3.10.29 \(\int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx\) [929]

Optimal. Leaf size=167 \[ -\frac {3 A \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (7+3 m);\frac {1}{6} (13+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+3 m) \sqrt {\sin ^2(c+d x)}} \]

[Out]

-3*A*cos(d*x+c)^(1+m)*(b*cos(d*x+c))^(1/3)*hypergeom([1/2, 2/3+1/2*m],[5/3+1/2*m],cos(d*x+c)^2)*sin(d*x+c)/d/(
4+3*m)/(sin(d*x+c)^2)^(1/2)-3*B*cos(d*x+c)^(2+m)*(b*cos(d*x+c))^(1/3)*hypergeom([1/2, 7/6+1/2*m],[13/6+1/2*m],
cos(d*x+c)^2)*sin(d*x+c)/d/(7+3*m)/(sin(d*x+c)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {20, 2827, 2722} \begin {gather*} -\frac {3 A \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+4);\frac {1}{6} (3 m+10);\cos ^2(c+d x)\right )}{d (3 m+4) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (3 m+7);\frac {1}{6} (3 m+13);\cos ^2(c+d x)\right )}{d (3 m+7) \sqrt {\sin ^2(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*A*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Hypergeometric2F1[1/2, (4 + 3*m)/6, (10 + 3*m)/6, Cos[c + d*
x]^2]*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]) - (3*B*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Hype
rgeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])

Rule 20

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

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]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rubi steps

\begin {align*} \int \cos ^m(c+d x) \sqrt [3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx &=\frac {\sqrt [3]{b \cos (c+d x)} \int \cos ^{\frac {1}{3}+m}(c+d x) (A+B \cos (c+d x)) \, dx}{\sqrt [3]{\cos (c+d x)}}\\ &=\frac {\left (A \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {1}{3}+m}(c+d x) \, dx}{\sqrt [3]{\cos (c+d x)}}+\frac {\left (B \sqrt [3]{b \cos (c+d x)}\right ) \int \cos ^{\frac {4}{3}+m}(c+d x) \, dx}{\sqrt [3]{\cos (c+d x)}}\\ &=-\frac {3 A \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {1}{6} (10+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (4+3 m) \sqrt {\sin ^2(c+d x)}}-\frac {3 B \cos ^{2+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {1}{6} (7+3 m);\frac {1}{6} (13+3 m);\cos ^2(c+d x)\right ) \sin (c+d x)}{d (7+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.31, size = 140, normalized size = 0.84 \begin {gather*} -\frac {3 \cos ^{1+m}(c+d x) \sqrt [3]{b \cos (c+d x)} \csc (c+d x) \left (A (7+3 m) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4+3 m);\frac {5}{3}+\frac {m}{2};\cos ^2(c+d x)\right )+B (4+3 m) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (7+3 m);\frac {1}{6} (13+3 m);\cos ^2(c+d x)\right )\right ) \sqrt {\sin ^2(c+d x)}}{d (4+3 m) (7+3 m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(A*(7 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6,
5/3 + m/2, Cos[c + d*x]^2] + B*(4 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c
+ d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + 3*m)*(7 + 3*m))

________________________________________________________________________________________

Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (A +B \cos \left (d x +c \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*cos(d*x + c) + A)*(b*cos(d*x + c))^(1/3)*cos(d*x + c)^m, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{b \cos {\left (c + d x \right )}} \left (A + B \cos {\left (c + d x \right )}\right ) \cos ^{m}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**m*(b*cos(d*x+c))**(1/3)*(A+B*cos(d*x+c)),x)

[Out]

Integral((b*cos(c + d*x))**(1/3)*(A + B*cos(c + d*x))*cos(c + d*x)**m, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________