3.3.66 \(\int x^3 (a+b \sin (c+d (f+g x)^n)) \, dx\) [266]

Optimal. Leaf size=519 \[ \frac {a x^4}{4}-\frac {i b e^{i c} f^3 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{-i c} f^3 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{i c} f^2 (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{-i c} f^2 (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{i c} f (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{-i c} f (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{i c} (f+g x)^4 \left (-i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {i b e^{-i c} (f+g x)^4 \left (i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )}{2 g^4 n} \]

[Out]

1/4*a*x^4-1/2*I*b*exp(I*c)*f^3*(g*x+f)*GAMMA(1/n,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(1/n))+1/2*I*b*f^3*(g
*x+f)*GAMMA(1/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/((I*d*(g*x+f)^n)^(1/n))+3/2*I*b*exp(I*c)*f^2*(g*x+f)^2*GAMMA(2/n
,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(2/n))-3/2*I*b*f^2*(g*x+f)^2*GAMMA(2/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/
((I*d*(g*x+f)^n)^(2/n))-3/2*I*b*exp(I*c)*f*(g*x+f)^3*GAMMA(3/n,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(3/n))+
3/2*I*b*f*(g*x+f)^3*GAMMA(3/n,I*d*(g*x+f)^n)/exp(I*c)/g^4/n/((I*d*(g*x+f)^n)^(3/n))+1/2*I*b*exp(I*c)*(g*x+f)^4
*GAMMA(4/n,-I*d*(g*x+f)^n)/g^4/n/((-I*d*(g*x+f)^n)^(4/n))-1/2*I*b*(g*x+f)^4*GAMMA(4/n,I*d*(g*x+f)^n)/exp(I*c)/
g^4/n/((I*d*(g*x+f)^n)^(4/n))

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Rubi [A]
time = 0.36, antiderivative size = 519, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 3514, 3446, 2239, 3504, 2250} \begin {gather*} -\frac {i b e^{i c} f^3 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{-i c} f^3 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{i c} f^2 (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{-i c} f^2 (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \text {Gamma}\left (\frac {2}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{i c} (f+g x)^4 \left (-i d (f+g x)^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{i c} f (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{-i c} f (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^4 n}-\frac {i b e^{-i c} (f+g x)^4 \left (i d (f+g x)^n\right )^{-4/n} \text {Gamma}\left (\frac {4}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {a x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Sin[c + d*(f + g*x)^n]),x]

[Out]

(a*x^4)/4 - ((I/2)*b*E^(I*c)*f^3*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^n^(-
1)) + ((I/2)*b*f^3*(f + g*x)*Gamma[n^(-1), I*d*(f + g*x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^n^(-1)) + (((3*I
)/2)*b*E^(I*c)*f^2*(f + g*x)^2*Gamma[2/n, (-I)*d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^(2/n)) - (((3*I)/2)
*b*f^2*(f + g*x)^2*Gamma[2/n, I*d*(f + g*x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(2/n)) - (((3*I)/2)*b*E^(I*c)
*f*(f + g*x)^3*Gamma[3/n, (-I)*d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^(3/n)) + (((3*I)/2)*b*f*(f + g*x)^3
*Gamma[3/n, I*d*(f + g*x)^n])/(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(3/n)) + ((I/2)*b*E^(I*c)*(f + g*x)^4*Gamma[4/n
, (-I)*d*(f + g*x)^n])/(g^4*n*((-I)*d*(f + g*x)^n)^(4/n)) - ((I/2)*b*(f + g*x)^4*Gamma[4/n, I*d*(f + g*x)^n])/
(E^(I*c)*g^4*n*(I*d*(f + g*x)^n)^(4/n))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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^3 \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx &=\int \left (a x^3+b x^3 \sin \left (c+d (f+g x)^n\right )\right ) \, dx\\ &=\frac {a x^4}{4}+b \int x^3 \sin \left (c+d (f+g x)^n\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {b \text {Subst}\left (\int \left (-f^3 \sin \left (c+d x^n\right )+3 f^2 x \sin \left (c+d x^n\right )-3 f x^2 \sin \left (c+d x^n\right )+x^3 \sin \left (c+d x^n\right )\right ) \, dx,x,f+g x\right )}{g^4}\\ &=\frac {a x^4}{4}+\frac {b \text {Subst}\left (\int x^3 \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^4}-\frac {(3 b f) \text {Subst}\left (\int x^2 \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^4}+\frac {\left (3 b f^2\right ) \text {Subst}\left (\int x \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^4}-\frac {\left (b f^3\right ) \text {Subst}\left (\int \sin \left (c+d x^n\right ) \, dx,x,f+g x\right )}{g^4}\\ &=\frac {a x^4}{4}+\frac {(i b) \text {Subst}\left (\int e^{-i c-i d x^n} x^3 \, dx,x,f+g x\right )}{2 g^4}-\frac {(i b) \text {Subst}\left (\int e^{i c+i d x^n} x^3 \, dx,x,f+g x\right )}{2 g^4}-\frac {(3 i b f) \text {Subst}\left (\int e^{-i c-i d x^n} x^2 \, dx,x,f+g x\right )}{2 g^4}+\frac {(3 i b f) \text {Subst}\left (\int e^{i c+i d x^n} x^2 \, dx,x,f+g x\right )}{2 g^4}+\frac {\left (3 i b f^2\right ) \text {Subst}\left (\int e^{-i c-i d x^n} x \, dx,x,f+g x\right )}{2 g^4}-\frac {\left (3 i b f^2\right ) \text {Subst}\left (\int e^{i c+i d x^n} x \, dx,x,f+g x\right )}{2 g^4}-\frac {\left (i b f^3\right ) \text {Subst}\left (\int e^{-i c-i d x^n} \, dx,x,f+g x\right )}{2 g^4}+\frac {\left (i b f^3\right ) \text {Subst}\left (\int e^{i c+i d x^n} \, dx,x,f+g x\right )}{2 g^4}\\ &=\frac {a x^4}{4}-\frac {i b e^{i c} f^3 (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{-i c} f^3 (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{i c} f^2 (f+g x)^2 \left (-i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{-i c} f^2 (f+g x)^2 \left (i d (f+g x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )}{2 g^4 n}-\frac {3 i b e^{i c} f (f+g x)^3 \left (-i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right )}{2 g^4 n}+\frac {3 i b e^{-i c} f (f+g x)^3 \left (i d (f+g x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )}{2 g^4 n}+\frac {i b e^{i c} (f+g x)^4 \left (-i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right )}{2 g^4 n}-\frac {i b e^{-i c} (f+g x)^4 \left (i d (f+g x)^n\right )^{-4/n} \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )}{2 g^4 n}\\ \end {align*}

