3.1.22 \(\int \frac {\sin ^4(a+b \log (c x^n))}{x} \, dx\) [22]

Optimal. Leaf size=73 \[ \frac {3 \log (x)}{8}-\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

[Out]

3/8*ln(x)-3/8*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/b/n-1/4*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))^3/b/n

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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8} \begin {gather*} -\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^4/x,x]

[Out]

(3*Log[x])/8 - (3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(8*b*n) - (Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[
c*x^n]]^3)/(4*b*n)

Rule 8

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps

\begin {align*} \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sin ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \sin ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=-\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac {3 \log (x)}{8}-\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 51, normalized size = 0.70 \begin {gather*} \frac {12 \left (a+b \log \left (c x^n\right )\right )-8 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*Log[c*x^n]]^4/x,x]

[Out]

(12*(a + b*Log[c*x^n]) - 8*Sin[2*(a + b*Log[c*x^n])] + Sin[4*(a + b*Log[c*x^n])])/(32*b*n)

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Maple [A]
time = 0.05, size = 61, normalized size = 0.84

method result size
derivativedivides \(\frac {-\frac {\left (\sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) \(61\)
default \(\frac {-\frac {\left (\sin ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) \(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)

[Out]

1/n/b*(-1/4*(sin(a+b*ln(c*x^n))^3+3/2*sin(a+b*ln(c*x^n)))*cos(a+b*ln(c*x^n))+3/8*b*ln(c*x^n)+3/8*a)

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Maxima [A]
time = 0.30, size = 93, normalized size = 1.27 \begin {gather*} \frac {12 \, b n \log \left (x\right ) + \cos \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) \sin \left (4 \, b \log \left (c\right )\right ) - 8 \, \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) - 8 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{32 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(12*b*n*log(x) + cos(4*b*log(x^n) + 4*a)*sin(4*b*log(c)) - 8*cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + co
s(4*b*log(c))*sin(4*b*log(x^n) + 4*a) - 8*cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(b*n)

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Fricas [A]
time = 1.22, size = 59, normalized size = 0.81 \begin {gather*} \frac {3 \, b n \log \left (x\right ) + {\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 5 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*b*n*log(x) + (2*cos(b*n*log(x) + b*log(c) + a)^3 - 5*cos(b*n*log(x) + b*log(c) + a))*sin(b*n*log(x) + b
*log(c) + a))/(b*n)

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Sympy [A]
time = 14.14, size = 100, normalized size = 1.37 \begin {gather*} - \frac {\begin {cases} \log {\left (x \right )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (2 a + 2 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\begin {cases} \log {\left (x \right )} \cos {\left (4 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (4 a + 4 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (4 a + 4 b \log {\left (c x^{n} \right )} \right )}}{4 b n} & \text {otherwise} \end {cases}}{8} + \frac {3 \log {\left (x \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*ln(c*x**n))**4/x,x)

[Out]

-Piecewise((log(x)*cos(2*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*cos(2*a + 2*b*log(c)), Eq(n, 0)), (sin
(2*a + 2*b*log(c*x**n))/(2*b*n), True))/2 + Piecewise((log(x)*cos(4*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (lo
g(x)*cos(4*a + 4*b*log(c)), Eq(n, 0)), (sin(4*a + 4*b*log(c*x**n))/(4*b*n), True))/8 + 3*log(x)/8

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*log(c*x^n) + a)^4/x, x)

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Mupad [B]
time = 2.58, size = 51, normalized size = 0.70 \begin {gather*} \frac {3\,\ln \left (x^n\right )}{8\,n}-\frac {\frac {\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}-\frac {\sin \left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*log(c*x^n))^4/x,x)

[Out]

(3*log(x^n))/(8*n) - (sin(2*a + 2*b*log(c*x^n))/4 - sin(4*a + 4*b*log(c*x^n))/32)/(b*n)

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