3.1.35 \(\int \sqrt [3]{c \sin (a+b x)} \, dx\) [35]

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

[Out]

3/4*c^(1/3)*EllipticF(2^(1/2)*(1-(c*sin(b*x+a))^(2/3)/c^(2/3))^(1/2)/(3-I*3^(1/2))^(1/2),((3*I+3^(1/2))/(3*I-3
^(1/2)))^(1/2))*sec(b*x+a)*(1-I*3^(1/2))*(1-(c*sin(b*x+a))^(2/3)/c^(2/3))^(1/2)*((I-3^(1/2))/(3*I-3^(1/2))+2*(
c*sin(b*x+a))^(2/3)/c^(2/3)/(3+I*3^(1/2)))^(1/2)*(3-I*3^(1/2))^(1/2)*(2*(c*sin(b*x+a))^(2/3)/c^(2/3)/(3-I*3^(1
/2))+(3^(1/2)+I)/(3*I+3^(1/2)))^(1/2)/b*2^(1/2)-3/2*c^(1/3)*EllipticE(2^(1/2)*(1-(c*sin(b*x+a))^(2/3)/c^(2/3))
^(1/2)/(3+I*3^(1/2))^(1/2),((3*I-3^(1/2))/(3*I+3^(1/2)))^(1/2))*sec(b*x+a)*(1-(c*sin(b*x+a))^(2/3)/c^(2/3))^(1
/2)*((I-3^(1/2))/(3*I-3^(1/2))+2*(c*sin(b*x+a))^(2/3)/c^(2/3)/(3+I*3^(1/2)))^(1/2)*(18-6*I*3^(1/2))^(1/2)*(2*(
c*sin(b*x+a))^(2/3)/c^(2/3)/(3-I*3^(1/2))+(3^(1/2)+I)/(3*I+3^(1/2)))^(1/2)/b

________________________________________________________________________________________

Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.01, 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 = {2722} \begin {gather*} \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)}} \end {gather*}

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 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[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] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.03, size = 55, normalized size = 0.11 \begin {gather*} \frac {3 \sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sin ^2(a+b x)\right ) \sqrt [3]{c \sin (a+b x)} \tan (a+b x)}{4 b} \end {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (c \sin \left (b x +a \right )\right )^{\frac {1}{3}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

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

[Out]

integral((c*sin(b*x + a))^(1/3), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{c \sin {\left (a + b x \right )}}\, dx \end {gather*}

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)

________________________________________________________________________________________

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

[Out]

integrate((c*sin(b*x + a))^(1/3), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x))^(1/3),x)

[Out]

int((c*sin(a + b*x))^(1/3), x)

________________________________________________________________________________________