\(\int (c \sin (a+b x))^{4/3} \, dx\) [33]

Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int (c \sin (a+b x))^{4/3} \, dx=\frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{7/3}}{7 b c \sqrt {\cos ^2(a+b x)}} \]

[Out]

3/7*cos(b*x+a)*hypergeom([1/2, 7/6],[13/6],sin(b*x+a)^2)*(c*sin(b*x+a))^(7/3)/b/c/(cos(b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int (c \sin (a+b x))^{4/3} \, dx=\frac {3 \cos (a+b x) (c \sin (a+b x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\sin ^2(a+b x)\right )}{7 b c \sqrt {\cos ^2(a+b x)}} \]

[In]

Int[(c*Sin[a + b*x])^(4/3),x]

[Out]

(3*Cos[a + b*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/3))/(7*b*c*Sqrt[Cos[a +
b*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{7/3}}{7 b c \sqrt {\cos ^2(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int (c \sin (a+b x))^{4/3} \, dx=\frac {3 \sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},\sin ^2(a+b x)\right ) (c \sin (a+b x))^{4/3} \tan (a+b x)}{7 b} \]

[In]

Integrate[(c*Sin[a + b*x])^(4/3),x]

[Out]

(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(4/3)*Tan[a + b*x])
/(7*b)

Maple [F]

\[\int \left (c \sin \left (b x +a \right )\right )^{\frac {4}{3}}d x\]

[In]

int((c*sin(b*x+a))^(4/3),x)

[Out]

int((c*sin(b*x+a))^(4/3),x)

Fricas [F]

\[ \int (c \sin (a+b x))^{4/3} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(4/3),x, algorithm="fricas")

[Out]

integral((c*sin(b*x + a))^(1/3)*c*sin(b*x + a), x)

Sympy [F]

\[ \int (c \sin (a+b x))^{4/3} \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{\frac {4}{3}}\, dx \]

[In]

integrate((c*sin(b*x+a))**(4/3),x)

[Out]

Integral((c*sin(a + b*x))**(4/3), x)

Maxima [F]

\[ \int (c \sin (a+b x))^{4/3} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(4/3),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(4/3), x)

Giac [F]

\[ \int (c \sin (a+b x))^{4/3} \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((c*sin(b*x+a))^(4/3),x, algorithm="giac")

[Out]

integrate((c*sin(b*x + a))^(4/3), x)

Mupad [F(-1)]

Timed out. \[ \int (c \sin (a+b x))^{4/3} \, dx=\int {\left (c\,\sin \left (a+b\,x\right )\right )}^{4/3} \,d x \]

[In]

int((c*sin(a + b*x))^(4/3),x)

[Out]

int((c*sin(a + b*x))^(4/3), x)