3.68 \(\int \frac{a+b \sin (c+d x^3)}{x^6} \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 i b e^{i c} d^2 x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac{3 i b e^{-i c} d^2 x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac{a}{5 x^5}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 b d \cos \left (c+d x^3\right )}{10 x^2} \]

[Out]

-a/(5*x^5) - (3*b*d*Cos[c + d*x^3])/(10*x^2) - (((3*I)/20)*b*d^2*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3
)^(1/3) + (((3*I)/20)*b*d^2*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3)) - (b*Sin[c + d*x^3])/(5*x^5)

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Rubi [A]  time = 0.0711301, antiderivative size = 126, normalized size of antiderivative = 1., 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, 3387, 3388, 3355, 2208} \[ -\frac{3 i b e^{i c} d^2 x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac{3 i b e^{-i c} d^2 x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac{a}{5 x^5}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}-\frac{3 b d \cos \left (c+d x^3\right )}{10 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x^3])/x^6,x]

[Out]

-a/(5*x^5) - (3*b*d*Cos[c + d*x^3])/(10*x^2) - (((3*I)/20)*b*d^2*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3
)^(1/3) + (((3*I)/20)*b*d^2*x*Gamma[1/3, I*d*x^3])/(E^(I*c)*(I*d*x^3)^(1/3)) - (b*Sin[c + d*x^3])/(5*x^5)

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 3387

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 3388

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 3355

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 2208

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]

Rubi steps

\begin{align*} \int \frac{a+b \sin \left (c+d x^3\right )}{x^6} \, dx &=\int \left (\frac{a}{x^6}+\frac{b \sin \left (c+d x^3\right )}{x^6}\right ) \, dx\\ &=-\frac{a}{5 x^5}+b \int \frac{\sin \left (c+d x^3\right )}{x^6} \, dx\\ &=-\frac{a}{5 x^5}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}+\frac{1}{5} (3 b d) \int \frac{\cos \left (c+d x^3\right )}{x^3} \, dx\\ &=-\frac{a}{5 x^5}-\frac{3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}-\frac{1}{10} \left (9 b d^2\right ) \int \sin \left (c+d x^3\right ) \, dx\\ &=-\frac{a}{5 x^5}-\frac{3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}-\frac{1}{20} \left (9 i b d^2\right ) \int e^{-i c-i d x^3} \, dx+\frac{1}{20} \left (9 i b d^2\right ) \int e^{i c+i d x^3} \, dx\\ &=-\frac{a}{5 x^5}-\frac{3 b d \cos \left (c+d x^3\right )}{10 x^2}-\frac{3 i b d^2 e^{i c} x \Gamma \left (\frac{1}{3},-i d x^3\right )}{20 \sqrt [3]{-i d x^3}}+\frac{3 i b d^2 e^{-i c} x \Gamma \left (\frac{1}{3},i d x^3\right )}{20 \sqrt [3]{i d x^3}}-\frac{b \sin \left (c+d x^3\right )}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.44704, size = 146, normalized size = 1.16 \[ \frac{3 b d^2 x^6 \sqrt [3]{i d x^3} (\sin (c)-i \cos (c)) \text{Gamma}\left (\frac{1}{3},-i d x^3\right )+3 b d^2 x^6 \sqrt [3]{-i d x^3} (\sin (c)+i \cos (c)) \text{Gamma}\left (\frac{1}{3},i d x^3\right )-2 \sqrt [3]{d^2 x^6} \left (2 a+2 b \sin \left (c+d x^3\right )+3 b d x^3 \cos \left (c+d x^3\right )\right )}{20 x^5 \sqrt [3]{d^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x^3])/x^6,x]

[Out]

(3*b*d^2*x^6*(I*d*x^3)^(1/3)*Gamma[1/3, (-I)*d*x^3]*((-I)*Cos[c] + Sin[c]) + 3*b*d^2*x^6*((-I)*d*x^3)^(1/3)*Ga
mma[1/3, I*d*x^3]*(I*Cos[c] + Sin[c]) - 2*(d^2*x^6)^(1/3)*(2*a + 3*b*d*x^3*Cos[c + d*x^3] + 2*b*Sin[c + d*x^3]
))/(20*x^5*(d^2*x^6)^(1/3))

