + m
*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 - b*d*n*x*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*log(abs(e)) + m*log(abs(x))) - b*d*n*x*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*log(abs(e)) + m*log(abs(x))) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(
x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/
2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log
(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))
)*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1
/2*a*d) - 2*x*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*log(abs(e)) + m*log(
abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/2*a*d) + 8*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*l
og(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d) - 8*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*
d*log(abs(c)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*s
gn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d
*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/2*a*d)^2 - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c)))*tan(1/2
*a*d)^2 + 2*x*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*log(abs(e)) + m*log(a
bs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(1/2*a*d)^2 - 2*x*e^(-1/2*pi*b*d*n*sgn(x) + 1/2*p
i*b*d*n - 1/2*pi*b*d*sgn(c) + 1/2*pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x
) - 1/2*pi*m)*tan(1/2*a*d)^2 + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) +
1/2*b*d*log(abs(c))) - 2*m*x*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*log(a
bs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sg
n(x) - 1/2*pi*m) + 2*m*x*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*log(abs(e)
) + m*log(abs(x)))*tan(1/2*a*d) + 2*m*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*a*d) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*n*log(abs(x)) + 1/2*b*d*log(abs(c))) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/2*b*d*
n*log(abs(x)) + 1/2*b*d*log(abs(c))) - 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m) + 2*x*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*log(abs(e)) + m*log(abs(x)))*tan(1/4*pi*m*sgn(e) +
1/4*pi*m*sgn(x) - 1/2*pi*m) + 2*x*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*l
og(abs(e)) + m*log(abs(x)))*tan(1/2*a*d) + 2*x*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*log(abs(e)) + m*log(abs(x)))*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(a
bs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + b^2*d^2*n^2*tan(1/2*b*d*n*log(a
bs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*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(e) + 1/4*pi*m*sgn(x) - 1/
2*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(e) + 1/4*pi*m*sgn(x) - 1/2*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(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + 2*m*tan(1/2*b*d*n*l
og(abs(x)) + 1/2*b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*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(e) + 1/4*pi*m*sgn(x) - 1/2*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(e) + 1/4*
pi*m*sgn(x) - 1/2*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
(e) + 1/4*pi*m*sgn(x) - 1/2*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*ta
n(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*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*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/2*a*d)^2 + m^2*tan(1/2*b*d*n*log(a
bs(x)) + 1/2*b*d*log(abs(c)))^2 + m^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*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(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 + m^2*tan(1/2*a*d)^2 + ta
n(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(e) + 1/4*pi*m*sgn(x) - 1/2*
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(e) + 1/
4*pi*m*sgn(x) - 1/2*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(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2 + tan(1/2*a*d)^2 + 2*m + 1)

Mupad [B] (verification not implemented)

Time = 28.70 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-a\,d\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,d\,1{}\mathrm {i}}}\,{\left (e\,x\right )}^m\,1{}\mathrm {i}}{2\,m+2-b\,d\,n\,2{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,d\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,1{}\mathrm {i}}\,{\left (e\,x\right )}^m}{m\,2{}\mathrm {i}-2\,b\,d\,n+2{}\mathrm {i}} \]

[In]

int(sin(d*(a + b*log(c*x^n)))*(e*x)^m,x)

[Out]

(x*exp(-a*d*1i)/(c*x^n)^(b*d*1i)*(e*x)^m*1i)/(2*m - b*d*n*2i + 2) + (x*exp(a*d*1i)*(c*x^n)^(b*d*1i)*(e*x)^m)/(
m*2i - 2*b*d*n + 2i)