3.1.73 \(\int x^4 (a+b \sin (c+d x^3))^2 \, dx\) [73]

Optimal. Leaf size=249 \[ \frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {2 a b e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac {i b^2 e^{2 i c} x^2 \Gamma \left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \Gamma \left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d} \]

[Out]

1/10*(2*a^2+b^2)*x^5-2/3*a*b*x^2*cos(d*x^3+c)/d-2/9*a*b*exp(I*c)*x^2*GAMMA(2/3,-I*d*x^3)/d/(-I*d*x^3)^(2/3)-2/
9*a*b*x^2*GAMMA(2/3,I*d*x^3)/d/exp(I*c)/(I*d*x^3)^(2/3)+1/72*I*b^2*exp(2*I*c)*x^2*GAMMA(2/3,-2*I*d*x^3)*2^(1/3
)/d/(-I*d*x^3)^(2/3)-1/72*I*b^2*x^2*GAMMA(2/3,2*I*d*x^3)*2^(1/3)/d/exp(2*I*c)/(I*d*x^3)^(2/3)-1/12*b^2*x^2*sin
(2*d*x^3+2*c)/d

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Rubi [A]
time = 0.14, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3484, 6, 3467, 3470, 2250, 3466, 3471} \begin {gather*} -\frac {2 a b e^{i c} x^2 \text {Gamma}\left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \text {Gamma}\left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac {i b^2 e^{2 i c} x^2 \text {Gamma}\left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \text {Gamma}\left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}}+\frac {1}{10} x^5 \left (2 a^2+b^2\right )-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*Sin[c + d*x^3])^2,x]

[Out]

((2*a^2 + b^2)*x^5)/10 - (2*a*b*x^2*Cos[c + d*x^3])/(3*d) - (2*a*b*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/(9*d*((
-I)*d*x^3)^(2/3)) - (2*a*b*x^2*Gamma[2/3, I*d*x^3])/(9*d*E^(I*c)*(I*d*x^3)^(2/3)) + ((I/36)*b^2*E^((2*I)*c)*x^
2*Gamma[2/3, (-2*I)*d*x^3])/(2^(2/3)*d*((-I)*d*x^3)^(2/3)) - ((I/36)*b^2*x^2*Gamma[2/3, (2*I)*d*x^3])/(2^(2/3)
*d*E^((2*I)*c)*(I*d*x^3)^(2/3)) - (b^2*x^2*Sin[2*c + 2*d*x^3])/(12*d)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 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 3467

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*
x^n]/(d*n)), x] - Dist[e^n*((m - n + 1)/(d*n)), Int[(e*x)^(m - n)*Sin[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 3471

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}, x] && IGtQ[n, 0]

