3.4.17 \(\int \frac {\sqrt [3]{c \sin ^3(a+b x)}}{x^3} \, dx\) [317]

Optimal. Leaf size=116 \[ -\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x}-\frac {1}{2} b^2 \text {Ci}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}-\frac {1}{2} b^2 \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x) \]

[Out]

-1/2*(c*sin(b*x+a)^3)^(1/3)/x^2-1/2*b*cot(b*x+a)*(c*sin(b*x+a)^3)^(1/3)/x-1/2*b^2*cos(a)*csc(b*x+a)*Si(b*x)*(c
*sin(b*x+a)^3)^(1/3)-1/2*b^2*Ci(b*x)*csc(b*x+a)*sin(a)*(c*sin(b*x+a)^3)^(1/3)

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Rubi [A]
time = 0.14, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, 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*} -\frac {1}{2} b^2 \sin (a) \text {CosIntegral}(b x) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}-\frac {1}{2} b^2 \cos (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)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(1/3)/x^3,x]

[Out]

-1/2*(c*Sin[a + b*x]^3)^(1/3)/x^2 - (b*Cot[a + b*x]*(c*Sin[a + b*x]^3)^(1/3))/(2*x) - (b^2*CosIntegral[b*x]*Cs
c[a + b*x]*Sin[a]*(c*Sin[a + b*x]^3)^(1/3))/2 - (b^2*Cos[a]*Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*SinIntegral[
b*x])/2

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^3} \, dx &=\left (\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (a+b x)}{x^3} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}+\frac {1}{2} \left (b \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (a+b x)}{x^2} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x}-\frac {1}{2} \left (b^2 \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (a+b x)}{x} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x}-\frac {1}{2} \left (b^2 \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\sin (b x)}{x} \, dx-\frac {1}{2} \left (b^2 \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}\right ) \int \frac {\cos (b x)}{x} \, dx\\ &=-\frac {\sqrt [3]{c \sin ^3(a+b x)}}{2 x^2}-\frac {b \cot (a+b x) \sqrt [3]{c \sin ^3(a+b x)}}{2 x}-\frac {1}{2} b^2 \text {Ci}(b x) \csc (a+b x) \sin (a) \sqrt [3]{c \sin ^3(a+b x)}-\frac {1}{2} b^2 \cos (a) \csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \text {Si}(b x)\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 69, normalized size = 0.59 \begin {gather*} -\frac {\csc (a+b x) \sqrt [3]{c \sin ^3(a+b x)} \left (b x \cos (a+b x)+b^2 x^2 \text {Ci}(b x) \sin (a)+\sin (a+b x)+b^2 x^2 \cos (a) \text {Si}(b x)\right )}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Csc[a + b*x]*(c*Sin[a + b*x]^3)^(1/3)*(b*x*Cos[a + b*x] + b^2*x^2*CosIntegral[b*x]*Sin[a] + Sin[a + b*x]
+ b^2*x^2*Cos[a]*SinIntegral[b*x]))/x^2

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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 183, normalized size = 1.58

method	result	size
risch	\(-\frac {b^{2} \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} \left (\frac {{\mathrm e}^{2 i \left (b x +a \right )}}{2 x^{2} b^{2}}+\frac {i {\mathrm e}^{2 i \left (b x +a \right )}}{2 b x}-\frac {\expIntegral \left (1, -i b x \right ) {\mathrm e}^{i \left (b x +2 a \right )}}{2}\right )}{2 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {1}{3}} b^{2} \left (\frac {1}{2 x^{2} b^{2}}-\frac {i}{2 b x}-\frac {{\mathrm e}^{i b x} \expIntegral \left (1, i b x \right )}{2}\right )}{2 \,{\mathrm e}^{2 i \left (b x +a \right )}-2}\)	\(183\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.58, size = 256, normalized size = 2.21 \begin {gather*} -\frac {{\left ({\left ({\left (\sqrt {3} - i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right )^{3} + {\left ({\left (\sqrt {3} - i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} + i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) \sin \left (a\right )^{2} + {\left ({\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \sin \left (a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{3}\left (i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right ) + {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \cos \left (a\right )^{2} + {\left (i \, \sqrt {3} - 1\right )} E_{3}\left (i \, b x\right ) + {\left (-i \, \sqrt {3} - 1\right )} E_{3}\left (-i \, b x\right )\right )} \sin \left (a\right )\right )} b^{2} c^{\frac {1}{3}}}{8 \, {\left (a^{2} \cos \left (a\right )^{2} + a^{2} \sin \left (a\right )^{2} + {\left (b x + a\right )}^{2} {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} - 2 \, {\left (a \cos \left (a\right )^{2} + a \sin \left (a\right )^{2}\right )} {\left (b x + a\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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 140, normalized size = 1.21 \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^{2} x^{2} \cos \left (a\right ) \operatorname {Si}\left (b x\right ) + 2 \cdot 4^{\frac {2}{3}} b x \cos \left (b x + a\right ) + {\left (4^{\frac {2}{3}} b^{2} x^{2} \operatorname {Ci}\left (b x\right ) + 4^{\frac {2}{3}} b^{2} x^{2} \operatorname {Ci}\left (-b x\right )\right )} \sin \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}}}{16 \, {\left (x^{2} \cos \left (b x + a\right )^{2} - x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*4^(1/3)*(2*4^(2/3)*cos(b*x + a)^2 - (2*4^(2/3)*b^2*x^2*cos(a)*sin_integral(b*x) + 2*4^(2/3)*b*x*cos(b*x
+ a) + (4^(2/3)*b^2*x^2*cos_integral(b*x) + 4^(2/3)*b^2*x^2*cos_integral(-b*x))*sin(a))*sin(b*x + a) - 2*4^(2/
3))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(1/3)/(x^2*cos(b*x + a)^2 - x^2)

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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^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

int((c*sin(a + b*x)^3)^(1/3)/x^3, x)

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