3.1.64 \(\int \frac {a+b \sin (c+d x^3)}{x^5} \, dx\) [64]

Optimal. Leaf size=130 \[ -\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {3 i b d^2 e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b d^2 e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {b \sin \left (c+d x^3\right )}{4 x^4} \]

[Out]

-1/4*a/x^4-3/4*b*d*cos(d*x^3+c)/x-3/8*I*b*d^2*exp(I*c)*x^2*GAMMA(2/3,-I*d*x^3)/(-I*d*x^3)^(2/3)+3/8*I*b*d^2*x^
2*GAMMA(2/3,I*d*x^3)/exp(I*c)/(I*d*x^3)^(2/3)-1/4*b*sin(d*x^3+c)/x^4

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {14, 3468, 3469, 3470, 2250} \begin {gather*} -\frac {3 i b e^{i c} d^2 x^2 \text {Gamma}\left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b e^{-i c} d^2 x^2 \text {Gamma}\left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x^3])/x^5,x]

[Out]

-1/4*a/x^4 - (3*b*d*Cos[c + d*x^3])/(4*x) - (((3*I)/8)*b*d^2*E^(I*c)*x^2*Gamma[2/3, (-I)*d*x^3])/((-I)*d*x^3)^
(2/3) + (((3*I)/8)*b*d^2*x^2*Gamma[2/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(2/3)) - (b*Sin[c + d*x^3])/(4*x^4)

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 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 3468

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

Rule 3469

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

Rubi steps

\begin {align*} \int \frac {a+b \sin \left (c+d x^3\right )}{x^5} \, dx &=\int \left (\frac {a}{x^5}+\frac {b \sin \left (c+d x^3\right )}{x^5}\right ) \, dx\\ &=-\frac {a}{4 x^4}+b \int \frac {\sin \left (c+d x^3\right )}{x^5} \, dx\\ &=-\frac {a}{4 x^4}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}+\frac {1}{4} (3 b d) \int \frac {\cos \left (c+d x^3\right )}{x^2} \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}-\frac {1}{4} \left (9 b d^2\right ) \int x \sin \left (c+d x^3\right ) \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}-\frac {1}{8} \left (9 i b d^2\right ) \int e^{-i c-i d x^3} x \, dx+\frac {1}{8} \left (9 i b d^2\right ) \int e^{i c+i d x^3} x \, dx\\ &=-\frac {a}{4 x^4}-\frac {3 b d \cos \left (c+d x^3\right )}{4 x}-\frac {3 i b d^2 e^{i c} x^2 \Gamma \left (\frac {2}{3},-i d x^3\right )}{8 \left (-i d x^3\right )^{2/3}}+\frac {3 i b d^2 e^{-i c} x^2 \Gamma \left (\frac {2}{3},i d x^3\right )}{8 \left (i d x^3\right )^{2/3}}-\frac {b \sin \left (c+d x^3\right )}{4 x^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.25, size = 143, normalized size = 1.10 \begin {gather*} \frac {3 b d^2 x^6 \left (i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},-i d x^3\right ) (-i \cos (c)+\sin (c))+3 b d^2 x^6 \left (-i d x^3\right )^{2/3} \Gamma \left (\frac {2}{3},i d x^3\right ) (i \cos (c)+\sin (c))-2 \left (d^2 x^6\right )^{2/3} \left (a+3 b d x^3 \cos \left (c+d x^3\right )+b \sin \left (c+d x^3\right )\right )}{8 x^4 \left (d^2 x^6\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x^3])/x^5,x]

[Out]

(3*b*d^2*x^6*(I*d*x^3)^(2/3)*Gamma[2/3, (-I)*d*x^3]*((-I)*Cos[c] + Sin[c]) + 3*b*d^2*x^6*((-I)*d*x^3)^(2/3)*Ga
mma[2/3, I*d*x^3]*(I*Cos[c] + Sin[c]) - 2*(d^2*x^6)^(2/3)*(a + 3*b*d*x^3*Cos[c + d*x^3] + b*Sin[c + d*x^3]))/(
8*x^4*(d^2*x^6)^(2/3))

________________________________________________________________________________________

Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \sin \left (d \,x^{3}+c \right )}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x^3+c))/x^5,x)

[Out]

int((a+b*sin(d*x^3+c))/x^5,x)

________________________________________________________________________________________

Maxima [A]
time = 0.32, size = 91, normalized size = 0.70 \begin {gather*} \frac {\left (d x^{3}\right )^{\frac {1}{3}} {\left ({\left ({\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {4}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {4}{3}, -i \, d x^{3}\right )\right )} \cos \left (c\right ) - {\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, -i \, d x^{3}\right )\right )} \sin \left (c\right )\right )} b d}{12 \, x} - \frac {a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x^3+c))/x^5,x, algorithm="maxima")

[Out]

1/12*(d*x^3)^(1/3)*(((sqrt(3) + I)*gamma(-4/3, I*d*x^3) + (sqrt(3) - I)*gamma(-4/3, -I*d*x^3))*cos(c) - ((I*sq
rt(3) - 1)*gamma(-4/3, I*d*x^3) + (-I*sqrt(3) - 1)*gamma(-4/3, -I*d*x^3))*sin(c))*b*d/x - 1/4*a/x^4

________________________________________________________________________________________

Fricas [A]
time = 0.13, size = 83, normalized size = 0.64 \begin {gather*} \frac {3 \, b \left (i \, d\right )^{\frac {1}{3}} d x^{4} e^{\left (-i \, c\right )} \Gamma \left (\frac {2}{3}, i \, d x^{3}\right ) + 3 \, b \left (-i \, d\right )^{\frac {1}{3}} d x^{4} e^{\left (i \, c\right )} \Gamma \left (\frac {2}{3}, -i \, d x^{3}\right ) - 6 \, b d x^{3} \cos \left (d x^{3} + c\right ) - 2 \, b \sin \left (d x^{3} + c\right ) - 2 \, a}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x^3+c))/x^5,x, algorithm="fricas")

[Out]

1/8*(3*b*(I*d)^(1/3)*d*x^4*e^(-I*c)*gamma(2/3, I*d*x^3) + 3*b*(-I*d)^(1/3)*d*x^4*e^(I*c)*gamma(2/3, -I*d*x^3)
- 6*b*d*x^3*cos(d*x^3 + c) - 2*b*sin(d*x^3 + c) - 2*a)/x^4

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \sin {\left (c + d x^{3} \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x**3+c))/x**5,x)

[Out]

Integral((a + b*sin(c + d*x**3))/x**5, x)

________________________________________________________________________________________

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((a+b*sin(d*x^3+c))/x^5,x, algorithm="giac")

[Out]

integrate((b*sin(d*x^3 + c) + a)/x^5, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\sin \left (d\,x^3+c\right )}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x^3))/x^5,x)

[Out]

int((a + b*sin(c + d*x^3))/x^5, x)

________________________________________________________________________________________