3.9 \(\int \cos ^{\frac{7}{2}}(a+b x) \, dx\)

Optimal. Leaf size=65 \[ \frac{10 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b}+\frac{2 \sin (a+b x) \cos ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{10 \sin (a+b x) \sqrt{\cos (a+b x)}}{21 b} \]

[Out]

(10*EllipticF[(a + b*x)/2, 2])/(21*b) + (10*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(21*b) + (2*Cos[a + b*x]^(5/2)*Si
n[a + b*x])/(7*b)

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Rubi [A]  time = 0.0316657, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2635, 2641} \[ \frac{10 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b}+\frac{2 \sin (a+b x) \cos ^{\frac{5}{2}}(a+b x)}{7 b}+\frac{10 \sin (a+b x) \sqrt{\cos (a+b x)}}{21 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(7/2),x]

[Out]

(10*EllipticF[(a + b*x)/2, 2])/(21*b) + (10*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(21*b) + (2*Cos[a + b*x]^(5/2)*Si
n[a + b*x])/(7*b)

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

Mathematica [A]  time = 0.109926, size = 51, normalized size = 0.78 \[ \frac{20 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+(23 \sin (a+b x)+3 \sin (3 (a+b x))) \sqrt{\cos (a+b x)}}{42 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(7/2),x]

[Out]

(20*EllipticF[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*(23*Sin[a + b*x] + 3*Sin[3*(a + b*x)]))/(42*b)

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Maple [B]  time = 2.068, size = 199, normalized size = 3.1 \begin{align*} -{\frac{2}{21\,b}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-120\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+128\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-72\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +16\,\cos \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^(7/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (b x + a\right )^{\frac{7}{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)^(7/2), 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(b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [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(b*x+a)^(7/2),x, algorithm="giac")

[Out]

Timed out