3.21 \(\int \sin ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=191 \[ \frac {x \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}-\frac {4 b n x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac {12 b^2 n^2 x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}-\frac {24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac {24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]

[Out]

24*b^4*n^4*x/(64*b^4*n^4+20*b^2*n^2+1)-24*b^3*n^3*x*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/(64*b^4*n^4+20*b^2*n
^2+1)+12*b^2*n^2*x*sin(a+b*ln(c*x^n))^2/(64*b^4*n^4+20*b^2*n^2+1)-4*b*n*x*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n)
)^3/(16*b^2*n^2+1)+x*sin(a+b*ln(c*x^n))^4/(16*b^2*n^2+1)

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Rubi [A]  time = 0.05, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4477, 8} \[ \frac {12 b^2 n^2 x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac {x \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}-\frac {4 b n x \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}-\frac {24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac {24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^4,x]

[Out]

(24*b^4*n^4*x)/(1 + 20*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 +
20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Sin[a + b*Log[c*x^n]]^2)/(1 + 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x*Co
s[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^3)/(1 + 16*b^2*n^2) + (x*Sin[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4477

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> Simp[(x*Sin[d*(a + b*Log[c*x^n])]^p)/(
b^2*d^2*n^2*p^2 + 1), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 + 1), Int[Sin[d*(a + b*Log[c*x^n])]^
(p - 2), x], x] - Simp[(b*d*n*p*x*Cos[d*(a + b*Log[c*x^n])]*Sin[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*n^2*p^
2 + 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + 1, 0]

Rubi steps

\begin {align*} \int \sin ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {4 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {x \sin ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {\left (12 b^2 n^2\right ) \int \sin ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1+16 b^2 n^2}\\ &=-\frac {24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {x \sin ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {\left (24 b^4 n^4\right ) \int 1 \, dx}{1+20 b^2 n^2+64 b^4 n^4}\\ &=\frac {24 b^4 n^4 x}{1+20 b^2 n^2+64 b^4 n^4}-\frac {24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {x \sin ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 168, normalized size = 0.88 \[ \frac {x \left (-128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (16 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (4 b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4+60 b^2 n^2+3\right )}{8 \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*Log[c*x^n]]^4,x]

[Out]

(x*(3 + 60*b^2*n^2 + 192*b^4*n^4 - 4*(1 + 16*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b
*Log[c*x^n])] - 8*b*n*Sin[2*(a + b*Log[c*x^n])] - 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 4*b*n*Sin[4*(a + b*L
og[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(1 + 20*b^2*n^2 + 64*b^4*n^4))

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fricas [A]  time = 0.46, size = 165, normalized size = 0.86 \[ \frac {{\left (4 \, b^{2} n^{2} + 1\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 2 \, {\left (10 \, b^{2} n^{2} + 1\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (24 \, b^{4} n^{4} + 16 \, b^{2} n^{2} + 1\right )} x + 4 \, {\left ({\left (4 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - {\left (10 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

((4*b^2*n^2 + 1)*x*cos(b*n*log(x) + b*log(c) + a)^4 - 2*(10*b^2*n^2 + 1)*x*cos(b*n*log(x) + b*log(c) + a)^2 +
(24*b^4*n^4 + 16*b^2*n^2 + 1)*x + 4*((4*b^3*n^3 + b*n)*x*cos(b*n*log(x) + b*log(c) + a)^3 - (10*b^3*n^3 + b*n)
*x*cos(b*n*log(x) + b*log(c) + a))*sin(b*n*log(x) + b*log(c) + a))/(64*b^4*n^4 + 20*b^2*n^2 + 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(sin(a+b*log(c*x^n))^4,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*ln(c*x^n))^4,x)

[Out]

int(sin(a+b*ln(c*x^n))^4,x)

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maxima [B]  time = 0.41, size = 1078, normalized size = 5.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*log(c*x^n))^4,x, algorithm="maxima")

[Out]

1/16*((16*(b^3*cos(4*b*log(c))*sin(8*b*log(c)) - b^3*cos(8*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c)))*n^
3 + 4*(b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*log(c))*sin(4*b*log(c)) + b^2*cos(4*b*log(c)))*n^2 +
4*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c)))*n + cos(8*b*log(
c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*a) - 4*(32*(b^
3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)) + b^3*cos(2*b*log(c))*sin(4*b*log(c))
- b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3 + 16*(b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*b*log(c))*cos
(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 2*(b*cos(4*b*l
og(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log
(c))*sin(2*b*log(c)))*n + cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*
sin(4*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + (16*(b^3*cos(8*b*log(c))*cos(4*
b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)))*n^3 - 4*(b^2*cos(4*b*log(c))*sin(8*b*lo
g(c)) - b^2*cos(8*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c)))*n^2 + 4*(b*cos(8*b*log(c))*cos(4*b*log(c))
+ b*sin(8*b*log(c))*sin(4*b*log(c)) + b*cos(4*b*log(c)))*n - cos(4*b*log(c))*sin(8*b*log(c)) + cos(8*b*log(c))
*sin(4*b*log(c)) - sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) - 4*(32*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b
^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c))
)*n^3 - 16*(b^2*cos(4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*log(c)) + b^2*cos(2*b*log(c))*si
n(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 + 2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*lo
g(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n - cos(4*b*log
(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)) - cos(2*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c))*sin
(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) + 6*(64*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 20*(b^2*
cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*x)/(64*(b^4*cos(4*b*lo
g(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 20*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + cos(4*b*log(c)
)^2 + sin(4*b*log(c))^2)

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mupad [B]  time = 2.86, size = 117, normalized size = 0.61 \[ \frac {3\,x}{8}-\frac {x\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+4{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{4+b\,n\,8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,4{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}\,1{}\mathrm {i}}{64\,b\,n+16{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}{16+b\,n\,64{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*log(c*x^n))^4,x)

[Out]

(3*x)/8 - (x*exp(-a*2i)/(c*x^n)^(b*2i)*1i)/(8*b*n + 4i) - (x*exp(a*2i)*(c*x^n)^(b*2i))/(b*n*8i + 4) + (x*exp(-
a*4i)/(c*x^n)^(b*4i)*1i)/(64*b*n + 16i) + (x*exp(a*4i)*(c*x^n)^(b*4i))/(b*n*64i + 16)

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sympy [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(sin(a+b*ln(c*x**n))**4,x)

[Out]

Timed out

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