3.2.71 \(\int (e+f x)^3 \sin (a+b (c+d x)^3) \, dx\) [171]

Optimal. Leaf size=434 \[ -\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b d^4}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4}-\frac {e^{i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{-i b (c+d x)^3}}+\frac {i e^{i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {e^{-i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{i b (c+d x)^3}}-\frac {i e^{-i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{2 d^4 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{2 d^4 \left (i b (c+d x)^3\right )^{2/3}} \]

[Out]

-f^2*(-c*f+d*e)*cos(a+b*(d*x+c)^3)/b/d^4-1/3*f^3*(d*x+c)*cos(a+b*(d*x+c)^3)/b/d^4-1/18*exp(I*a)*f^3*(d*x+c)*GA
MMA(1/3,-I*b*(d*x+c)^3)/b/d^4/(-I*b*(d*x+c)^3)^(1/3)+1/6*I*exp(I*a)*(-c*f+d*e)^3*(d*x+c)*GAMMA(1/3,-I*b*(d*x+c
)^3)/d^4/(-I*b*(d*x+c)^3)^(1/3)-1/18*f^3*(d*x+c)*GAMMA(1/3,I*b*(d*x+c)^3)/b/d^4/exp(I*a)/(I*b*(d*x+c)^3)^(1/3)
-1/6*I*(-c*f+d*e)^3*(d*x+c)*GAMMA(1/3,I*b*(d*x+c)^3)/d^4/exp(I*a)/(I*b*(d*x+c)^3)^(1/3)+1/2*I*exp(I*a)*f*(-c*f
+d*e)^2*(d*x+c)^2*GAMMA(2/3,-I*b*(d*x+c)^3)/d^4/(-I*b*(d*x+c)^3)^(2/3)-1/2*I*f*(-c*f+d*e)^2*(d*x+c)^2*GAMMA(2/
3,I*b*(d*x+c)^3)/d^4/exp(I*a)/(I*b*(d*x+c)^3)^(2/3)

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Rubi [A]
time = 0.32, antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3514, 3436, 2239, 3470, 2250, 3460, 2718, 3466, 3437} \begin {gather*} \frac {i e^{i a} f (c+d x)^2 (d e-c f)^2 \text {Gamma}\left (\frac {2}{3},-i b (c+d x)^3\right )}{2 d^4 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (c+d x)^2 (d e-c f)^2 \text {Gamma}\left (\frac {2}{3},i b (c+d x)^3\right )}{2 d^4 \left (i b (c+d x)^3\right )^{2/3}}+\frac {i e^{i a} (c+d x) (d e-c f)^3 \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) (d e-c f)^3 \text {Gamma}\left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{i b (c+d x)^3}}-\frac {e^{i a} f^3 (c+d x) \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {e^{-i a} f^3 (c+d x) \text {Gamma}\left (\frac {1}{3},i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{i b (c+d x)^3}}-\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b d^4}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((f^2*(d*e - c*f)*Cos[a + b*(c + d*x)^3])/(b*d^4)) - (f^3*(c + d*x)*Cos[a + b*(c + d*x)^3])/(3*b*d^4) - (E^(I
*a)*f^3*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(18*b*d^4*((-I)*b*(c + d*x)^3)^(1/3)) + ((I/6)*E^(I*a)*(d*e
- c*f)^3*(c + d*x)*Gamma[1/3, (-I)*b*(c + d*x)^3])/(d^4*((-I)*b*(c + d*x)^3)^(1/3)) - (f^3*(c + d*x)*Gamma[1/3
, I*b*(c + d*x)^3])/(18*b*d^4*E^(I*a)*(I*b*(c + d*x)^3)^(1/3)) - ((I/6)*(d*e - c*f)^3*(c + d*x)*Gamma[1/3, I*b
*(c + d*x)^3])/(d^4*E^(I*a)*(I*b*(c + d*x)^3)^(1/3)) + ((I/2)*E^(I*a)*f*(d*e - c*f)^2*(c + d*x)^2*Gamma[2/3, (
-I)*b*(c + d*x)^3])/(d^4*((-I)*b*(c + d*x)^3)^(2/3)) - ((I/2)*f*(d*e - c*f)^2*(c + d*x)^2*Gamma[2/3, I*b*(c +
d*x)^3])/(d^4*E^(I*a)*(I*b*(c + d*x)^3)^(2/3))

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 2718

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

Rule 3436

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}, x] && IGtQ[n, 2]

Rule 3437

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

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[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]))

Rule 3466

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^(n - 1))*(e*x)^(m - n + 1)*(Cos[c +
d*x^n]/(d*n)), x] + Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}
, x] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3470

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}, x] && IGtQ[n, 0]

