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

Optimal. Leaf size=65 \[ \frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.0136397, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {17, 2637} \[ \frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*x*Sqrt[Cos[c + d*x]])/(b^2*Sqrt[b*Cos[c + d*x]]) + (B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0433792, size = 45, normalized size = 0.69 \[ \frac{\sqrt{\cos (c+d x)} (A (c+d x)+B \sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.23, size = 39, normalized size = 0.6 \begin{align*}{\frac{A \left ( dx+c \right ) +B\sin \left ( dx+c \right ) }{d} \left ( \cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \left ( b\cos \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.86252, size = 54, normalized size = 0.83 \begin{align*} \frac{\frac{2 \, A \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac{5}{2}}} + \frac{B \sin \left (d x + c\right )}{b^{\frac{5}{2}}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.92784, size = 520, normalized size = 8. \begin{align*} \left [-\frac{A \sqrt{-b} \cos \left (d x + c\right ) \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 \, \sqrt{b \cos \left (d x + c\right )} B \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b^{3} d \cos \left (d x + c\right )}, \frac{A \sqrt{b} \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 ) \cos \left (d x + c\right ) + \sqrt{b \cos \left (d x + c\right )} B \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b^{3} d \cos \left (d x + c\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(A*sqrt(-b)*cos(d*x + c)*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) - 2*sqrt(b*cos(d*x + c))*B*sqrt(cos(d*x + c))*sin(d*x + c))/(b^3*d*cos(d*x + c)), (A*sqrt(b)*ar
ctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt(b)*cos(d*x + c)^(3/2)))*cos(d*x + c) + sqrt(b*cos(d*x + c))*B*sqr
t(cos(d*x + c))*sin(d*x + c))/(b^3*d*cos(d*x + c))]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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