3.4.41 \(\int \frac {(c \sin ^3(a+b x))^{2/3}}{x^3} \, dx\) [341]

Optimal. Leaf size=119 \[ -\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x}+b^2 \cos (2 a) \text {Ci}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \csc ^2(a+b x) \sin (2 a) \left (c \sin ^3(a+b x)\right )^{2/3} \text {Si}(2 b x) \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6852, 3395, 29, 3393, 3384, 3380, 3383} \begin {gather*} b^2 \cos (2 a) \text {CosIntegral}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-b^2 \sin (2 a) \text {Si}(2 b x) \csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}-\frac {\left (c \sin ^3(a+b x)\right )^{2/3}}{2 x^2}-\frac {b \cot (a+b x) \left (c \sin ^3(a+b x)\right )^{2/3}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x]^3)^(2/3)/x^3,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3395

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*Si
n[e + f*x])^n/(d*(m + 1))), x] + (Dist[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[f^2*(n^2/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1)*(m + 2))), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

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

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Mathematica [A]
time = 0.14, size = 85, normalized size = 0.71 \begin {gather*} \frac {\csc ^2(a+b x) \left (c \sin ^3(a+b x)\right )^{2/3} \left (-1+\cos (2 (a+b x))+4 b^2 x^2 \cos (2 a) \text {Ci}(2 b x)-2 b x \sin (2 (a+b x))-4 b^2 x^2 \sin (2 a) \text {Si}(2 b x)\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x]^2*(c*Sin[a + b*x]^3)^(2/3)*(-1 + Cos[2*(a + b*x)] + 4*b^2*x^2*Cos[2*a]*CosIntegral[2*b*x] - 2*b*
x*Sin[2*(a + b*x)] - 4*b^2*x^2*Sin[2*a]*SinIntegral[2*b*x]))/(4*x^2)

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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 238, normalized size = 2.00

method	result	size
risch	\(-\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} b^{2} \left (\frac {1}{2 x^{2} b^{2}}-\frac {i}{b x}-2 \,{\mathrm e}^{2 i b x} \expIntegral \left (1, 2 i b x \right )\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}-\frac {b^{2} \left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} \left (\frac {{\mathrm e}^{4 i \left (b x +a \right )}}{2 x^{2} b^{2}}+\frac {i {\mathrm e}^{4 i \left (b x +a \right )}}{x b}-2 \expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{2 i \left (b x +2 a \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} {\mathrm e}^{-3 i \left (b x +a \right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b x +a \right )}}{4 x^{2} \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{2}}\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(2/3)/(exp(2*I*(b*x+a))-1)^2*b^2*(1/2/x^2/b^2-I/b/x-2*exp(
2*I*b*x)*Ei(1,2*I*b*x))-1/4*b^2/(exp(2*I*(b*x+a))-1)^2*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(2/3)*(1
/2/x^2/b^2*exp(4*I*(b*x+a))+I/x/b*exp(4*I*(b*x+a))-2*Ei(1,-2*I*b*x)*exp(2*I*(b*x+2*a)))+1/4/x^2/(exp(2*I*(b*x+
a))-1)^2*(I*c*(exp(2*I*(b*x+a))-1)^3*exp(-3*I*(b*x+a)))^(2/3)*exp(2*I*(b*x+a))

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Maxima [C] Result contains complex when optimal does not.
time = 0.63, size = 296, normalized size = 2.49 \begin {gather*} -\frac {{\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} - {\left ({\left (\sqrt {3} + i\right )} E_{3}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + {\left ({\left ({\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 2\right )} \sin \left (2 \, a\right )^{2} + {\left ({\left (i \, \sqrt {3} + 1\right )} E_{3}\left (2 i \, b x\right ) + {\left (-i \, \sqrt {3} + 1\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) - 2 \, \cos \left (2 \, a\right )^{2} - {\left ({\left ({\left (\sqrt {3} + i\right )} E_{3}\left (2 i \, b x\right ) + {\left (\sqrt {3} - i\right )} E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - {\left (\sqrt {3} - i\right )} E_{3}\left (2 i \, b x\right ) - {\left (\sqrt {3} + i\right )} E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} b^{2} c^{\frac {2}{3}}}{16 \, {\left (a^{2} \cos \left (2 \, a\right )^{2} + a^{2} \sin \left (2 \, a\right )^{2} + {\left (b x + a\right )}^{2} {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} - 2 \, {\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, 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)^(2/3)/x^3,x, algorithm="maxima")

[Out]

-1/16*(((-I*sqrt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (I*sqrt(3) + 1)*exp_integral_e(3, -2*I*b*x))*cos(2*a)^3
- ((sqrt(3) + I)*exp_integral_e(3, 2*I*b*x) + (sqrt(3) - I)*exp_integral_e(3, -2*I*b*x))*sin(2*a)^3 + (((-I*sq
rt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (I*sqrt(3) + 1)*exp_integral_e(3, -2*I*b*x))*cos(2*a) - 2)*sin(2*a)^2
+ ((I*sqrt(3) + 1)*exp_integral_e(3, 2*I*b*x) + (-I*sqrt(3) + 1)*exp_integral_e(3, -2*I*b*x))*cos(2*a) - 2*cos
(2*a)^2 - (((sqrt(3) + I)*exp_integral_e(3, 2*I*b*x) + (sqrt(3) - I)*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 -
(sqrt(3) - I)*exp_integral_e(3, 2*I*b*x) - (sqrt(3) + I)*exp_integral_e(3, -2*I*b*x))*sin(2*a))*b^2*c^(2/3)/(
a^2*cos(2*a)^2 + a^2*sin(2*a)^2 + (b*x + a)^2*(cos(2*a)^2 + sin(2*a)^2) - 2*(a*cos(2*a)^2 + a*sin(2*a)^2)*(b*x
+ a))

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Fricas [A]
time = 0.39, size = 142, normalized size = 1.19 \begin {gather*} \frac {4^{\frac {2}{3}} {\left (2 \cdot 4^{\frac {1}{3}} b^{2} x^{2} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right ) + 2 \cdot 4^{\frac {1}{3}} b x \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 4^{\frac {1}{3}} \cos \left (b x + a\right )^{2} - {\left (4^{\frac {1}{3}} b^{2} x^{2} \operatorname {Ci}\left (2 \, b x\right ) + 4^{\frac {1}{3}} b^{2} x^{2} \operatorname {Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right ) + 4^{\frac {1}{3}}\right )} \left (-{\left (c \cos \left (b x + a\right )^{2} - c\right )} \sin \left (b x + a\right )\right )^{\frac {2}{3}}}{8 \, {\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)^(2/3)/x^3,x, algorithm="fricas")

[Out]

1/8*4^(2/3)*(2*4^(1/3)*b^2*x^2*sin(2*a)*sin_integral(2*b*x) + 2*4^(1/3)*b*x*cos(b*x + a)*sin(b*x + a) - 4^(1/3
)*cos(b*x + a)^2 - (4^(1/3)*b^2*x^2*cos_integral(2*b*x) + 4^(1/3)*b^2*x^2*cos_integral(-2*b*x))*cos(2*a) + 4^(
1/3))*(-(c*cos(b*x + a)^2 - c)*sin(b*x + a))^(2/3)/(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 {\left (c \sin ^{3}{\left (a + b x \right )}\right )^{\frac {2}{3}}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*sin(b*x + a)^3)^(2/3)/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 )}^{2/3}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x)^3)^(2/3)/x^3,x)

[Out]

int((c*sin(a + b*x)^3)^(2/3)/x^3, x)

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