3.129 \(\int \frac{\cos ^6(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=100 \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}} \]

[Out]

(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*S
in[c + d*x])/(21*b^3*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)

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Rubi [A]  time = 0.0601368, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {16, 2635, 2642, 2641} \[ \frac{2 \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6/(b*Cos[c + d*x])^(5/2),x]

[Out]

(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b^2*d*Sqrt[b*Cos[c + d*x]]) + (10*Sqrt[b*Cos[c + d*x]]*S
in[c + d*x])/(21*b^3*d) + (2*(b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b^5*d)

Rule 16

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac{\int (b \cos (c+d x))^{7/2} \, dx}{b^6}\\ &=\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{5 \int (b \cos (c+d x))^{3/2} \, dx}{7 b^4}\\ &=\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{5 \int \frac{1}{\sqrt{b \cos (c+d x)}} \, dx}{21 b^2}\\ &=\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}+\frac{\left (5 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{21 b^2 \sqrt{b \cos (c+d x)}}\\ &=\frac{10 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{10 \sqrt{b \cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac{2 (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b^5 d}\\ \end{align*}

Mathematica [A]  time = 0.0535276, size = 66, normalized size = 0.66 \[ \frac{26 \sin (2 (c+d x))+3 \sin (4 (c+d x))+40 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{84 b^2 d \sqrt{b \cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6/(b*Cos[c + d*x])^(5/2),x]

[Out]

(40*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 26*Sin[2*(c + d*x)] + 3*Sin[4*(c + d*x)])/(84*b^2*d*Sqrt[b*
Cos[c + d*x]])

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Maple [A]  time = 2.086, size = 210, normalized size = 2.1 \begin{align*} -{\frac{2}{21\,{b}^{2}d}\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-120\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+128\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}-72\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +16\,\cos \left ( 1/2\,dx+c/2 \right ) \right ){\frac{1}{\sqrt{-b \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{b \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6/(b*cos(d*x+c))^(5/2),x)

[Out]

-2/21*(b*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/b^2*(48*cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1
/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-72*cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)
^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*cos(1/2*d*x+1/2*c))/(-b*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d
*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c)/(b*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{6}}{\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)^6/(b*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^6/(b*cos(d*x + c))^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{3}}{b^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6/(b*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

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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)**6/(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{\cos \left (d x + c\right )^{6}}{\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)^6/(b*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^6/(b*cos(d*x + c))^(5/2), x)