3.144 \(\int x^m \sin ^3(a+b x^n) \, dx\)

Optimal. Leaf size=237 \[ \frac{3 i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{8 n}-\frac{3 i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{8 n}-\frac{i e^{3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )}{8 n}+\frac{i e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )}{8 n} \]

[Out]

(((3*I)/8)*E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(n*((-I)*b*x^n)^((1 + m)/n)) - (((3*I)/8)*x^(1 + m)
*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*n*(I*b*x^n)^((1 + m)/n)) - ((I/8)*E^((3*I)*a)*x^(1 + m)*Gamma[(1 + m)/n,
(-3*I)*b*x^n])/(3^((1 + m)/n)*n*((-I)*b*x^n)^((1 + m)/n)) + ((I/8)*x^(1 + m)*Gamma[(1 + m)/n, (3*I)*b*x^n])/(3
^((1 + m)/n)*E^((3*I)*a)*n*(I*b*x^n)^((1 + m)/n))

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Rubi [A]  time = 0.23821, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3425, 3423, 2218} \[ \frac{3 i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{8 n}-\frac{3 i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{8 n}-\frac{i e^{3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )}{8 n}+\frac{i e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sin[a + b*x^n]^3,x]

[Out]

(((3*I)/8)*E^(I*a)*x^(1 + m)*Gamma[(1 + m)/n, (-I)*b*x^n])/(n*((-I)*b*x^n)^((1 + m)/n)) - (((3*I)/8)*x^(1 + m)
*Gamma[(1 + m)/n, I*b*x^n])/(E^(I*a)*n*(I*b*x^n)^((1 + m)/n)) - ((I/8)*E^((3*I)*a)*x^(1 + m)*Gamma[(1 + m)/n,
(-3*I)*b*x^n])/(3^((1 + m)/n)*n*((-I)*b*x^n)^((1 + m)/n)) + ((I/8)*x^(1 + m)*Gamma[(1 + m)/n, (3*I)*b*x^n])/(3
^((1 + m)/n)*E^((3*I)*a)*n*(I*b*x^n)^((1 + m)/n))

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]

Rule 3423

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

Rule 2218

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

Rubi steps

\begin{align*} \int x^m \sin ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac{3}{4} x^m \sin \left (a+b x^n\right )-\frac{1}{4} x^m \sin \left (3 a+3 b x^n\right )\right ) \, dx\\ &=-\left (\frac{1}{4} \int x^m \sin \left (3 a+3 b x^n\right ) \, dx\right )+\frac{3}{4} \int x^m \sin \left (a+b x^n\right ) \, dx\\ &=-\left (\frac{1}{8} i \int e^{-3 i a-3 i b x^n} x^m \, dx\right )+\frac{1}{8} i \int e^{3 i a+3 i b x^n} x^m \, dx+\frac{3}{8} i \int e^{-i a-i b x^n} x^m \, dx-\frac{3}{8} i \int e^{i a+i b x^n} x^m \, dx\\ &=\frac{3 i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-i b x^n\right )}{8 n}-\frac{3 i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},i b x^n\right )}{8 n}-\frac{i 3^{-\frac{1+m}{n}} e^{3 i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-3 i b x^n\right )}{8 n}+\frac{i 3^{-\frac{1+m}{n}} e^{-3 i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},3 i b x^n\right )}{8 n}\\ \end{align*}

Mathematica [A]  time = 0.579187, size = 225, normalized size = 0.95 \[ \frac{i e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left (e^{2 i a} \left (-3^{\frac{m+n+1}{n}}\right ) \left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )+e^{4 i a} 3^{\frac{m+n+1}{n}} \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )-e^{6 i a} \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sin[a + b*x^n]^3,x]

[Out]

((I/8)*x^(1 + m)*(3^((1 + m + n)/n)*E^((4*I)*a)*(I*b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, (-I)*b*x^n] - 3^((1 + m
+ n)/n)*E^((2*I)*a)*((-I)*b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, I*b*x^n] - E^((6*I)*a)*(I*b*x^n)^((1 + m)/n)*Ga
mma[(1 + m)/n, (-3*I)*b*x^n] + ((-I)*b*x^n)^((1 + m)/n)*Gamma[(1 + m)/n, (3*I)*b*x^n]))/(3^((1 + m)/n)*E^((3*I
)*a)*n*(b^2*x^(2*n))^((1 + m)/n))

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Maple [F]  time = 0.217, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \sin \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sin(a+b*x^n)^3,x)

[Out]

int(x^m*sin(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sin \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sin(a+b*x^n)^3,x, algorithm="maxima")

[Out]

integrate(x^m*sin(b*x^n + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (x^{m} \cos \left (b x^{n} + a\right )^{2} - x^{m}\right )} \sin \left (b x^{n} + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sin(a+b*x^n)^3,x, algorithm="fricas")

[Out]

integral(-(x^m*cos(b*x^n + a)^2 - x^m)*sin(b*x^n + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sin ^{3}{\left (a + b x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*sin(a+b*x**n)**3,x)

[Out]

Integral(x**m*sin(a + b*x**n)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \sin \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sin(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(x^m*sin(b*x^n + a)^3, x)