3.2.46 \(\int x^{-1+2 n} \cos (a+b x^n) \, dx\) [146]

Optimal. Leaf size=34 \[ \frac {\cos \left (a+b x^n\right )}{b^2 n}+\frac {x^n \sin \left (a+b x^n\right )}{b n} \]

[Out]

cos(a+b*x^n)/b^2/n+x^n*sin(a+b*x^n)/b/n

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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3461, 3377, 2718} \begin {gather*} \frac {\cos \left (a+b x^n\right )}{b^2 n}+\frac {x^n \sin \left (a+b x^n\right )}{b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)*Cos[a + b*x^n],x]

[Out]

Cos[a + b*x^n]/(b^2*n) + (x^n*Sin[a + b*x^n])/(b*n)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 29, normalized size = 0.85 \begin {gather*} \frac {\cos \left (a+b x^n\right )+b x^n \sin \left (a+b x^n\right )}{b^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 2*n)*Cos[a + b*x^n],x]

[Out]

(Cos[a + b*x^n] + b*x^n*Sin[a + b*x^n])/(b^2*n)

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Maple [A]
time = 0.06, size = 44, normalized size = 1.29

method	result	size
risch	\(\frac {\cos \left (a +b \,x^{n}\right )}{b^{2} n}+\frac {x^{n} \sin \left (a +b \,x^{n}\right )}{b n}\)	\(35\)
default	\(\frac {\cos \left (a +b \,x^{n}\right )+\sin \left (a +b \,x^{n}\right ) \left (a +b \,x^{n}\right )-\sin \left (a +b \,x^{n}\right ) a}{n \,b^{2}}\)	\(44\)
meijerg	error in int/gproduct: numeric exception: division by zero\	N/A

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)*cos(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 29, normalized size = 0.85 \begin {gather*} \frac {b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*cos(a+b*x^n),x, algorithm="maxima")

[Out]

(b*x^n*sin(b*x^n + a) + cos(b*x^n + a))/(b^2*n)

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Fricas [A]
time = 0.36, size = 29, normalized size = 0.85 \begin {gather*} \frac {b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)*cos(a+b*x^n),x, algorithm="fricas")

[Out]

(b*x^n*sin(b*x^n + a) + cos(b*x^n + a))/(b^2*n)

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Sympy [A]
time = 29.00, size = 53, normalized size = 1.56 \begin {gather*} \begin {cases} \log {\left (x \right )} \cos {\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{2 n} \cos {\left (a \right )}}{2 n} & \text {for}\: b = 0 \\\log {\left (x \right )} \cos {\left (a + b \right )} & \text {for}\: n = 0 \\\frac {x^{n} \sin {\left (a + b x^{n} \right )}}{b n} + \frac {\cos {\left (a + b x^{n} \right )}}{b^{2} n} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+2*n)*cos(a+b*x**n),x)

[Out]

Piecewise((log(x)*cos(a), Eq(b, 0) & Eq(n, 0)), (x**(2*n)*cos(a)/(2*n), Eq(b, 0)), (log(x)*cos(a + b), Eq(n, 0
)), (x**n*sin(a + b*x**n)/(b*n) + cos(a + b*x**n)/(b**2*n), True))

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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(x^(-1+2*n)*cos(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)*cos(b*x^n + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int x^{2\,n-1}\,\cos \left (a+b\,x^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n - 1)*cos(a + b*x^n),x)

[Out]

int(x^(2*n - 1)*cos(a + b*x^n), x)

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