3.2.36 \(\int \frac {\sin ^2(a+b x^n)}{x} \, dx\) [136]

Optimal. Leaf size=43 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n]^2/x,x]

[Out]

-1/2*(Cos[2*a]*CosIntegral[2*b*x^n])/n + Log[x]/2 + (Sin[2*a]*SinIntegral[2*b*x^n])/(2*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 ^2\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac {1}{2 x}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \int \frac {\cos \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {\log (x)}{2}-\frac {1}{2} \cos (2 a) \int \frac {\cos \left (2 b x^n\right )}{x} \, dx+\frac {1}{2} \sin (2 a) \int \frac {\sin \left (2 b x^n\right )}{x} \, dx\\ &=-\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right )}{2 n}+\frac {\log (x)}{2}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 37, normalized size = 0.86 \begin {gather*} \frac {-\cos (2 a) \text {Ci}\left (2 b x^n\right )+n \log (x)+\sin (2 a) \text {Si}\left (2 b x^n\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^n]^2/x,x]

[Out]

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

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Maple [A]
time = 0.06, size = 40, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (b \,x^{n}\right )}{2}+\frac {\sinIntegral \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}-\frac {\cosineIntegral \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}}{n}\)	\(40\)
default	\(\frac {\frac {\ln \left (b \,x^{n}\right )}{2}+\frac {\sinIntegral \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}-\frac {\cosineIntegral \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}}{n}\)	\(40\)
risch	\(-\frac {i {\mathrm e}^{-2 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{4 n}+\frac {i {\mathrm e}^{-2 i a} \sinIntegral \left (2 b \,x^{n}\right )}{2 n}+\frac {{\mathrm e}^{-2 i a} \expIntegral \left (1, -2 i b \,x^{n}\right )}{4 n}+\frac {{\mathrm e}^{2 i a} \expIntegral \left (1, -2 i b \,x^{n}\right )}{4 n}+\frac {\ln \left (x \right )}{2}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*x^n)^2/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.44, size = 100, normalized size = 2.33 \begin {gather*} -\frac {{\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 ) - 4 \, n \log \left (x\right ) - {\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 )}{8 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*x^n)^2/x,x, algorithm="maxima")

[Out]

-1/8*((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) - 4*n*log(x) - (-I*Ei(2*I*b*x^n) + I*Ei(-2*I*b*x^n) - I*Ei(2*I*b*e^(n*conjugate(log(x)))) + I*Ei(-2
*I*b*e^(n*conjugate(log(x)))))*sin(2*a))/n

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Fricas [A]
time = 0.39, size = 48, normalized size = 1.12 \begin {gather*} -\frac {\cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) - 2 \, n \log \left (x\right ) - 2 \, \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )}{4 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*x^n)^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{2}{\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)**2/x,x)

[Out]

Integral(sin(a + b*x**n)**2/x, x)

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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*x^n)^2/x,x, algorithm="giac")

[Out]

integrate(sin(b*x^n + a)^2/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\sin \left (a+b\,x^n\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)^2/x,x)

[Out]

int(sin(a + b*x^n)^2/x, x)

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