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Maple [F]  time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\sin \left ( d{x}^{3}+c \right ) }{{x}^{6}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(d*x^3+c))/x^6,x)

[Out]

int((a+b*sin(d*x^3+c))/x^6,x)

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Maxima [B]  time = 1.16373, size = 369, normalized size = 2.93 \begin{align*} -\frac{\left (x^{3}{\left | d \right |}\right )^{\frac{2}{3}}{\left ({\left ({\left (i \, \Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) - i \, \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) +{\left (i \, \Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) - i \, \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) -{\left (\Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) + \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) + \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) +{\left ({\left (\Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) + \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) + \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) +{\left (i \, \Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) - i \, \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right ) +{\left (-i \, \Gamma \left (-\frac{5}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (-\frac{5}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{5}{6} \, \pi + \frac{5}{3} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} b{\left | d \right |}}{12 \, x^{2}} - \frac{a}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x^3+c))/x^6,x, algorithm="maxima")

[Out]

-1/12*(x^3*abs(d))^(2/3)*(((I*gamma(-5/3, I*d*x^3) - I*gamma(-5/3, -I*d*x^3))*cos(5/6*pi + 5/3*arctan2(0, d))
+ (I*gamma(-5/3, I*d*x^3) - I*gamma(-5/3, -I*d*x^3))*cos(-5/6*pi + 5/3*arctan2(0, d)) - (gamma(-5/3, I*d*x^3)
+ gamma(-5/3, -I*d*x^3))*sin(5/6*pi + 5/3*arctan2(0, d)) + (gamma(-5/3, I*d*x^3) + gamma(-5/3, -I*d*x^3))*sin(
-5/6*pi + 5/3*arctan2(0, d)))*cos(c) + ((gamma(-5/3, I*d*x^3) + gamma(-5/3, -I*d*x^3))*cos(5/6*pi + 5/3*arctan
2(0, d)) + (gamma(-5/3, I*d*x^3) + gamma(-5/3, -I*d*x^3))*cos(-5/6*pi + 5/3*arctan2(0, d)) + (I*gamma(-5/3, I*
d*x^3) - I*gamma(-5/3, -I*d*x^3))*sin(5/6*pi + 5/3*arctan2(0, d)) + (-I*gamma(-5/3, I*d*x^3) + I*gamma(-5/3, -
I*d*x^3))*sin(-5/6*pi + 5/3*arctan2(0, d)))*sin(c))*b*abs(d)/x^2 - 1/5*a/x^5

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Fricas [A]  time = 1.71829, size = 231, normalized size = 1.83 \begin{align*} \frac{3 \, b \left (i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (-i \, c\right )} \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + 3 \, b \left (-i \, d\right )^{\frac{2}{3}} d x^{5} e^{\left (i \, c\right )} \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right ) - 6 \, b d x^{3} \cos \left (d x^{3} + c\right ) - 4 \, b \sin \left (d x^{3} + c\right ) - 4 \, a}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x^3+c))/x^6,x, algorithm="fricas")

[Out]

1/20*(3*b*(I*d)^(2/3)*d*x^5*e^(-I*c)*gamma(1/3, I*d*x^3) + 3*b*(-I*d)^(2/3)*d*x^5*e^(I*c)*gamma(1/3, -I*d*x^3)
- 6*b*d*x^3*cos(d*x^3 + c) - 4*b*sin(d*x^3 + c) - 4*a)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \sin{\left (c + d x^{3} \right )}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x**3+c))/x**6,x)

[Out]

Integral((a + b*sin(c + d*x**3))/x**6, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \sin \left (d x^{3} + c\right ) + a}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(d*x^3+c))/x^6,x, algorithm="giac")

[Out]

integrate((b*sin(d*x^3 + c) + a)/x^6, x)