Rule 3484

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}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int x^4 \left (a+b \sin \left (c+d x^3\right )\right )^2 \, dx &=\int \left (a^2 x^4+\frac {b^2 x^4}{2}-\frac {1}{2} b^2 x^4 \cos \left (2 c+2 d x^3\right )+2 a b x^4 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\int \left (\left (a^2+\frac {b^2}{2}\right ) x^4-\frac {1}{2} b^2 x^4 \cos \left (2 c+2 d x^3\right )+2 a b x^4 \sin \left (c+d x^3\right )\right ) \, dx\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5+(2 a b) \int x^4 \sin \left (c+d x^3\right ) \, dx-\frac {1}{2} b^2 \int x^4 \cos \left (2 c+2 d x^3\right ) \, dx\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(4 a b) \int x \cos \left (c+d x^3\right ) \, dx}{3 d}+\frac {b^2 \int x \sin \left (2 c+2 d x^3\right ) \, dx}{6 d}\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}+\frac {(2 a b) \int e^{-i c-i d x^3} x \, dx}{3 d}+\frac {(2 a b) \int e^{i c+i d x^3} x \, dx}{3 d}+\frac {\left (i b^2\right ) \int e^{-2 i c-2 i d x^3} x \, dx}{12 d}-\frac {\left (i b^2\right ) \int e^{2 i c+2 i d x^3} x \, dx}{12 d}\\ &=\frac {1}{10} \left (2 a^2+b^2\right ) x^5-\frac {2 a b x^2 \cos \left (c+d x^3\right )}{3 d}-\frac {2 a b e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{9 d \left (-i d x^3\right )^{2/3}}-\frac {2 a b e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{9 d \left (i d x^3\right )^{2/3}}+\frac {i b^2 e^{2 i c} x^2 \Gamma \left (\frac {2}{3},-2 i d x^3\right )}{36\ 2^{2/3} d \left (-i d x^3\right )^{2/3}}-\frac {i b^2 e^{-2 i c} x^2 \Gamma \left (\frac {2}{3},2 i d x^3\right )}{36\ 2^{2/3} d \left (i d x^3\right )^{2/3}}-\frac {b^2 x^2 \sin \left (2 c+2 d x^3\right )}{12 d}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 339, normalized size = 1.36 \begin {gather*} \frac {d x^8 \left (72 a^2 d x^3 \left (d^2 x^6\right )^{2/3}+36 b^2 d x^3 \left (d^2 x^6\right )^{2/3}-240 a b \left (d^2 x^6\right )^{2/3} \cos \left (c+d x^3\right )+5 i \sqrt [3]{2} b^2 \left (i d x^3\right )^{2/3} \cos (2 c) \Gamma \left (\frac {2}{3},-2 i d x^3\right )-5 i \sqrt [3]{2} b^2 \left (-i d x^3\right )^{2/3} \cos (2 c) \Gamma \left (\frac {2}{3},2 i d x^3\right )-80 a b \left (-i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},i d x^3\right ) (\cos (c)-i \sin (c))-80 a b \left (i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},-i d x^3\right ) (\cos (c)+i \sin (c))-5 \sqrt [3]{2} b^2 \left (i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},-2 i d x^3\right ) \sin (2 c)-5 \sqrt [3]{2} b^2 \left (-i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},2 i d x^3\right ) \sin (2 c)-30 b^2 \left (d^2 x^6\right )^{2/3} \sin \left (2 \left (c+d x^3\right )\right )\right )}{360 \left (d^2 x^6\right )^{5/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*Sin[c + d*x^3])^2,x]

[Out]

(d*x^8*(72*a^2*d*x^3*(d^2*x^6)^(2/3) + 36*b^2*d*x^3*(d^2*x^6)^(2/3) - 240*a*b*(d^2*x^6)^(2/3)*Cos[c + d*x^3] +
(5*I)*2^(1/3)*b^2*(I*d*x^3)^(2/3)*Cos[2*c]*Gamma[2/3, (-2*I)*d*x^3] - (5*I)*2^(1/3)*b^2*((-I)*d*x^3)^(2/3)*Co
s[2*c]*Gamma[2/3, (2*I)*d*x^3] - 80*a*b*((-I)*d*x^3)^(2/3)*Gamma[2/3, I*d*x^3]*(Cos[c] - I*Sin[c]) - 80*a*b*(I
*d*x^3)^(2/3)*Gamma[2/3, (-I)*d*x^3]*(Cos[c] + I*Sin[c]) - 5*2^(1/3)*b^2*(I*d*x^3)^(2/3)*Gamma[2/3, (-2*I)*d*x
^3]*Sin[2*c] - 5*2^(1/3)*b^2*((-I)*d*x^3)^(2/3)*Gamma[2/3, (2*I)*d*x^3]*Sin[2*c] - 30*b^2*(d^2*x^6)^(2/3)*Sin[
2*(c + d*x^3)]))/(360*(d^2*x^6)^(5/3))

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int x^{4} \left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*sin(d*x^3+c))^2,x)

[Out]

int(x^4*(a+b*sin(d*x^3+c))^2,x)