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Mathematica [A]
time = 10.91, size = 539, normalized size = 1.04 \begin {gather*} \frac {1}{4} \left (a x^4-\frac {2 i b (f+g x) \left (d^2 (f+g x)^{2 n}\right )^{-4/n} \left (-f^3 \left (-i d (f+g x)^n\right )^{4/n} \left (i d (f+g x)^n\right )^{3/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )-(f+g x) \left (-3 f^2 \left (-i d (f+g x)^n\right )^{4/n} \left (i d (f+g x)^n\right )^{2/n} \Gamma \left (\frac {2}{n},i d (f+g x)^n\right )-(f+g x) \left (-3 f \left (-i d (f+g x)^n\right )^{4/n} \left (i d (f+g x)^n\right )^{\frac {1}{n}} \Gamma \left (\frac {3}{n},i d (f+g x)^n\right )-(f+g x) \left (-\left (-i d (f+g x)^n\right )^{4/n} \Gamma \left (\frac {4}{n},i d (f+g x)^n\right )+\left (i d (f+g x)^n\right )^{4/n} \Gamma \left (\frac {4}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right )+3 f \left (-i d (f+g x)^n\right )^{\frac {1}{n}} \left (i d (f+g x)^n\right )^{4/n} \Gamma \left (\frac {3}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right )+3 f^2 \left (-i d (f+g x)^n\right )^{2/n} \left (i d (f+g x)^n\right )^{4/n} \Gamma \left (\frac {2}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right )+f^3 \left (-i d (f+g x)^n\right )^{3/n} \left (i d (f+g x)^n\right )^{4/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right ) (\cos (c)+i \sin (c))^2\right ) (\cos (c)-i \sin (c))}{g^4 n}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Sin[c + d*(f + g*x)^n]),x]

[Out]

(a*x^4 - ((2*I)*b*(f + g*x)*(-(f^3*((-I)*d*(f + g*x)^n)^(4/n)*(I*d*(f + g*x)^n)^(3/n)*Gamma[n^(-1), I*d*(f + g
*x)^n]) - (f + g*x)*(-3*f^2*((-I)*d*(f + g*x)^n)^(4/n)*(I*d*(f + g*x)^n)^(2/n)*Gamma[2/n, I*d*(f + g*x)^n] - (
f + g*x)*(-3*f*((-I)*d*(f + g*x)^n)^(4/n)*(I*d*(f + g*x)^n)^n^(-1)*Gamma[3/n, I*d*(f + g*x)^n] - (f + g*x)*(-(
((-I)*d*(f + g*x)^n)^(4/n)*Gamma[4/n, I*d*(f + g*x)^n]) + (I*d*(f + g*x)^n)^(4/n)*Gamma[4/n, (-I)*d*(f + g*x)^
n]*(Cos[c] + I*Sin[c])^2) + 3*f*((-I)*d*(f + g*x)^n)^n^(-1)*(I*d*(f + g*x)^n)^(4/n)*Gamma[3/n, (-I)*d*(f + g*x
)^n]*(Cos[c] + I*Sin[c])^2) + 3*f^2*((-I)*d*(f + g*x)^n)^(2/n)*(I*d*(f + g*x)^n)^(4/n)*Gamma[2/n, (-I)*d*(f +
g*x)^n]*(Cos[c] + I*Sin[c])^2) + f^3*((-I)*d*(f + g*x)^n)^(3/n)*(I*d*(f + g*x)^n)^(4/n)*Gamma[n^(-1), (-I)*d*(
f + g*x)^n]*(Cos[c] + I*Sin[c])^2)*(Cos[c] - I*Sin[c]))/(g^4*n*(d^2*(f + g*x)^(2*n))^(4/n)))/4

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*sin(c+d*(g*x+f)^n)),x)

[Out]

int(x^3*(a+b*sin(c+d*(g*x+f)^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^3*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="maxima")

[Out]

1/4*a*x^4 + b*integrate(x^3*sin((g*x + f)^n*d + c), 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^3*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="fricas")

[Out]

integral(b*x^3*sin((g*x + f)^n*d + c) + a*x^3, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*sin(c+d*(g*x+f)**n)),x)

[Out]

Integral(x**3*(a + b*sin(c + d*(f + g*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^3*(a+b*sin(c+d*(g*x+f)^n)),x, algorithm="giac")

[Out]

integrate((b*sin((g*x + f)^n*d + c) + a)*x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,\left (a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*sin(c + d*(f + g*x)^n)),x)

[Out]

int(x^3*(a + b*sin(c + d*(f + g*x)^n)), x)

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