\(\int x^m \cos ^2(a+b x^n) \, dx\) [77]

Optimal result

Integrand size = 14, antiderivative size = 141 \[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\frac {x^{1+m}}{2 (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )}{n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )}{n} \]

[Out]

1/2*x^(1+m)/(1+m)-exp(2*I*a)*x^(1+m)*GAMMA((1+m)/n,-2*I*b*x^n)/(2^((1+m+2*n)/n))/n/((-I*b*x^n)^((1+m)/n))-x^(1
+m)*GAMMA((1+m)/n,2*I*b*x^n)/(2^((1+m+2*n)/n))/exp(2*I*a)/n/((I*b*x^n)^((1+m)/n))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3507, 3505, 2250} \[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=-\frac {e^{2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 i b x^n\right )}{n}-\frac {e^{-2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 i b x^n\right )}{n}+\frac {x^{m+1}}{2 (m+1)} \]

[In]

Int[x^m*Cos[a + b*x^n]^2,x]

[Out]

x^(1 + m)/(2*(1 + m)) - (E^((2*I)*a)*x^(1 + m)*Gamma[(1 + m)/n, (-2*I)*b*x^n])/(2^((1 + m + 2*n)/n)*n*((-I)*b*
x^n)^((1 + m)/n)) - (x^(1 + m)*Gamma[(1 + m)/n, (2*I)*b*x^n])/(2^((1 + m + 2*n)/n)*E^((2*I)*a)*n*(I*b*x^n)^((1
+ m)/n))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3505

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, n}, x]

Rule 3507

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^m}{2}+\frac {1}{2} x^m \cos \left (2 a+2 b x^n\right )\right ) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cos \left (2 a+2 b x^n\right ) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-2 i a-2 i b x^n} x^m \, dx+\frac {1}{4} \int e^{2 i a+2 i b x^n} x^m \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )}{n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.91 \[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=-\frac {x^{1+m} \left (-2 n+2^{-\frac {1+m}{n}} e^{2 i a} (1+m) \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )+2^{-\frac {1+m}{n}} e^{-2 i a} (1+m) \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )\right )}{4 (1+m) n} \]

[In]

Integrate[x^m*Cos[a + b*x^n]^2,x]

[Out]

-1/4*(x^(1 + m)*(-2*n + (E^((2*I)*a)*(1 + m)*Gamma[(1 + m)/n, (-2*I)*b*x^n])/(2^((1 + m)/n)*((-I)*b*x^n)^((1 +
m)/n)) + ((1 + m)*Gamma[(1 + m)/n, (2*I)*b*x^n])/(2^((1 + m)/n)*E^((2*I)*a)*(I*b*x^n)^((1 + m)/n))))/((1 + m)
*n)

Maple [F]

\[\int x^{m} \left (\cos ^{2}\left (a +b \,x^{n}\right )\right )d x\]

[In]

int(x^m*cos(a+b*x^n)^2,x)

[Out]

int(x^m*cos(a+b*x^n)^2,x)

Fricas [F]

\[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\int { x^{m} \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

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

[Out]

integral(x^m*cos(b*x^n + a)^2, x)

Sympy [F]

\[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\int x^{m} \cos ^{2}{\left (a + b x^{n} \right )}\, dx \]

[In]

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

[Out]

Integral(x**m*cos(a + b*x**n)**2, x)

Maxima [F]

\[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\int { x^{m} \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\int { x^{m} \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(x^m*cos(b*x^n + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^m \cos ^2\left (a+b x^n\right ) \, dx=\int x^m\,{\cos \left (a+b\,x^n\right )}^2 \,d x \]

[In]

int(x^m*cos(a + b*x^n)^2,x)

[Out]

int(x^m*cos(a + b*x^n)^2, x)