3.4.55 \(\int \frac {(c \sin ^3(a+b x^n))^{2/3}}{x} \, dx\) [355]

Optimal. Leaf size=121 \[ -\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \text {Si}\left (2 b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6852, 3506, 3459, 3457, 3456} \begin {gather*} -\frac {\cos (2 a) \text {CosIntegral}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {\sin (2 a) \text {Si}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \log (x) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(2/3)/x,x]

[Out]

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

Rule 3456

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3457

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3459

Int[Cos[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Cos[c], Int[Cos[d*x^n]/x, x], x] - Dist[Sin[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3506

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 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 {\left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{x} \, dx &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \frac {\sin ^2\left (a+b x^n\right )}{x} \, dx\\ &=\left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \left (\frac {1}{2 x}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x}\right ) \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac {1}{2} \left (\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \frac {\cos \left (2 a+2 b x^n\right )}{x} \, dx\\ &=\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}-\frac {1}{2} \left (\cos (2 a) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \frac {\cos \left (2 b x^n\right )}{x} \, dx+\frac {1}{2} \left (\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}\right ) \int \frac {\sin \left (2 b x^n\right )}{x} \, dx\\ &=-\frac {\cos (2 a) \text {Ci}\left (2 b x^n\right ) \csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 n}+\frac {1}{2} \csc ^2\left (a+b x^n\right ) \log (x) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}+\frac {\csc ^2\left (a+b x^n\right ) \sin (2 a) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \text {Si}\left (2 b x^n\right )}{2 n}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 63, normalized size = 0.52 \begin {gather*} \frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3} \left (-\cos (2 a) \text {Ci}\left (2 b x^n\right )+n \log (x)+\sin (2 a) \text {Si}\left (2 b x^n\right )\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3)*(-(Cos[2*a]*CosIntegral[2*b*x^n]) + n*Log[x] + Sin[2*a]*SinIntegr
al[2*b*x^n]))/(2*n)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.30, size = 343, normalized size = 2.83

method	result	size
risch	\(\frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{4 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \sinIntegral \left (2 b \,x^{n}\right )}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {\left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i b \,x^{n}} \expIntegral \left (1, -2 i b \,x^{n}\right )}{4 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2} n}-\frac {\expIntegral \left (1, -2 i b \,x^{n}\right ) \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (b \,x^{n}+2 a \right )}}{4 n \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2}}-\frac {\ln \left (x \right ) \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {2}{3}} {\mathrm e}^{2 i \left (a +b \,x^{n}\right )}}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{2}}\)	\(343\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I*(I*c*(exp(2*I*(a+b*x^n))-1)^3*exp(-3*I*(a+b*x^n)))^(2/3)/(exp(2*I*(a+b*x^n))-1)^2*exp(2*I*b*x^n)/n*Pi*cs
gn(b*x^n)-1/2*I*(I*c*(exp(2*I*(a+b*x^n))-1)^3*exp(-3*I*(a+b*x^n)))^(2/3)/(exp(2*I*(a+b*x^n))-1)^2*exp(2*I*b*x^
n)/n*Si(2*b*x^n)-1/4*(I*c*(exp(2*I*(a+b*x^n))-1)^3*exp(-3*I*(a+b*x^n)))^(2/3)/(exp(2*I*(a+b*x^n))-1)^2*exp(2*I
*b*x^n)/n*Ei(1,-2*I*b*x^n)-1/4*Ei(1,-2*I*b*x^n)/n/(exp(2*I*(a+b*x^n))-1)^2*(I*c*(exp(2*I*(a+b*x^n))-1)^3*exp(-
3*I*(a+b*x^n)))^(2/3)*exp(2*I*(b*x^n+2*a))-1/2*ln(x)/(exp(2*I*(a+b*x^n))-1)^2*(I*c*(exp(2*I*(a+b*x^n))-1)^3*ex
p(-3*I*(a+b*x^n)))^(2/3)*exp(2*I*(a+b*x^n))

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.69, size = 153, normalized size = 1.26 \begin {gather*} \frac {{\left ({\left ({\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (2 \, a\right ) - 4 \, n \log \left (x\right ) - {\left ({\left (\sqrt {3} - i\right )} {\rm Ei}\left (2 i \, b x^{n}\right ) - {\left (\sqrt {3} - i\right )} {\rm Ei}\left (-2 i \, b x^{n}\right ) - {\left (\sqrt {3} + i\right )} {\rm Ei}\left (2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (\sqrt {3} + i\right )} {\rm Ei}\left (-2 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (2 \, a\right )\right )} c^{\frac {2}{3}}}{16 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(((I*sqrt(3) + 1)*Ei(2*I*b*x^n) + (I*sqrt(3) + 1)*Ei(-2*I*b*x^n) + (-I*sqrt(3) + 1)*Ei(2*I*b*e^(n*conjuga
te(log(x)))) + (-I*sqrt(3) + 1)*Ei(-2*I*b*e^(n*conjugate(log(x)))))*cos(2*a) - 4*n*log(x) - ((sqrt(3) - I)*Ei(
2*I*b*x^n) - (sqrt(3) - I)*Ei(-2*I*b*x^n) - (sqrt(3) + I)*Ei(2*I*b*e^(n*conjugate(log(x)))) + (sqrt(3) + I)*Ei
(-2*I*b*e^(n*conjugate(log(x)))))*sin(2*a))*c^(2/3)/n

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Fricas [A]
time = 0.38, size = 106, normalized size = 0.88 \begin {gather*} \frac {4^{\frac {2}{3}} {\left (4^{\frac {1}{3}} \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + 4^{\frac {1}{3}} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) - 2 \cdot 4^{\frac {1}{3}} n \log \left (x\right ) - 2 \cdot 4^{\frac {1}{3}} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right )\right )} \left (-{\left (c \cos \left (b x^{n} + a\right )^{2} - c\right )} \sin \left (b x^{n} + a\right )\right )^{\frac {2}{3}}}{16 \, {\left (n \cos \left (b x^{n} + a\right )^{2} - n\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{n} \right )}\right )^{\frac {2}{3}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x^n)^3)^(2/3)/x,x)

[Out]

int((c*sin(a + b*x^n)^3)^(2/3)/x, x)

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