3.140 \(\int \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} (A+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=95 \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^3 d}-\frac{3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right )}{130 b^3 d \sqrt{\sin ^2(c+d x)}} \]

[Out]

(3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b^3*d) - (3*(13*A + 10*C)*(b*Cos[c + d*x])^(10/3)*Hypergeometri
c2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(130*b^3*d*Sqrt[Sin[c + d*x]^2])

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Rubi [A]  time = 0.0709075, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {16, 3014, 2643} \[ \frac{3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^3 d}-\frac{3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right )}{130 b^3 d \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*C*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x])/(13*b^3*d) - (3*(13*A + 10*C)*(b*Cos[c + d*x])^(10/3)*Hypergeometri
c2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2]*Sin[c + d*x])/(130*b^3*d*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 3014

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[
e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*
x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

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 \cos ^2(c+d x) \sqrt [3]{b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (b \cos (c+d x))^{7/3} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac{3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^3 d}+\frac{(13 A+10 C) \int (b \cos (c+d x))^{7/3} \, dx}{13 b^2}\\ &=\frac{3 C (b \cos (c+d x))^{10/3} \sin (c+d x)}{13 b^3 d}-\frac{3 (13 A+10 C) (b \cos (c+d x))^{10/3} \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{130 b^3 d \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.111706, size = 96, normalized size = 1.01 \[ -\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt [3]{b \cos (c+d x)} \left (8 A \cos ^2(c+d x) \, _2F_1\left (\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right )+5 C \cos ^4(c+d x) \, _2F_1\left (\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right )\right )}{80 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(8*A*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] +
5*C*Cos[c + d*x]^4*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(80*d)

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Maple [F]  time = 0.408, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt [3]{b\cos \left ( dx+c \right ) } \left ( A+C \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (C \cos \left (d x + c\right )^{4} + A \cos \left (d x + c\right )^{2}\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((C*cos(d*x + c)^4 + A*cos(d*x + c)^2)*(b*cos(d*x + c))^(1/3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \cos \left (d x + c\right )^{2} + A\right )} \left (b \cos \left (d x + c\right )\right )^{\frac{1}{3}} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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