\(\int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\) [688]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\text {Int}\left (\frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \]

[In]

Int[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]),x]

[Out]

Defer[Int][1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 83.83 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \]

[In]

Integrate[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]),x]

[Out]

Integrate[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]), x]

Maple [N/A] (verified)

Not integrable

Time = 0.57 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {1}{\cos \left (d x +c \right )^{\frac {5}{3}} \sqrt {a +\cos \left (d x +c \right ) b}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {5}{3}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 23.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \cos {\left (c + d x \right )}} \cos ^{\frac {5}{3}}{\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(a + b*cos(c + d*x))*cos(c + d*x)**(5/3)), x)

Maxima [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {5}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 19.52 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {5}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\cos ^{\frac {5}{3}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{5/3}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]

[In]

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

[Out]

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