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

Optimal. Leaf size=58 \[ \frac {3 b^2 \sin (c+d x) \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {16, 2643} \[ \frac {3 b^2 \sin (c+d x) \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Cos[c + d*x])^(7/3)*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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx &=b^3 \int \frac {1}{(b \cos (c+d x))^{10/3}} \, dx\\ &=\frac {3 b^2 \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 58, normalized size = 1.00 \[ \frac {3 b^2 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right )}{7 d (b \cos (c+d x))^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3}}{b \cos \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}\left (d x +c \right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{3}}{\left (b \cos \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (c + d x \right )}}{\sqrt [3]{b \cos {\left (c + d x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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