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Maxima [A]
time = 0.36, size = 234, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, a^{2} x^{5} - \frac {{\left (6 \, d x^{3} \cos \left (d x^{3} + c\right ) - \left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right )\right )} \cos \left (c\right ) + {\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right )\right )} \sin \left (c\right )\right )}\right )} a b}{9 \, d^{2} x} + \frac {{\left (72 \, d^{2} x^{6} - 60 \, d x^{3} \sin \left (2 \, d x^{3} + 2 \, c\right ) - 5 \cdot 2^{\frac {1}{3}} \left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \cos \left (2 \, c\right ) + {\left ({\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (i \, \sqrt {3} + 1\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \sin \left (2 \, c\right )\right )}\right )} b^{2}}{720 \, d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*sin(d*x^3+c))^2,x, algorithm="maxima")

[Out]

1/5*a^2*x^5 - 1/9*(6*d*x^3*cos(d*x^3 + c) - (d*x^3)^(1/3)*(((I*sqrt(3) - 1)*gamma(2/3, I*d*x^3) + (-I*sqrt(3)
- 1)*gamma(2/3, -I*d*x^3))*cos(c) + ((sqrt(3) + I)*gamma(2/3, I*d*x^3) + (sqrt(3) - I)*gamma(2/3, -I*d*x^3))*s
in(c)))*a*b/(d^2*x) + 1/720*(72*d^2*x^6 - 60*d*x^3*sin(2*d*x^3 + 2*c) - 5*2^(1/3)*(d*x^3)^(1/3)*(((sqrt(3) + I
)*gamma(2/3, 2*I*d*x^3) + (sqrt(3) - I)*gamma(2/3, -2*I*d*x^3))*cos(2*c) + ((-I*sqrt(3) + 1)*gamma(2/3, 2*I*d*
x^3) + (I*sqrt(3) + 1)*gamma(2/3, -2*I*d*x^3))*sin(2*c)))*b^2/(d^2*x)

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Fricas [A]
time = 0.14, size = 150, normalized size = 0.60 \begin {gather*} \frac {36 \, {\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{5} - 60 \, b^{2} d x^{2} \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) - 240 \, a b d x^{2} \cos \left (d x^{3} + c\right ) - 5 \, b^{2} \left (2 i \, d\right )^{\frac {1}{3}} e^{\left (-2 i \, c\right )} \Gamma \left (\frac {2}{3}, 2 i \, d x^{3}\right ) + 80 i \, a b \left (i \, d\right )^{\frac {1}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) - 80 i \, a b \left (-i \, d\right )^{\frac {1}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right ) - 5 \, b^{2} \left (-2 i \, d\right )^{\frac {1}{3}} e^{\left (2 i \, c\right )} \Gamma \left (\frac {2}{3}, -2 i \, d x^{3}\right )}{360 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*sin(d*x^3+c))^2,x, algorithm="fricas")

[Out]

1/360*(36*(2*a^2 + b^2)*d^2*x^5 - 60*b^2*d*x^2*cos(d*x^3 + c)*sin(d*x^3 + c) - 240*a*b*d*x^2*cos(d*x^3 + c) -
5*b^2*(2*I*d)^(1/3)*e^(-2*I*c)*gamma(2/3, 2*I*d*x^3) + 80*I*a*b*(I*d)^(1/3)*e^(-I*c)*gamma(2/3, I*d*x^3) - 80*
I*a*b*(-I*d)^(1/3)*e^(I*c)*gamma(2/3, -I*d*x^3) - 5*b^2*(-2*I*d)^(1/3)*e^(2*I*c)*gamma(2/3, -2*I*d*x^3))/d^2

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*sin(d*x**3+c))**2,x)

[Out]

Integral(x**4*(a + b*sin(c + d*x**3))**2, 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^4*(a+b*sin(d*x^3+c))^2,x, algorithm="giac")

[Out]

integrate((b*sin(d*x^3 + c) + a)^2*x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*sin(c + d*x^3))^2,x)

[Out]

int(x^4*(a + b*sin(c + d*x^3))^2, x)

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