\(\int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx\) [228]

Optimal result

Integrand size = 21, antiderivative size = 56 \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 b \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}} \]

[Out]

3/4*b*hypergeom([-2/3, 1/2],[1/3],cos(d*x+c)^2)*sin(d*x+c)/d/(b*cos(d*x+c))^(4/3)/(sin(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 56, 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} \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 b \sin (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]

[In]

Int[Sec[c + d*x]^2/(b*Cos[c + d*x])^(1/3),x]

[Out]

(3*b*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Cos[c + d*x])^(4/3)*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^2 \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx \\ & = \frac {3 b \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\frac {3 b^2 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{4 d (b \cos (c+d x))^{7/3}} \]

[In]

Integrate[Sec[c + d*x]^2/(b*Cos[c + d*x])^(1/3),x]

[Out]

(3*b^2*Cot[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*
x])^(7/3))

Maple [F]

\[\int \frac {\sec ^{2}\left (d x +c \right )}{\left (\cos \left (d x +c \right ) b \right )^{\frac {1}{3}}}d x\]

[In]

int(sec(d*x+c)^2/(cos(d*x+c)*b)^(1/3),x)

[Out]

int(sec(d*x+c)^2/(cos(d*x+c)*b)^(1/3),x)

Fricas [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\sqrt [3]{b \cos {\left (c + d x \right )}}}\, dx \]

[In]

integrate(sec(d*x+c)**2/(b*cos(d*x+c))**(1/3),x)

[Out]

Integral(sec(c + d*x)**2/(b*cos(c + d*x))**(1/3), x)

Maxima [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="maxima")

[Out]

integrate(sec(d*x + c)^2/(b*cos(d*x + c))^(1/3), x)

Giac [F]

\[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(b*cos(d*x+c))^(1/3),x, algorithm="giac")

[Out]

integrate(sec(d*x + c)^2/(b*cos(d*x + c))^(1/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^2(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \]

[In]

int(1/(cos(c + d*x)^2*(b*cos(c + d*x))^(1/3)),x)

[Out]

int(1/(cos(c + d*x)^2*(b*cos(c + d*x))^(1/3)), x)