3.165 \(\int \frac{(b \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=27 \[ \frac{b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]

[Out]

(b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0027092, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 8} \[ \frac{b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]

[Out]

(b^2*x*Sqrt[b*Cos[c + d*x]])/Sqrt[Cos[c + d*x]]

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v]
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(b \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{\left (b^2 \sqrt{b \cos (c+d x)}\right ) \int 1 \, dx}{\sqrt{\cos (c+d x)}}\\ &=\frac{b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0123551, size = 24, normalized size = 0.89 \[ \frac{x (b \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]

[Out]

(x*(b*Cos[c + d*x])^(5/2))/Cos[c + d*x]^(5/2)

________________________________________________________________________________________

Maple [A]  time = 0.131, size = 28, normalized size = 1. \begin{align*}{\frac{dx+c}{d} \left ( b\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(d*x+c)*(b*cos(d*x+c))^(5/2)/cos(d*x+c)^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.53812, size = 35, normalized size = 1.3 \begin{align*} \frac{2 \, b^{\frac{5}{2}} \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^(5/2)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/d

________________________________________________________________________________________

Fricas [A]  time = 2.15123, size = 270, normalized size = 10. \begin{align*} \left [\frac{\sqrt{-b} b^{2} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b \cos \left (d x + c\right )} \sqrt{-b} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac{b^{\frac{5}{2}} \arctan \left (\frac{\sqrt{b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{b} \cos \left (d x + c\right )^{\frac{3}{2}}}\right )}{d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-b)*b^2*log(2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c))*sin(d*x + c) - b
)/d, b^(5/2)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt(b)*cos(d*x + c)^(3/2)))/d]

________________________________________________________________________________________

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((b*cos(d*x+c))**(5/2)/cos(d*x+c)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*cos(d*x + c))^(5/2)/cos(d*x + c)^(5/2), x)