3.138 \(\int \frac {\sin ^4(a+b x^n)}{x} \, dx\)

Optimal. Leaf size=79 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \text {Ci}\left (4 b x^n\right )}{8 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3425, 3378, 3376, 3375} \[ -\frac {\cos (2 a) \text {CosIntegral}\left (2 b x^n\right )}{2 n}+\frac {\cos (4 a) \text {CosIntegral}\left (4 b x^n\right )}{8 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}-\frac {\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n]^4/x,x]

[Out]

-(Cos[2*a]*CosIntegral[2*b*x^n])/(2*n) + (Cos[4*a]*CosIntegral[4*b*x^n])/(8*n) + (3*Log[x])/8 + (Sin[2*a]*SinI
ntegral[2*b*x^n])/(2*n) - (Sin[4*a]*SinIntegral[4*b*x^n])/(8*n)

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3378

Int[Cos[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cos[c], Int[Cos[d*x^n]/x, x], x] - Dist[Sin[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3425

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 66, normalized size = 0.84 \[ \frac {-4 \cos (2 a) \text {Ci}\left (2 b x^n\right )+\cos (4 a) \text {Ci}\left (4 b x^n\right )+4 \sin (2 a) \text {Si}\left (2 b x^n\right )-\sin (4 a) \text {Si}\left (4 b x^n\right )}{8 n}+\frac {3 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^n]^4/x,x]

[Out]

(3*Log[x])/8 + (-4*Cos[2*a]*CosIntegral[2*b*x^n] + Cos[4*a]*CosIntegral[4*b*x^n] + 4*Sin[2*a]*SinIntegral[2*b*
x^n] - Sin[4*a]*SinIntegral[4*b*x^n])/(8*n)

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 87, normalized size = 1.10 \[ \frac {\cos \left (4 \, a\right ) \operatorname {Ci}\left (4 \, b x^{n}\right ) - 4 \, \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) - 4 \, \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) + \cos \left (4 \, a\right ) \operatorname {Ci}\left (-4 \, b x^{n}\right ) + 6 \, n \log \relax (x) - 2 \, \sin \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x^{n}\right ) + 8 \, \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )}{16 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(cos(4*a)*cos_integral(4*b*x^n) - 4*cos(2*a)*cos_integral(2*b*x^n) - 4*cos(2*a)*cos_integral(-2*b*x^n) +
cos(4*a)*cos_integral(-4*b*x^n) + 6*n*log(x) - 2*sin(4*a)*sin_integral(4*b*x^n) + 8*sin(2*a)*sin_integral(2*b*
x^n))/n

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x^{n} + a\right )^{4}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x^n + a)^4/x, x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 77, normalized size = 0.97 \[ \frac {3 \ln \left (b \,x^{n}\right )}{8 n}-\frac {\Si \left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8 n}+\frac {\Ci \left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8 n}+\frac {\Si \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2 n}-\frac {\Ci \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 1.32, size = 188, normalized size = 2.38 \[ \frac {{\left ({\rm Ei}\left (4 i \, b x^{n}\right ) + {\rm Ei}\left (-4 i \, b x^{n}\right ) + {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (4 \, a\right ) - 4 \, {\left ({\rm Ei}\left (2 i \, b x^{n}\right ) + {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \left (2 \, a\right ) + 12 \, n \log \relax (x) + {\left (i \, {\rm Ei}\left (4 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-4 i \, b x^{n}\right ) + i \, {\rm Ei}\left (4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - i \, {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (4 \, a\right ) + {\left (-4 i \, {\rm Ei}\left (2 i \, b x^{n}\right ) + 4 i \, {\rm Ei}\left (-2 i \, b x^{n}\right ) - 4 i \, {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + 4 i \, {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \left (2 \, a\right )}{32 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*((Ei(4*I*b*x^n) + Ei(-4*I*b*x^n) + Ei(4*I*b*e^(n*conjugate(log(x)))) + Ei(-4*I*b*e^(n*conjugate(log(x))))
)*cos(4*a) - 4*(Ei(2*I*b*x^n) + Ei(-2*I*b*x^n) + Ei(2*I*b*e^(n*conjugate(log(x)))) + Ei(-2*I*b*e^(n*conjugate(
log(x)))))*cos(2*a) + 12*n*log(x) + (I*Ei(4*I*b*x^n) - I*Ei(-4*I*b*x^n) + I*Ei(4*I*b*e^(n*conjugate(log(x))))
- I*Ei(-4*I*b*e^(n*conjugate(log(x)))))*sin(4*a) + (-4*I*Ei(2*I*b*x^n) + 4*I*Ei(-2*I*b*x^n) - 4*I*Ei(2*I*b*e^(
n*conjugate(log(x)))) + 4*I*Ei(-2*I*b*e^(n*conjugate(log(x)))))*sin(2*a))/n

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,x^n\right )}^4}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)^4/x,x)

[Out]

int(sin(a + b*x^n)^4/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x**n)**4/x, x)

________________________________________________________________________________________