3.4.56 \(\int \frac {(c \sin ^3(a+b x^n))^{2/3}}{x^2} \, dx\) [356]

Optimal. Leaf size=180 \[ -\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {2^{-2+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {2^{-2+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x} \]

[Out]

-1/2*csc(a+b*x^n)^2*(c*sin(a+b*x^n)^3)^(2/3)/x+2^(-2+1/n)*exp(2*I*a)*(-I*b*x^n)^(1/n)*csc(a+b*x^n)^2*GAMMA(-1/
n,-2*I*b*x^n)*(c*sin(a+b*x^n)^3)^(2/3)/n/x+2^(-2+1/n)*(I*b*x^n)^(1/n)*csc(a+b*x^n)^2*GAMMA(-1/n,2*I*b*x^n)*(c*
sin(a+b*x^n)^3)^(2/3)/exp(2*I*a)/n/x

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Rubi [A]
time = 0.19, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6852, 3506, 3505, 2250} \begin {gather*} \frac {e^{2 i a} 2^{\frac {1}{n}-2} \left (-i b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},-2 i 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}}{n x}+\frac {e^{-2 i a} 2^{\frac {1}{n}-2} \left (i b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (-\frac {1}{n},2 i 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}}{n x}-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Sin[a + b*x^n]^3)^(2/3)/x^2,x]

[Out]

-1/2*(Csc[a + b*x^n]^2*(c*Sin[a + b*x^n]^3)^(2/3))/x + (2^(-2 + n^(-1))*E^((2*I)*a)*((-I)*b*x^n)^n^(-1)*Csc[a
+ b*x^n]^2*Gamma[-n^(-1), (-2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(n*x) + (2^(-2 + n^(-1))*(I*b*x^n)^n^(-1)*
Csc[a + b*x^n]^2*Gamma[-n^(-1), (2*I)*b*x^n]*(c*Sin[a + b*x^n]^3)^(2/3))/(E^((2*I)*a)*n*x)

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3505

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] + Dist[1/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m, 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^2} \, 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^2} \, 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^2}-\frac {\cos \left (2 a+2 b x^n\right )}{2 x^2}\right ) \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}-\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^2} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}-\frac {1}{4} \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 {e^{-2 i a-2 i b x^n}}{x^2} \, dx-\frac {1}{4} \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 {e^{2 i a+2 i b x^n}}{x^2} \, dx\\ &=-\frac {\csc ^2\left (a+b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{2 x}+\frac {2^{-2+\frac {1}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},-2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}+\frac {2^{-2+\frac {1}{n}} e^{-2 i a} \left (i b x^n\right )^{\frac {1}{n}} \csc ^2\left (a+b x^n\right ) \Gamma \left (-\frac {1}{n},2 i b x^n\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{n x}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 125, normalized size = 0.69 \begin {gather*} \frac {e^{-2 i a} \csc ^2\left (a+b x^n\right ) \left (-2 e^{2 i a} n+2^{\frac {1}{n}} e^{4 i a} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},-2 i b x^n\right )+2^{\frac {1}{n}} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (-\frac {1}{n},2 i b x^n\right )\right ) \left (c \sin ^3\left (a+b x^n\right )\right )^{2/3}}{4 n x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x^n]^2*(-2*E^((2*I)*a)*n + 2^n^(-1)*E^((4*I)*a)*((-I)*b*x^n)^n^(-1)*Gamma[-n^(-1), (-2*I)*b*x^n] +
2^n^(-1)*(I*b*x^n)^n^(-1)*Gamma[-n^(-1), (2*I)*b*x^n])*(c*Sin[a + b*x^n]^3)^(2/3))/(4*E^((2*I)*a)*n*x)

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (c \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\right )^{\frac {2}{3}}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to 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^2,x, algorithm="maxima")

[Out]

1/4*(x*integrate(cos(2*b*x^n + 2*a)/x^2, x) + 1)*c^(2/3)/x

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Fricas [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^2,x, algorithm="fricas")

[Out]

integral((-(c*cos(b*x^n + a)^2 - c)*sin(b*x^n + a))^(2/3)/x^2, x)

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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^{2}}\, 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**2,x)

[Out]

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

[Out]

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

[Out]

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

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