3.3.61 \(\int x^2 \sin (a+b (c+d x)^n) \, dx\) [261]

Optimal. Leaf size=369 \[ \frac {i c^2 e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i c^2 e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^3 n}-\frac {i c e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i b (c+d x)^n\right )}{d^3 n}+\frac {i c e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i b (c+d x)^n\right )}{d^3 n}+\frac {i e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^3 n} \]

[Out]

1/2*I*c^2*exp(I*a)*(d*x+c)*GAMMA(1/n,-I*b*(d*x+c)^n)/d^3/n/((-I*b*(d*x+c)^n)^(1/n))-1/2*I*c^2*(d*x+c)*GAMMA(1/
n,I*b*(d*x+c)^n)/d^3/exp(I*a)/n/((I*b*(d*x+c)^n)^(1/n))-I*c*exp(I*a)*(d*x+c)^2*GAMMA(2/n,-I*b*(d*x+c)^n)/d^3/n
/((-I*b*(d*x+c)^n)^(2/n))+I*c*(d*x+c)^2*GAMMA(2/n,I*b*(d*x+c)^n)/d^3/exp(I*a)/n/((I*b*(d*x+c)^n)^(2/n))+1/2*I*
exp(I*a)*(d*x+c)^3*GAMMA(3/n,-I*b*(d*x+c)^n)/d^3/n/((-I*b*(d*x+c)^n)^(3/n))-1/2*I*(d*x+c)^3*GAMMA(3/n,I*b*(d*x
+c)^n)/d^3/exp(I*a)/n/((I*b*(d*x+c)^n)^(3/n))

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Rubi [A]
time = 0.17, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3514, 3446, 2239, 3504, 2250} \begin {gather*} \frac {i e^{i a} c^2 (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{-i a} c^2 (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^3 n}+\frac {i e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{i a} c (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i b (c+d x)^n\right )}{d^3 n}+\frac {i e^{-i a} c (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i b (c+d x)^n\right )}{d^3 n}-\frac {i e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[a + b*(c + d*x)^n],x]

[Out]

((I/2)*c^2*E^(I*a)*(c + d*x)*Gamma[n^(-1), (-I)*b*(c + d*x)^n])/(d^3*n*((-I)*b*(c + d*x)^n)^n^(-1)) - ((I/2)*c
^2*(c + d*x)*Gamma[n^(-1), I*b*(c + d*x)^n])/(d^3*E^(I*a)*n*(I*b*(c + d*x)^n)^n^(-1)) - (I*c*E^(I*a)*(c + d*x)
^2*Gamma[2/n, (-I)*b*(c + d*x)^n])/(d^3*n*((-I)*b*(c + d*x)^n)^(2/n)) + (I*c*(c + d*x)^2*Gamma[2/n, I*b*(c + d
*x)^n])/(d^3*E^(I*a)*n*(I*b*(c + d*x)^n)^(2/n)) + ((I/2)*E^(I*a)*(c + d*x)^3*Gamma[3/n, (-I)*b*(c + d*x)^n])/(
d^3*n*((-I)*b*(c + d*x)^n)^(3/n)) - ((I/2)*(c + d*x)^3*Gamma[3/n, I*b*(c + d*x)^n])/(d^3*E^(I*a)*n*(I*b*(c + d
*x)^n)^(3/n))

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/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 3446

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

Rule 3504

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 3514

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[k/f^(m + 1), Subst[Int[ExpandIntegrand[(a + b*Sin[c +
d*x^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f,
g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]

Rubi steps

\begin {align*} \int x^2 \sin \left (a+b (c+d x)^n\right ) \, dx &=\frac {\text {Subst}\left (\int \left (c^2 \sin \left (a+b x^n\right )-2 c x \sin \left (a+b x^n\right )+x^2 \sin \left (a+b x^n\right )\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {\text {Subst}\left (\int x^2 \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}-\frac {(2 c) \text {Subst}\left (\int x \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}+\frac {c^2 \text {Subst}\left (\int \sin \left (a+b x^n\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {i \text {Subst}\left (\int e^{-i a-i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^3}-\frac {i \text {Subst}\left (\int e^{i a+i b x^n} x^2 \, dx,x,c+d x\right )}{2 d^3}-\frac {(i c) \text {Subst}\left (\int e^{-i a-i b x^n} x \, dx,x,c+d x\right )}{d^3}+\frac {(i c) \text {Subst}\left (\int e^{i a+i b x^n} x \, dx,x,c+d x\right )}{d^3}+\frac {\left (i c^2\right ) \text {Subst}\left (\int e^{-i a-i b x^n} \, dx,x,c+d x\right )}{2 d^3}-\frac {\left (i c^2\right ) \text {Subst}\left (\int e^{i a+i b x^n} \, dx,x,c+d x\right )}{2 d^3}\\ &=\frac {i c^2 e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i c^2 e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b (c+d x)^n\right )}{2 d^3 n}-\frac {i c e^{i a} (c+d x)^2 \left (-i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i b (c+d x)^n\right )}{d^3 n}+\frac {i c e^{-i a} (c+d x)^2 \left (i b (c+d x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i b (c+d x)^n\right )}{d^3 n}+\frac {i e^{i a} (c+d x)^3 \left (-i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i b (c+d x)^n\right )}{2 d^3 n}-\frac {i e^{-i a} (c+d x)^3 \left (i b (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i b (c+d x)^n\right )}{2 d^3 n}\\ \end {align*}

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Mathematica [F]
time = 5.20, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \sin \left (a+b (c+d x)^n\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2*Sin[a + b*(c + d*x)^n],x]

[Out]

Integrate[x^2*Sin[a + b*(c + d*x)^n], x]

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{2} \sin \left (a +b \left (d x +c \right )^{n}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(a+b*(d*x+c)^n),x)

[Out]

int(x^2*sin(a+b*(d*x+c)^n),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*sin((d*x + c)^n*b + a), x)

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Fricas [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^2*sin(a+b*(d*x+c)^n),x, algorithm="fricas")

[Out]

integral(x^2*sin((d*x + c)^n*b + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sin {\left (a + b \left (c + d x\right )^{n} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sin(a+b*(d*x+c)**n),x)

[Out]

Integral(x**2*sin(a + b*(c + d*x)**n), 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(x^2*sin(a+b*(d*x+c)^n),x, algorithm="giac")

[Out]

integrate(x^2*sin((d*x + c)^n*b + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(a + b*(c + d*x)^n),x)

[Out]

int(x^2*sin(a + b*(c + d*x)^n), x)

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