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 (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx &=\frac {\text {Subst}\left (\int \left (d^3 e^3 \left (1-\frac {c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) \sin \left (a+b x^3\right )+3 d^2 e^2 f \left (1+\frac {c f (-2 d e+c f)}{d^2 e^2}\right ) x \sin \left (a+b x^3\right )+3 d e f^2 \left (1-\frac {c f}{d e}\right ) x^2 \sin \left (a+b x^3\right )+f^3 x^3 \sin \left (a+b x^3\right )\right ) \, dx,x,c+d x\right )}{d^4}\\ &=\frac {f^3 \text {Subst}\left (\int x^3 \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (3 f^2 (d e-c f)\right ) \text {Subst}\left (\int x^2 \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^4}+\frac {\left (3 f (d e-c f)^2\right ) \text {Subst}\left (\int x \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^4}+\frac {(d e-c f)^3 \text {Subst}\left (\int \sin \left (a+b x^3\right ) \, dx,x,c+d x\right )}{d^4}\\ &=-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4}+\frac {f^3 \text {Subst}\left (\int \cos \left (a+b x^3\right ) \, dx,x,c+d x\right )}{3 b d^4}+\frac {\left (f^2 (d e-c f)\right ) \text {Subst}\left (\int \sin (a+b x) \, dx,x,(c+d x)^3\right )}{d^4}+\frac {\left (3 i f (d e-c f)^2\right ) \text {Subst}\left (\int e^{-i a-i b x^3} x \, dx,x,c+d x\right )}{2 d^4}-\frac {\left (3 i f (d e-c f)^2\right ) \text {Subst}\left (\int e^{i a+i b x^3} x \, dx,x,c+d x\right )}{2 d^4}+\frac {\left (i (d e-c f)^3\right ) \text {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,c+d x\right )}{2 d^4}-\frac {\left (i (d e-c f)^3\right ) \text {Subst}\left (\int e^{i a+i b x^3} \, dx,x,c+d x\right )}{2 d^4}\\ &=-\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b d^4}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4}+\frac {i e^{i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{2 d^4 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{2 d^4 \left (i b (c+d x)^3\right )^{2/3}}+\frac {f^3 \text {Subst}\left (\int e^{-i a-i b x^3} \, dx,x,c+d x\right )}{6 b d^4}+\frac {f^3 \text {Subst}\left (\int e^{i a+i b x^3} \, dx,x,c+d x\right )}{6 b d^4}\\ &=-\frac {f^2 (d e-c f) \cos \left (a+b (c+d x)^3\right )}{b d^4}-\frac {f^3 (c+d x) \cos \left (a+b (c+d x)^3\right )}{3 b d^4}-\frac {e^{i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{-i b (c+d x)^3}}+\frac {i e^{i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{-i b (c+d x)^3}}-\frac {e^{-i a} f^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{18 b d^4 \sqrt [3]{i b (c+d x)^3}}-\frac {i e^{-i a} (d e-c f)^3 (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d^4 \sqrt [3]{i b (c+d x)^3}}+\frac {i e^{i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},-i b (c+d x)^3\right )}{2 d^4 \left (-i b (c+d x)^3\right )^{2/3}}-\frac {i e^{-i a} f (d e-c f)^2 (c+d x)^2 \Gamma \left (\frac {2}{3},i b (c+d x)^3\right )}{2 d^4 \left (i b (c+d x)^3\right )^{2/3}}\\ \end {align*}

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Mathematica [F]
time = 101.48, size = 0, normalized size = 0.00 \begin {gather*} \int (e+f x)^3 \sin \left (a+b (c+d x)^3\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (f x +e \right )^{3} \sin \left (a +b \left (d x +c \right )^{3}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [A]
time = 0.11, size = 424, normalized size = 0.98 \begin {gather*} \frac {{\left (3 \, b c^{3} f^{3} - 9 \, b c^{2} d f^{2} e + 9 \, b c d^{2} f e^{2} - 3 \, b d^{3} e^{3} + i \, f^{3}\right )} \left (i \, b d^{3}\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + {\left (3 \, b c^{3} f^{3} - 9 \, b c^{2} d f^{2} e + 9 \, b c d^{2} f e^{2} - 3 \, b d^{3} e^{3} - i \, f^{3}\right )} \left (-i \, b d^{3}\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) - 9 \, {\left (b c^{2} d f^{3} - 2 \, b c d^{2} f^{2} e + b d^{3} f e^{2}\right )} \left (i \, b d^{3}\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) - 9 \, {\left (b c^{2} d f^{3} - 2 \, b c d^{2} f^{2} e + b d^{3} f e^{2}\right )} \left (-i \, b d^{3}\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right ) - 6 \, {\left (b d^{3} f^{3} x - 2 \, b c d^{2} f^{3} + 3 \, b d^{3} f^{2} e\right )} \cos \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{18 \, b^{2} d^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sin(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

1/18*((3*b*c^3*f^3 - 9*b*c^2*d*f^2*e + 9*b*c*d^2*f*e^2 - 3*b*d^3*e^3 + I*f^3)*(I*b*d^3)^(2/3)*e^(-I*a)*gamma(1
/3, I*b*d^3*x^3 + 3*I*b*c*d^2*x^2 + 3*I*b*c^2*d*x + I*b*c^3) + (3*b*c^3*f^3 - 9*b*c^2*d*f^2*e + 9*b*c*d^2*f*e^
2 - 3*b*d^3*e^3 - I*f^3)*(-I*b*d^3)^(2/3)*e^(I*a)*gamma(1/3, -I*b*d^3*x^3 - 3*I*b*c*d^2*x^2 - 3*I*b*c^2*d*x -
I*b*c^3) - 9*(b*c^2*d*f^3 - 2*b*c*d^2*f^2*e + b*d^3*f*e^2)*(I*b*d^3)^(1/3)*e^(-I*a)*gamma(2/3, I*b*d^3*x^3 + 3
*I*b*c*d^2*x^2 + 3*I*b*c^2*d*x + I*b*c^3) - 9*(b*c^2*d*f^3 - 2*b*c*d^2*f^2*e + b*d^3*f*e^2)*(-I*b*d^3)^(1/3)*e
^(I*a)*gamma(2/3, -I*b*d^3*x^3 - 3*I*b*c*d^2*x^2 - 3*I*b*c^2*d*x - I*b*c^3) - 6*(b*d^3*f^3*x - 2*b*c*d^2*f^3 +
3*b*d^3*f^2*e)*cos(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(b^2*d^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e + f x\right )^{3} \sin {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e + f*x)**3*sin(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3), 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((f*x+e)^3*sin(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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