3.71 \(\int (e x)^m \sin ^3(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=256 \[ \frac{(m+1) (e x)^{m+1} \sin ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (9 b^2 d^2 n^2+(m+1)^2\right )}+\frac{6 b^2 d^2 (m+1) n^2 (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right ) \left (9 b^2 d^2 n^2+(m+1)^2\right )}-\frac{6 b^3 d^3 n^3 (e x)^{m+1} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right ) \left (9 b^2 d^2 n^2+(m+1)^2\right )}-\frac{3 b d n (e x)^{m+1} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (9 b^2 d^2 n^2+(m+1)^2\right )} \]

[Out]

(-6*b^3*d^3*n^3*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2 + 9*b^2*d^2*n
^2)) + (6*b^2*d^2*(1 + m)*n^2*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2
+ 9*b^2*d^2*n^2)) - (3*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^2)/(e*((1 + m)
^2 + 9*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^3)/(e*((1 + m)^2 + 9*b^2*d^2*n^2))

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Rubi [A]  time = 0.117907, antiderivative size = 256, 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, 4485} \[ \frac{(m+1) (e x)^{m+1} \sin ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (9 b^2 d^2 n^2+(m+1)^2\right )}+\frac{6 b^2 d^2 (m+1) n^2 (e x)^{m+1} \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right ) \left (9 b^2 d^2 n^2+(m+1)^2\right )}-\frac{6 b^3 d^3 n^3 (e x)^{m+1} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (b^2 d^2 n^2+(m+1)^2\right ) \left (9 b^2 d^2 n^2+(m+1)^2\right )}-\frac{3 b d n (e x)^{m+1} \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left (9 b^2 d^2 n^2+(m+1)^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^3,x]

[Out]

(-6*b^3*d^3*n^3*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2 + 9*b^2*d^2*n
^2)) + (6*b^2*d^2*(1 + m)*n^2*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])])/(e*((1 + m)^2 + b^2*d^2*n^2)*((1 + m)^2
+ 9*b^2*d^2*n^2)) - (3*b*d*n*(e*x)^(1 + m)*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^2)/(e*((1 + m)
^2 + 9*b^2*d^2*n^2)) + ((1 + m)*(e*x)^(1 + m)*Sin[d*(a + b*Log[c*x^n])]^3)/(e*((1 + m)^2 + 9*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 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 ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=-\frac{3 b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+9 b^2 d^2 n^2\right )}+\frac{(1+m) (e x)^{1+m} \sin ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+9 b^2 d^2 n^2\right )}+\frac{\left (6 b^2 d^2 n^2\right ) \int (e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx}{(1+m)^2+9 b^2 d^2 n^2}\\ &=-\frac{6 b^3 d^3 n^3 (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 ) \left ((1+m)^2+9 b^2 d^2 n^2\right )}+\frac{6 b^2 d^2 (1+m) n^2 (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 ) \left ((1+m)^2+9 b^2 d^2 n^2\right )}-\frac{3 b d n (e x)^{1+m} \cos \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+9 b^2 d^2 n^2\right )}+\frac{(1+m) (e x)^{1+m} \sin ^3\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e \left ((1+m)^2+9 b^2 d^2 n^2\right )}\\ \end{align*}

Mathematica [A]  time = 1.20835, size = 326, normalized size = 1.27 \[ \frac{1}{4} x (e x)^m \left (\frac{3 \cos (b d n \log (x)) \left ((m+1) \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-b d n \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{b^2 d^2 n^2+m^2+2 m+1}+\frac{3 \sin (b d n \log (x)) \left ((m+1) \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{b^2 d^2 n^2+m^2+2 m+1}-\frac{\cos (3 b d n \log (x)) \left ((m+1) \sin \left (3 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-3 b d n \cos \left (3 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{9 b^2 d^2 n^2+m^2+2 m+1}-\frac{\sin (3 b d n \log (x)) \left ((m+1) \cos \left (3 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+3 b d n \sin \left (3 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{9 b^2 d^2 n^2+m^2+2 m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Sin[d*(a + b*Log[c*x^n])]^3,x]

[Out]

(x*(e*x)^m*((3*Cos[b*d*n*Log[x]]*(-(b*d*n*Cos[d*(a - b*n*Log[x] + b*Log[c*x^n])]) + (1 + m)*Sin[d*(a - b*n*Log
[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*n^2) + (3*Sin[b*d*n*Log[x]]*((1 + m)*Cos[d*(a - b*n*Log[x] + b
*Log[c*x^n])] + b*d*n*Sin[d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + b^2*d^2*n^2) - (Cos[3*b*d*n*Lo
g[x]]*(-3*b*d*n*Cos[3*d*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[3*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/
(1 + 2*m + m^2 + 9*b^2*d^2*n^2) - (Sin[3*b*d*n*Log[x]]*((1 + m)*Cos[3*d*(a - b*n*Log[x] + b*Log[c*x^n])] + 3*b
*d*n*Sin[3*d*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 9*b^2*d^2*n^2)))/4

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Maple [F]  time = 0.092, 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 ) ^{3}\, 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)))^3,x)

[Out]

int((e*x)^m*sin(d*(a+b*ln(c*x^n)))^3,x)

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Maxima [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)))^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.548708, size = 695, normalized size = 2.71 \begin{align*} -\frac{{\left ({\left (m^{3} +{\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )^{2} -{\left (m^{3} + 7 \,{\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x\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 ) - 3 \,{\left ({\left (b^{3} d^{3} n^{3} +{\left (b d m^{2} + 2 \, b d m + b d\right )} n\right )} x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )^{3} -{\left (3 \, b^{3} d^{3} n^{3} +{\left (b d m^{2} + 2 \, b d m + b d\right )} n\right )} x \cos \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )\right )} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{9 \, b^{4} d^{4} n^{4} + m^{4} + 4 \, m^{3} + 10 \,{\left (b^{2} d^{2} m^{2} + 2 \, b^{2} d^{2} m + b^{2} d^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, 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)))^3,x, algorithm="fricas")

[Out]

-(((m^3 + (b^2*d^2*m + b^2*d^2)*n^2 + 3*m^2 + 3*m + 1)*x*cos(b*d*n*log(x) + b*d*log(c) + a*d)^2 - (m^3 + 7*(b^
2*d^2*m + b^2*d^2)*n^2 + 3*m^2 + 3*m + 1)*x)*e^(m*log(e) + m*log(x))*sin(b*d*n*log(x) + b*d*log(c) + a*d) - 3*
((b^3*d^3*n^3 + (b*d*m^2 + 2*b*d*m + b*d)*n)*x*cos(b*d*n*log(x) + b*d*log(c) + a*d)^3 - (3*b^3*d^3*n^3 + (b*d*
m^2 + 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)))/(9*b^4*d^4*n^4 + m^4
+ 4*m^3 + 10*(b^2*d^2*m^2 + 2*b^2*d^2*m + b^2*d^2)*n^2 + 6*m^2 + 4*m + 1)

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Sympy [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*ln(c*x**n)))**3,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \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)))^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError