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

Optimal. Leaf size=107 \[ \frac{i e^{i a} (c+d x) \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac{i e^{-i a} (c+d x) \text{Gamma}\left (\frac{1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}} \]

[Out]

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

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Rubi [A]  time = 0.028851, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3355, 2208} \[ \frac{i e^{i a} (c+d x) \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac{i e^{-i a} (c+d x) \text{Gamma}\left (\frac{1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^3],x]

[Out]

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

Mathematica [A]  time = 0.0168614, size = 115, normalized size = 1.07 \[ \frac{i (c+d x) \left ((\cos (a)+i \sin (a)) \sqrt [3]{i b (c+d x)^3} \text{Gamma}\left (\frac{1}{3},-i b (c+d x)^3\right )-(\cos (a)-i \sin (a)) \sqrt [3]{-i b (c+d x)^3} \text{Gamma}\left (\frac{1}{3},i b (c+d x)^3\right )\right )}{6 d \sqrt [3]{b^2 (c+d x)^6}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^3],x]

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( a+ \left ( dx+c \right ) ^{3}b \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin((d*x + c)^3*b + a), x)

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Fricas [A]  time = 1.80001, size = 277, normalized size = 2.59 \begin{align*} -\frac{\left (i \, b d^{3}\right )^{\frac{2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac{2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right )}{6 \, b d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*(c + d*x)**3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin((d*x + c)^3*b + a), x)