3.1.72 \(\int (e x)^m \sin ^2(d (a+b \log (c x^n))) \, dx\) [72]

Optimal. Leaf size=154 \[ \frac {2 b^2 d^2 n^2 (e x)^{1+m}}{e (1+m) \left ((1+m)^2+4 b^2 d^2 n^2\right )}-\frac {2 b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )}+\frac {(1+m) (e x)^{1+m} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+4 b^2 d^2 n^2\right )} \]

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 = {4575, 32} \begin {gather*} \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 )} \end {gather*}

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 4575

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*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.32, size = 102, normalized size = 0.66 \begin {gather*} -\frac {x (e x)^m \left (-1-2 m-m^2-4 b^2 d^2 n^2+(1+m)^2 \cos \left (2 d \left (a+b \log \left (c x^n\right )\right )\right )+2 b d (1+m) n \sin \left (2 d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 (1+m) (1+m-2 i b d n) (1+m+2 i b d n)} \end {gather*}

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))

________________________________________________________________________________________

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)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2517 vs. \(2 (154) = 308\).
time = 0.38, size = 2517, normalized size = 16.34 \begin {gather*} \text {Too large to display} \end {gather*}

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*((2*(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(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*lo
g(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*c
os(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)))*m + ((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)))*n*e^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) - 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))*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))*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) - 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)*x*x^m*cos(2*b*d*log(x^n)) + (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(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)))*m + ((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)))*n*e^m - (
(((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))*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))*m + ((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)*x*x^m*sin(2*b*d*log(x^n)) - 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)*n^2*e^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)*m^2 + (c
os(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 + 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)*m)*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 + sin(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*s
in(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...

________________________________________________________________________________________

Fricas [A]
time = 1.83, size = 149, normalized size = 0.97 \begin {gather*} -\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 (x\right ) + m\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 (x\right ) + m\right )}}{m^{3} + 4 \, {\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1} \end {gather*}

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(x) + m)*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(x
) + m))/(m^3 + 4*(b^2*d^2*m + b^2*d^2)*n^2 + 3*m^2 + 3*m + 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\begin {cases} \frac {\log {\left (x \right )} \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 n x \left (e x\right )^{m} \sin {\left (2 a d + 2 b d \log {\left (c x^{n} \right )} \right )}}{4 b^{2} d^{2} n^{2} + m^{2} + 2 m + 1} + \frac {m x \left (e x\right )^{m} \cos {\left (2 a d + 2 b d \log {\left (c x^{n} \right )} \right )}}{4 b^{2} d^{2} n^{2} + m^{2} + 2 m + 1} + \frac {x \left (e x\right )^{m} \cos {\left (2 a d + 2 b d \log {\left (c x^{n} \right )} \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} \end {gather*}

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*n*x*(e*x)**m*sin(2*a*d + 2*b*d*log(c*x**n))/(4*b**2*d**2*n**2 + m
**2 + 2*m + 1) + m*x*(e*x)**m*cos(2*a*d + 2*b*d*log(c*x**n))/(4*b**2*d**2*n**2 + m**2 + 2*m + 1) + x*(e*x)**m*
cos(2*a*d + 2*b*d*log(c*x**n))/(4*b**2*d**2*n**2 + m**2 + 2*m + 1), True))/2 + Piecewise(((e*x)**(m + 1)/(m +
1), Ne(m, -1)), (log(e*x), True))/(2*e)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30585 vs. \(2 (154) = 308\).
time = 1.23, size = 30585, normalized size = 198.60 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/4*(8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e)
- 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/
2*pi*m)^2*tan(a*d)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*f
loor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4
*pi*m*sgn(x) - 1/2*pi*m)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(
pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 - 8*(abs
(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + 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(a*d)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4
*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 +
4*b*d*m*n*x*e^(pi*b*d*n*sgn(x) - pi*b*d*n + pi*b*d*sgn(c) - pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*
log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn
(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d) + 4*b*d*m*n*x*e^(-pi*b*d*n*sgn(
x) + pi*b*d*n - pi*b*d*sgn(c) + pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)
))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi
*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d) - 4*b*d*m*n*x*e^(pi*b*d*n*sgn(x) - pi*b*d*n + pi*b*d*sgn(c)
- pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(
e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) -
1/2*pi*m)*tan(a*d)^2 + 4*b*d*m*n*x*e^(-pi*b*d*n*sgn(x) + pi*b*d*n - pi*b*d*sgn(c) + pi*b*d + m*log(abs(e)) +
m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*p
i*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(a*d)^2 + 4*b*
d*m*n*x*e^(pi*b*d*n*sgn(x) - pi*b*d*n + pi*b*d*sgn(c) - pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(
abs(x)) + b*d*log(abs(c)))*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) -
1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 + 4*b*d*m*n*x*e^(-pi*b*d*n*sgn(x) +
pi*b*d*n - pi*b*d*sgn(c) + pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))*t
an(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn
(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d
*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2
- 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m
*sgn(x) - 1/2*pi*m)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*p
i*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^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^(
pi*b*d*n*sgn(x) - pi*b*d*n + pi*b*d*sgn(c) - pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b
*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)
^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d) + 4*b*d*n*x*e^(-pi*b*d*n*sgn(x) + pi*b*d*n - p
i*b*d*sgn(c) + pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*flo
or(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*p
i*m*sgn(x) - 1/2*pi*m)^2*tan(a*d) - 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))
^2*tan(a*d)^2 + 8*(abs(e)*abs(x))^m*b^2*d^2*n^2*x*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(
e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 - 4*b*d*n*x*e^(pi*b*d*n*sgn(x) - pi*b*d*n + pi*b*d*sgn(c) - pi*b
*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/
4*sgn(x) + 1) + 1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi
*m)*tan(a*d)^2 + 4*b*d*n*x*e^(-pi*b*d*n*sgn(x) + pi*b*d*n - pi*b*d*sgn(c) + pi*b*d + m*log(abs(e)) + m*log(abs
(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(c)))^2*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + 1) + 1/4*pi*m*sgn(e
) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)*tan(a*d)^2 - 8*(abs(e)*abs
(x))^m*b^2*d^2*n^2*x*tan(1/4*pi*m*sgn(e) + 1/4*pi*m*sgn(x) - 1/2*pi*m)^2*tan(a*d)^2 + 4*b*d*n*x*e^(pi*b*d*n*sg
n(x) - pi*b*d*n + pi*b*d*sgn(c) - pi*b*d + m*log(abs(e)) + m*log(abs(x)))*tan(b*d*n*log(abs(x)) + b*d*log(abs(
c)))*tan(pi*m*floor(-1/4*sgn(e) - 1/4*sgn(x) + ...

________________________________________________________________________________________

Mupad [B]
time = 3.05, size = 95, normalized size = 0.62 \begin {gather*} \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}} \end {gather*}

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)

________________________________________________________________________________________