3.215 \(\int (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx\)

Optimal. Leaf size=58 \[ \frac {3 b^2 \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]

[Out]

3/4*b^2*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)

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Rubi [A]  time = 0.04, 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} \[ \frac {3 b^2 \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*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 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 (b \cos (c+d x))^{2/3} \sec ^3(c+d x) \, dx &=b^3 \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx\\ &=\frac {3 b^2 \, _2F_1\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*}

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Mathematica [A]  time = 0.04, size = 63, normalized size = 1.09 \[ \frac {3 \sqrt {\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*Sec[c + d*x]^2*Sqrt[S
in[c + d*x]^2])/(4*d)

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fricas [F]  time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \cos \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}} \left (\sec ^{3}\left (d x +c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (b\,\cos \left (c+d\,x\right )\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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