3.4.31 \(\int \frac {\sqrt [3]{c \sin ^3(a+b x^n)}}{x} \, dx\) [331]

Optimal. Leaf size=73 \[ \frac {\text {Ci}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sin (a) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac {\cos (a) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \text {Si}\left (b x^n\right )}{n} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6852, 3458, 3457, 3456} \begin {gather*} \frac {\sin (a) \text {CosIntegral}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n}+\frac {\cos (a) \text {Si}\left (b x^n\right ) \csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )}}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(1/3)/x,x]

[Out]

(CosIntegral[b*x^n]*Csc[a + b*x^n]*Sin[a]*(c*Sin[a + b*x^n]^3)^(1/3))/n + (Cos[a]*Csc[a + b*x^n]*(c*Sin[a + b*
x^n]^3)^(1/3)*SinIntegral[b*x^n])/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 3458

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

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

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Mathematica [A]
time = 0.05, size = 47, normalized size = 0.64 \begin {gather*} \frac {\csc \left (a+b x^n\right ) \sqrt [3]{c \sin ^3\left (a+b x^n\right )} \left (\text {Ci}\left (b x^n\right ) \sin (a)+\cos (a) \text {Si}\left (b x^n\right )\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.30, size = 280, normalized size = 3.84

method	result	size
risch	\(-\frac {\expIntegral \left (1, -i b \,x^{n}\right ) \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {1}{3}} {\mathrm e}^{i \left (b \,x^{n}+2 a \right )}}{2 n \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )}-\frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{n}} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right ) n}+\frac {i \left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{n}} \sinIntegral \left (b \,x^{n}\right )}{\left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right ) n}+\frac {\left (i c \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right )^{3} {\mathrm e}^{-3 i \left (a +b \,x^{n}\right )}\right )^{\frac {1}{3}} {\mathrm e}^{i b \,x^{n}} \expIntegral \left (1, -i b \,x^{n}\right )}{2 \left ({\mathrm e}^{2 i \left (a +b \,x^{n}\right )}-1\right ) n}\)	\(280\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.66, size = 144, normalized size = 1.97 \begin {gather*} \frac {{\left ({\left ({\left (\sqrt {3} + i\right )} {\rm Ei}\left (i \, b x^{n}\right ) - {\left (\sqrt {3} + i\right )} {\rm Ei}\left (-i \, b x^{n}\right ) - {\left (\sqrt {3} - i\right )} {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (\sqrt {3} - i\right )} {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (a\right ) - {\left ({\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (i \, b x^{n}\right ) + {\left (-i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-i \, b x^{n}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\left (i \, \sqrt {3} + 1\right )} {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (a\right )\right )} c^{\frac {1}{3}}}{8 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(((sqrt(3) + I)*Ei(I*b*x^n) - (sqrt(3) + I)*Ei(-I*b*x^n) - (sqrt(3) - I)*Ei(I*b*e^(n*conjugate(log(x)))) +
(sqrt(3) - I)*Ei(-I*b*e^(n*conjugate(log(x)))))*cos(a) - ((-I*sqrt(3) + 1)*Ei(I*b*x^n) + (-I*sqrt(3) + 1)*Ei(
-I*b*x^n) + (I*sqrt(3) + 1)*Ei(I*b*e^(n*conjugate(log(x)))) + (I*sqrt(3) + 1)*Ei(-I*b*e^(n*conjugate(log(x))))
)*sin(a))*c^(1/3)/n

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Fricas [A]
time = 0.35, size = 98, normalized size = 1.34 \begin {gather*} -\frac {4^{\frac {1}{3}} {\left (4^{\frac {2}{3}} \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) + 4^{\frac {2}{3}} \operatorname {Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \cdot 4^{\frac {2}{3}} \cos \left (a\right ) \operatorname {Si}\left (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 {1}{3}} \sin \left (b x^{n} + a\right )}{8 \, {\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)^(1/3)/x,x, algorithm="fricas")

[Out]

-1/8*4^(1/3)*(4^(2/3)*cos_integral(b*x^n)*sin(a) + 4^(2/3)*cos_integral(-b*x^n)*sin(a) + 2*4^(2/3)*cos(a)*sin_
integral(b*x^n))*(-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(1/3)*sin(b*x^n + a)/(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 {\sqrt [3]{c \sin ^{3}{\left (a + b x^{n} \right )}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((c*sin(b*x^n + a)^3)^(1/3)/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 )}^{1/3}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(a + b*x^n)^3)^(1/3)/x,x)

[Out]

int((c*sin(a + b*x^n)^3)^(1/3)/x, x)

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