\(\int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx\) [230]

   Optimal result
   Rubi [A] (verified)
   Mathematica [A] (verified)
   Maple [F]
   Fricas [F]
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 21, antiderivative size = 82 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 82, 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 = {20, 2722} \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {3 \sin (c+d x) \cos ^{m+1}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (3 m+1),\frac {1}{6} (3 m+7),\cos ^2(c+d x)\right )}{d (3 m+1) \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}} \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^{\frac {2}{3}}(c+d x) \int \cos ^{-\frac {2}{3}+m}(c+d x) \, dx}{(b \cos (c+d x))^{2/3}} \\ & = -\frac {3 \cos ^{1+m}(c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{6} (1+3 m),\frac {1}{6} (7+3 m),\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+3 m) (b \cos (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^m(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx=-\frac {\cos ^{1+m}(c+d x) \csc (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2} \left (\frac {1}{3}+m\right ),\frac {1}{2} \left (\frac {7}{3}+m\right ),\cos ^2(c+d x)\right ) \sqrt {\sin ^2(c+d x)}}{d \left (\frac {1}{3}+m\right ) (b \cos (c+d x))^{2/3}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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