\(\int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx\) [243]

Optimal result

Integrand size = 21, antiderivative size = 58 \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\frac {3 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3} \sqrt {\sin ^2(c+d x)}} \]

[Out]

3/10*b^2*hypergeom([-5/3, 1/2],[-2/3],cos(d*x+c)^2)*sin(d*x+c)/d/(b*cos(d*x+c))^(10/3)/(sin(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {16, 2722} \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\frac {3 b^2 \sin (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},\cos ^2(c+d x)\right )}{10 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{10/3}} \]

[In]

Int[Sec[c + d*x]^3/(b*Cos[c + d*x])^(4/3),x]

[Out]

(3*b^2*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Cos[c + d*x])^(10/3)*Sqrt[Sin
[c + d*x]^2])

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

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}& = b^3 \int \frac {1}{(b \cos (c+d x))^{13/3}} \, dx \\ & = \frac {3 b^2 \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\frac {3 b^2 \csc (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},\frac {1}{2},-\frac {2}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{10 d (b \cos (c+d x))^{10/3}} \]

[In]

Integrate[Sec[c + d*x]^3/(b*Cos[c + d*x])^(4/3),x]

[Out]

(3*b^2*Csc[c + d*x]*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(10*d*(b*Cos[c +
d*x])^(10/3))

Maple [F]

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

[In]

int(sec(d*x+c)^3/(cos(d*x+c)*b)^(4/3),x)

[Out]

int(sec(d*x+c)^3/(cos(d*x+c)*b)^(4/3),x)

Fricas [F]

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

[In]

integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c))^(2/3)*sec(d*x + c)^3/(b^2*cos(d*x + c)^2), x)

Sympy [F]

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

[In]

integrate(sec(d*x+c)**3/(b*cos(d*x+c))**(4/3),x)

[Out]

Integral(sec(c + d*x)**3/(b*cos(c + d*x))**(4/3), x)

Maxima [F]

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

[In]

integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate(sec(d*x + c)^3/(b*cos(d*x + c))^(4/3), x)

Giac [F]

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

[In]

integrate(sec(d*x+c)^3/(b*cos(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^3/(b*cos(d*x + c))^(4/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \]

[In]

int(1/(cos(c + d*x)^3*(b*cos(c + d*x))^(4/3)),x)

[Out]

int(1/(cos(c + d*x)^3*(b*cos(c + d*x))^(4/3)), x)