\(\int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx\) [681]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Int}\left (\frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}},x\right ) \]

[Out]

Unintegrable(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \]

[In]

Int[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

Defer[Int][Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 55.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx \]

[In]

Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]

[Out]

Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 0.50 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {\cos ^{\frac {5}{3}}\left (d x +c \right )}{\sqrt {a +\cos \left (d x +c \right ) b}}d x\]

[In]

int(cos(d*x+c)^(5/3)/(a+cos(d*x+c)*b)^(1/2),x)

[Out]

int(cos(d*x+c)^(5/3)/(a+cos(d*x+c)*b)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**(5/3)/(a+b*cos(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

Giac [N/A]

Not integrable

Time = 23.57 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{\frac {5}{3}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^(5/3)/(a+b*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(5/3)/sqrt(b*cos(d*x + c) + a), x)

Mupad [N/A]

Not integrable

Time = 15.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^{\frac {5}{3}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/3}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]

[In]

int(cos(c + d*x)^(5/3)/(a + b*cos(c + d*x))^(1/2),x)

[Out]

int(cos(c + d*x)^(5/3)/(a + b*cos(c + d*x))^(1/2), x)