3.2.38 \(\int \frac {\sin ^4(a+b x^n)}{x} \, dx\) [138]

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

[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 = 0.08, 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 = {3506, 3459, 3457, 3456} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n]^4/x,x]

[Out]

-1/2*(Cos[2*a]*CosIntegral[2*b*x^n])/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 3456

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

Rule 3457

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

Rule 3459

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 3506

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 = 0.08, size = 66, normalized size = 0.84 \begin {gather*} \frac {3 \log (x)}{8}+\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} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.06, size = 66, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {3 \ln \left (b \,x^{n}\right )}{8}-\frac {\sinIntegral \left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8}+\frac {\cosineIntegral \left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8}+\frac {\sinIntegral \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}-\frac {\cosineIntegral \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}}{n}\)	\(66\)
default	\(\frac {\frac {3 \ln \left (b \,x^{n}\right )}{8}-\frac {\sinIntegral \left (4 b \,x^{n}\right ) \sin \left (4 a \right )}{8}+\frac {\cosineIntegral \left (4 b \,x^{n}\right ) \cos \left (4 a \right )}{8}+\frac {\sinIntegral \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}-\frac {\cosineIntegral \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}}{n}\)	\(66\)
risch	\(\frac {i {\mathrm e}^{-4 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{16 n}-\frac {i {\mathrm e}^{-4 i a} \sinIntegral \left (4 b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{-4 i a} \expIntegral \left (1, -4 i b \,x^{n}\right )}{16 n}-\frac {{\mathrm e}^{4 i a} \expIntegral \left (1, -4 i b \,x^{n}\right )}{16 n}+\frac {3 \ln \left (x \right )}{8}-\frac {i {\mathrm e}^{-2 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{4 n}+\frac {i {\mathrm e}^{-2 i a} \sinIntegral \left (2 b \,x^{n}\right )}{2 n}+\frac {{\mathrm e}^{-2 i a} \expIntegral \left (1, -2 i b \,x^{n}\right )}{4 n}+\frac {{\mathrm e}^{2 i a} \expIntegral \left (1, -2 i b \,x^{n}\right )}{4 n}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.52, size = 189, normalized size = 2.39 \begin {gather*} \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 \left (x\right )}\right )}\right ) + {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \left (x\right )}\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 \left (x\right )}\right )}\right ) + {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (2 \, a\right ) + 12 \, n \log \left (x\right ) + {\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 \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-4 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (4 \, a\right ) - 4 \, {\left (i \, {\rm Ei}\left (2 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-2 i \, b x^{n}\right ) + i \, {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (2 \, a\right )}{32 \, n} \end {gather*}

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) - I*Ei(-2*I*b*x^n) + I*Ei(2*I*b*e^(n*con
jugate(log(x)))) - I*Ei(-2*I*b*e^(n*conjugate(log(x)))))*sin(2*a))/n

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 87, normalized size = 1.10 \begin {gather*} \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 \left (x\right ) - 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} \end {gather*}

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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*x^n)^4/x,x, algorithm="giac")

[Out]

integrate(sin(b*x^n + a)^4/x, x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________