3.66 \(\int (a+b \sin (c+d x^3)) \, dx\)

Optimal. Leaf size=82 \[ \frac{i b e^{i c} x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{6 \sqrt [3]{-i d x^3}}-\frac{i b e^{-i c} x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{6 \sqrt [3]{i d x^3}}+a x \]

[Out]

a*x + ((I/6)*b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) - ((I/6)*b*x*Gamma[1/3, I*d*x^3])/(E^(I*c)
*(I*d*x^3)^(1/3))

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Rubi [A]  time = 0.0294216, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3355, 2208} \[ \frac{i b e^{i c} x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{6 \sqrt [3]{-i d x^3}}-\frac{i b e^{-i c} x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{6 \sqrt [3]{i d x^3}}+a x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sin[c + d*x^3],x]

[Out]

a*x + ((I/6)*b*E^(I*c)*x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) - ((I/6)*b*x*Gamma[1/3, I*d*x^3])/(E^(I*c)
*(I*d*x^3)^(1/3))

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

Mathematica [A]  time = 0.100292, size = 138, normalized size = 1.68 \[ -\frac{1}{2} i b \cos (c) \left (\frac{x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}-\frac{x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}\right )+\frac{1}{2} b \sin (c) \left (-\frac{x \text{Gamma}\left (\frac{1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}-\frac{x \text{Gamma}\left (\frac{1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}\right )+a x \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sin[c + d*x^3],x]

[Out]

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

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int a+b\sin \left ( d{x}^{3}+c \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.12915, size = 363, normalized size = 4.43 \begin{align*} \frac{{\left ({\left ({\left (-i \, \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) +{\left (-i \, \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) -{\left (\Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right )\right )} \cos \left (c\right ) -{\left ({\left (\Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \cos \left (\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) +{\left (\Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \cos \left (-\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) -{\left (i \, \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) - i \, \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \sin \left (\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right ) -{\left (-i \, \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + i \, \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right )\right )} \sin \left (-\frac{1}{6} \, \pi + \frac{1}{3} \, \arctan \left (0, d\right )\right )\right )} \sin \left (c\right )\right )} b x}{12 \, \left (x^{3}{\left | d \right |}\right )^{\frac{1}{3}}} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62296, size = 147, normalized size = 1.79 \begin{align*} -\frac{b \left (i \, d\right )^{\frac{2}{3}} e^{\left (-i \, c\right )} \Gamma \left (\frac{1}{3}, i \, d x^{3}\right ) + b \left (-i \, d\right )^{\frac{2}{3}} e^{\left (i \, c\right )} \Gamma \left (\frac{1}{3}, -i \, d x^{3}\right ) - 6 \, a d x}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*sin(c + d*x**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*sin(d*x^3 + c) + a, x)