3.4.42 \(\int x^m (c \sin ^3(a+b x^2))^{2/3} \, dx\) [342]

Optimal. Leaf size=209 \[ \frac {x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}+2^{-\frac {7}{2}-\frac {m}{2}} e^{2 i a} x^{1+m} \left (-i b x^2\right )^{\frac {1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+2^{-\frac {7}{2}-\frac {m}{2}} e^{-2 i a} x^{1+m} \left (i b x^2\right )^{\frac {1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \]

[Out]

1/2*x^(1+m)*csc(b*x^2+a)^2*(c*sin(b*x^2+a)^3)^(2/3)/(1+m)+2^(-7/2-1/2*m)*exp(2*I*a)*x^(1+m)*(-I*b*x^2)^(-1/2-1
/2*m)*csc(b*x^2+a)^2*GAMMA(1/2+1/2*m,-2*I*b*x^2)*(c*sin(b*x^2+a)^3)^(2/3)+2^(-7/2-1/2*m)*x^(1+m)*(I*b*x^2)^(-1
/2-1/2*m)*csc(b*x^2+a)^2*GAMMA(1/2+1/2*m,2*I*b*x^2)*(c*sin(b*x^2+a)^3)^(2/3)/exp(2*I*a)

________________________________________________________________________________________

Rubi [A]
time = 0.22, antiderivative size = 209, 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, 3484, 3471, 2250} \begin {gather*} e^{2 i a} 2^{-\frac {m}{2}-\frac {7}{2}} x^{m+1} \left (-i b x^2\right )^{\frac {1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text {Gamma}\left (\frac {m+1}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+e^{-2 i a} 2^{-\frac {m}{2}-\frac {7}{2}} x^{m+1} \left (i b x^2\right )^{\frac {1}{2} (-m-1)} \csc ^2\left (a+b x^2\right ) \text {Gamma}\left (\frac {m+1}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+\frac {x^{m+1} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*(c*Sin[a + b*x^2]^3)^(2/3),x]

[Out]

(x^(1 + m)*Csc[a + b*x^2]^2*(c*Sin[a + b*x^2]^3)^(2/3))/(2*(1 + m)) + 2^(-7/2 - m/2)*E^((2*I)*a)*x^(1 + m)*((-
I)*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/3) + (2^(-7/2 -
m/2)*x^(1 + m)*(I*b*x^2)^((-1 - m)/2)*Csc[a + b*x^2]^2*Gamma[(1 + m)/2, (2*I)*b*x^2]*(c*Sin[a + b*x^2]^3)^(2/
3))/E^((2*I)*a)

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 3471

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}, x] && IGtQ[n, 0]

Rule 3484

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}, x] && IGtQ[p, 1] && IGtQ[n, 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 x^m \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3} \, dx &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^m \sin ^2\left (a+b x^2\right ) \, dx\\ &=\left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int \left (\frac {x^m}{2}-\frac {1}{2} x^m \cos \left (2 a+2 b x^2\right )\right ) \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}-\frac {1}{2} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int x^m \cos \left (2 a+2 b x^2\right ) \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}-\frac {1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int e^{-2 i a-2 i b x^2} x^m \, dx-\frac {1}{4} \left (\csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\right ) \int e^{2 i a+2 i b x^2} x^m \, dx\\ &=\frac {x^{1+m} \csc ^2\left (a+b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{2 (1+m)}+2^{-\frac {7}{2}-\frac {m}{2}} e^{2 i a} x^{1+m} \left (-i b x^2\right )^{\frac {1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},-2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}+2^{-\frac {7}{2}-\frac {m}{2}} e^{-2 i a} x^{1+m} \left (i b x^2\right )^{\frac {1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \Gamma \left (\frac {1+m}{2},2 i b x^2\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.60, size = 189, normalized size = 0.90 \begin {gather*} \frac {2^{\frac {1}{2} (-7-m)} x^{1+m} \left (b^2 x^4\right )^{\frac {1}{2} (-1-m)} \csc ^2\left (a+b x^2\right ) \left (2^{\frac {5+m}{2}} \left (b^2 x^4\right )^{\frac {1+m}{2}}+(1+m) \left (-i b x^2\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1+m}{2},2 i b x^2\right ) (\cos (2 a)-i \sin (2 a))+(1+m) \left (i b x^2\right )^{\frac {1+m}{2}} \Gamma \left (\frac {1+m}{2},-2 i b x^2\right ) (\cos (2 a)+i \sin (2 a))\right ) \left (c \sin ^3\left (a+b x^2\right )\right )^{2/3}}{1+m} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2^((-7 - m)/2)*x^(1 + m)*(b^2*x^4)^((-1 - m)/2)*Csc[a + b*x^2]^2*(2^((5 + m)/2)*(b^2*x^4)^((1 + m)/2) + (1 +
m)*((-I)*b*x^2)^((1 + m)/2)*Gamma[(1 + m)/2, (2*I)*b*x^2]*(Cos[2*a] - I*Sin[2*a]) + (1 + m)*(I*b*x^2)^((1 + m)
/2)*Gamma[(1 + m)/2, (-2*I)*b*x^2]*(Cos[2*a] + I*Sin[2*a]))*(c*Sin[a + b*x^2]^3)^(2/3))/(1 + m)

________________________________________________________________________________________

Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int x^{m} \left (c \left (\sin ^{3}\left (b \,x^{2}+a \right )\right )\right )^{\frac {2}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.11, size = 130, normalized size = 0.62 \begin {gather*} -\frac {{\left (8 \, b x x^{m} - {\left (i \, m + i\right )} e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, 2 i \, b x^{2}\right ) - {\left (-i \, m - i\right )} e^{\left (-\frac {1}{2} \, {\left (m - 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (\frac {1}{2} \, m + \frac {1}{2}, -2 i \, b x^{2}\right )\right )} \left (-{\left (c \cos \left (b x^{2} + a\right )^{2} - c\right )} \sin \left (b x^{2} + a\right )\right )^{\frac {2}{3}}}{16 \, {\left ({\left (b m + b\right )} \cos \left (b x^{2} + a\right )^{2} - b m - b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(8*b*x*x^m - (I*m + I)*e^(-1/2*(m - 1)*log(2*I*b) - 2*I*a)*gamma(1/2*m + 1/2, 2*I*b*x^2) - (-I*m - I)*e^
(-1/2*(m - 1)*log(-2*I*b) + 2*I*a)*gamma(1/2*m + 1/2, -2*I*b*x^2))*(-(c*cos(b*x^2 + a)^2 - c)*sin(b*x^2 + a))^
(2/3)/((b*m + b)*cos(b*x^2 + a)^2 - b*m - b)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \left (c \sin ^{3}{\left (a + b x^{2} \right )}\right )^{\frac {2}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^m\,{\left (c\,{\sin \left (b\,x^2+a\right )}^3\right )}^{2/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c*sin(a + b*x^2)^3)^(2/3),x)

[Out]

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

________________________________________________________________________________________