\(\int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx\) [38]

Optimal result

Integrand size = 12, antiderivative size = 56 \[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=-\frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]

[Out]

-3*cos(b*x+a)*hypergeom([-1/6, 1/2],[5/6],sin(b*x+a)^2)/b/c/(c*sin(b*x+a))^(1/3)/(cos(b*x+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, 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} \[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=-\frac {3 \cos (a+b x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \]

[In]

Int[(c*Sin[a + b*x])^(-4/3),x]

[Out]

(-3*Cos[a + b*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sin[a + b*x]^2])/(b*c*Sqrt[Cos[a + b*x]^2]*(c*Sin[a + b*x])
^(1/3))

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}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right )}{b c \sqrt {\cos ^2(a+b x)} \sqrt [3]{c \sin (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=-\frac {3 \sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\sin ^2(a+b x)\right ) \tan (a+b x)}{b (c \sin (a+b x))^{4/3}} \]

[In]

Integrate[(c*Sin[a + b*x])^(-4/3),x]

[Out]

(-3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(c*Sin[a + b*x])^(
4/3))

Maple [F]

\[\int \frac {1}{\left (c \sin \left (b x +a \right )\right )^{\frac {4}{3}}}d x\]

[In]

int(1/(c*sin(b*x+a))^(4/3),x)

[Out]

int(1/(c*sin(b*x+a))^(4/3),x)

Fricas [F]

\[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]

[In]

integrate(1/(c*sin(b*x+a))^(4/3),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=\int \frac {1}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \]

[In]

integrate(1/(c*sin(b*x+a))**(4/3),x)

[Out]

Integral((c*sin(a + b*x))**(-4/3), x)

Maxima [F]

\[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]

[In]

integrate(1/(c*sin(b*x+a))^(4/3),x, algorithm="maxima")

[Out]

integrate((c*sin(b*x + a))^(-4/3), x)

Giac [F]

\[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \sin \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]

[In]

integrate(1/(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. \[ \int \frac {1}{(c \sin (a+b x))^{4/3}} \, dx=\int \frac {1}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{4/3}} \,d x \]

[In]

int(1/(c*sin(a + b*x))^(4/3),x)

[Out]

int(1/(c*sin(a + b*x))^(4/3), x)