3.4.16 \(\int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx\) [316]

Optimal. Leaf size=77 \[ -\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x}+b \cos (a) \text {Ci}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x) \]

[Out]

-(c*sin(b*x+a)^3)^(1/3)/x+b*Ci(b*x)*cos(a)*csc(b*x+a)*(c*sin(b*x+a)^3)^(1/3)-b*csc(b*x+a)*Si(b*x)*sin(a)*(c*si
n(b*x+a)^3)^(1/3)

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Rubi [A]
time = 0.13, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6852, 3378, 3384, 3380, 3383} \begin {gather*} b \cos (a) \text {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \sin (a) \text {Si}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3)/x^2,x]

[Out]

-((c*Sin[a + b*x]^3)^(1/3)/x) + b*Cos[a]*CosIntegral[b*x]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3) - b*Csc[a + b*
x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3)*SinIntegral[b*x]

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 6852

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m)^FracPart[p]/v^(m*FracPart[p])), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^2} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (a+b x)}{x^2} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x}+\left (b \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (a+b x)}{x} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x}+\left (b \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (b x)}{x} \, dx-\left (b \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (b x)}{x} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{x}+b \cos (a) \text {Ci}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-b \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x)\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 51, normalized size = 0.66 \begin {gather*} \frac {\sqrt [3]{c \sin ^3(a+b x)} (-1+b x \cos (a) \text {Ci}(b x) \csc (a+b x)-b x \csc (a+b x) \sin (a) \text {Si}(b x))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Sin[a + b*x]^3)^(1/3)/x^2,x]

[Out]

((c*Sin[a + b*x]^3)^(1/3)*(-1 + b*x*Cos[a]*CosIntegral[b*x]*Csc[a + b*x] - b*x*Csc[a + b*x]*Sin[a]*SinIntegral
[b*x]))/x

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Maple [C] Result contains complex when optimal does not.
time = 0.13, size = 155, normalized size = 2.01

method	result	size
risch	\(\frac {i b \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} \left (\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{b x}-\expIntegral \left (1, -i b x \right ) {\mathrm e}^{i \left (b x +2 a \right )}\right )}{2 \,{\mathrm e}^{2 i \left (b x +a \right )}-2}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} b \left (\frac {i}{b x}+{\mathrm e}^{i b x} \expIntegral \left (1, i b x \right )\right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a)^3)^(1/3)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/2*I*b/(exp(2*I*(b*x+a))-1)*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)*(I/b/x*exp(2*I*(b*x+a))-Ei(1
,-I*b*x)*exp(I*(b*x+2*a)))-1/2*I*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(1/3)/(exp(2*I*(b*x+a))-1)*b*(
I/b/x+exp(I*b*x)*Ei(1,I*b*x))

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Maxima [C] Result contains complex when optimal does not.
time = 0.69, size = 229, normalized size = 2.97 \begin {gather*} \frac {{\left ({\left ({\left (\sqrt {3} - i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} + {\left ({\left (\sqrt {3} - i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + {\left ({\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \sin \left (a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{2}\left (i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right ) + {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \cos \left (a\right )^{2} + {\left (i \, \sqrt {3} - 1\right )} E_{2}\left (i \, b x\right ) + {\left (-i \, \sqrt {3} - 1\right )} E_{2}\left (-i \, b x\right )\right )} \sin \left (a\right )\right )} b c^{\frac {1}{3}}}{8 \, {\left (a \cos \left (a\right )^{2} + a \sin \left (a\right )^{2} - {\left (b x + a\right )} {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="maxima")

[Out]

1/8*(((sqrt(3) - I)*exp_integral_e(2, I*b*x) + (sqrt(3) + I)*exp_integral_e(2, -I*b*x))*cos(a)^3 + ((sqrt(3) -
I)*exp_integral_e(2, I*b*x) + (sqrt(3) + I)*exp_integral_e(2, -I*b*x))*cos(a)*sin(a)^2 + ((-I*sqrt(3) - 1)*ex
p_integral_e(2, I*b*x) + (I*sqrt(3) - 1)*exp_integral_e(2, -I*b*x))*sin(a)^3 - ((sqrt(3) + I)*exp_integral_e(2
, I*b*x) + (sqrt(3) - I)*exp_integral_e(2, -I*b*x))*cos(a) + (((-I*sqrt(3) - 1)*exp_integral_e(2, I*b*x) + (I*
sqrt(3) - 1)*exp_integral_e(2, -I*b*x))*cos(a)^2 + (I*sqrt(3) - 1)*exp_integral_e(2, I*b*x) + (-I*sqrt(3) - 1)
*exp_integral_e(2, -I*b*x))*sin(a))*b*c^(1/3)/(a*cos(a)^2 + a*sin(a)^2 - (b*x + a)*(cos(a)^2 + sin(a)^2))

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Fricas [A]
time = 0.36, size = 112, normalized size = 1.45 \begin {gather*} -\frac {4^{\frac {1}{3}} {\left (2 \cdot 4^{\frac {2}{3}} \cos \left (b x + a\right )^{2} - {\left (2 \cdot 4^{\frac {2}{3}} b x \sin \left (a\right ) \operatorname {Si}\left (b x\right ) - {\left (4^{\frac {2}{3}} b x \operatorname {Ci}\left (b x\right ) + 4^{\frac {2}{3}} b x \operatorname {Ci}\left (-b x\right )\right )} \cos \left (a\right )\right )} \sin \left (b x + a\right ) - 2 \cdot 4^{\frac {2}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {1}{3}}}{8 \, {\left (x \cos \left (b x + a\right )^{2} - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="fricas")

[Out]

-1/8*4^(1/3)*(2*4^(2/3)*cos(b*x + a)^2 - (2*4^(2/3)*b*x*sin(a)*sin_integral(b*x) - (4^(2/3)*b*x*cos_integral(b
*x) + 4^(2/3)*b*x*cos_integral(-b*x))*cos(a))*sin(b*x + a) - 2*4^(2/3))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))
^(1/3)/(x*cos(b*x + a)^2 - x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{c \sin ^{3}{\left (a + b x \right )}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a)**3)**(1/3)/x**2,x)

[Out]

Integral((c*sin(a + b*x)**3)**(1/3)/x**2, 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((c*sin(b*x+a)^3)^(1/3)/x^2,x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a)^3)^(1/3)/x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,{\sin \left (a+b\,x\right )}^3\right )}^{1/3}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x)^3)^(1/3)/x^2,x)

[Out]

int((c*sin(a + b*x)^3)^(1/3)/x^2, x)

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