3.1.66 \(\int (a+b \sin (c+d x^3)) \, dx\) [66]

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

[Out]

a*x+1/6*I*b*exp(I*c)*x*GAMMA(1/3,-I*d*x^3)/(-I*d*x^3)^(1/3)-1/6*I*b*x*GAMMA(1/3,I*d*x^3)/exp(I*c)/(I*d*x^3)^(1
/3)

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Rubi [A]
time = 0.02, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {3436, 2239} \begin {gather*} \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 \end {gather*}

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 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 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]

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*}

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Mathematica [A]
time = 0.07, size = 138, normalized size = 1.68 \begin {gather*} a x-\frac {1}{2} i b \cos (c) \left (-\frac {x \Gamma \left (\frac {1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}+\frac {x \Gamma \left (\frac {1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}\right )+\frac {1}{2} b \left (-\frac {x \Gamma \left (\frac {1}{3},-i d x^3\right )}{3 \sqrt [3]{-i d x^3}}-\frac {x \Gamma \left (\frac {1}{3},i d x^3\right )}{3 \sqrt [3]{i d x^3}}\right ) \sin (c) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sin[c + d*x^3],x]

[Out]

a*x - (I/2)*b*Cos[c]*(-1/3*(x*Gamma[1/3, (-I)*d*x^3])/((-I)*d*x^3)^(1/3) + (x*Gamma[1/3, I*d*x^3])/(3*(I*d*x^3
)^(1/3))) + (b*(-1/3*(x*Gamma[1/3, (-I)*d*x^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 = 0.01, size = 0, normalized size = 0.00 \[\int a +b \sin \left (d \,x^{3}+c \right )\, dx\]

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 [A]
time = 0.31, size = 85, normalized size = 1.04 \begin {gather*} \frac {{\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right )\right )} \cos \left (c\right ) - {\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right )\right )} \sin \left (c\right )\right )} b x}{12 \, \left (d x^{3}\right )^{\frac {1}{3}}} + a x \end {gather*}

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*sqrt(3) - 1)*gamma(1/3, I*d*x^3) + (I*sqrt(3) - 1)*gamma(1/3, -I*d*x^3))*cos(c) - ((sqrt(3) - I)*ga
mma(1/3, I*d*x^3) + (sqrt(3) + I)*gamma(1/3, -I*d*x^3))*sin(c))*b*x/(d*x^3)^(1/3) + a*x

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Fricas [A]
time = 0.10, size = 49, normalized size = 0.60 \begin {gather*} -\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 {gather*}

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

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 = 0.00, 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, algorithm="giac")

[Out]

integrate(b*sin(d*x^3 + c) + a, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*sin(c + d*x^3),x)

[Out]

int(a + b*sin(c + d*x^3), x)

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