3.35 \(\int \sqrt [3]{c \sin (a+b x)} \, dx\)

Optimal. Leaf size=517 \[ \frac{3 \left (1-i \sqrt{3}\right ) \sqrt{3-i \sqrt{3}} \sqrt [3]{c} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt{3}\right ) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt{3}\right ) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3-i \sqrt{3}}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{2 \sqrt{2} b}-\frac{3 \sqrt{\frac{3}{2} \left (3-i \sqrt{3}\right )} \sqrt [3]{c} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3-i \sqrt{3}\right ) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left (3+i \sqrt{3}\right ) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} E\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3+i \sqrt{3}}}\right )|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right )}{b} \]

[Out]

(-3*Sqrt[(3*(3 - I*Sqrt[3]))/2]*c^(1/3)*EllipticE[ArcSin[(Sqrt[2]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)])/Sq
rt[3 + I*Sqrt[3]]], (3*I - Sqrt[3])/(3*I + Sqrt[3])]*Sec[a + b*x]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)]*Sqr
t[(I + Sqrt[3])/(3*I + Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 - I*Sqrt[3])*c^(2/3))]*Sqrt[(I - Sqrt[3])/(3*
I - Sqrt[3]) + (2*(c*Sin[a + b*x])^(2/3))/((3 + I*Sqrt[3])*c^(2/3))])/b + (3*(1 - I*Sqrt[3])*Sqrt[3 - I*Sqrt[3
]]*c^(1/3)*EllipticF[ArcSin[(Sqrt[2]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)])/Sqrt[3 - I*Sqrt[3]]], (3*I + Sq
rt[3])/(3*I - Sqrt[3])]*Sec[a + b*x]*Sqrt[1 - (c*Sin[a + b*x])^(2/3)/c^(2/3)]*Sqrt[(I + Sqrt[3])/(3*I + Sqrt[3
]) + (2*(c*Sin[a + b*x])^(2/3))/((3 - I*Sqrt[3])*c^(2/3))]*Sqrt[(I - Sqrt[3])/(3*I - Sqrt[3]) + (2*(c*Sin[a +
b*x])^(2/3))/((3 + I*Sqrt[3])*c^(2/3))])/(2*Sqrt[2]*b)

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Rubi [C]  time = 0.0135465, antiderivative size = 58, normalized size of antiderivative = 0.11, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2643} \[ \frac{3 \cos (a+b x) (c \sin (a+b x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right )}{4 b c \sqrt{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x])^(1/3),x]

[Out]

(3*Cos[a + b*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(4/3))/(4*b*c*Sqrt[Cos[a + b
*x]^2])

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \sqrt [3]{c \sin (a+b x)} \, dx &=\frac{3 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{4/3}}{4 b c \sqrt{\cos ^2(a+b x)}}\\ \end{align*}

Mathematica [C]  time = 0.0333308, size = 55, normalized size = 0.11 \[ \frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \sqrt [3]{c \sin (a+b x)} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right )}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x])^(1/3),x]

[Out]

(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1/3)*Tan[a + b*x])/
(4*b)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{c\sin \left ( bx+a \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a))^(1/3),x)

[Out]

int((c*sin(b*x+a))^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/3),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (c \sin \left (b x + a\right )\right )^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/3),x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))**(1/3),x)

[Out]

Integral((c*sin(a + b*x))**(1/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )\right )^{\frac{1}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/3),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(1/3), x)