3.17 \(\int (c \cos (a+b x))^{7/2} \, dx\)

Optimal. Leaf size=98 \[ \frac{10 c^3 \sin (a+b x) \sqrt{c \cos (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{5/2}}{7 b} \]

[Out]

(10*c^4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(21*b*Sqrt[c*Cos[a + b*x]]) + (10*c^3*Sqrt[c*Cos[a + b*x
]]*Sin[a + b*x])/(21*b) + (2*c*(c*Cos[a + b*x])^(5/2)*Sin[a + b*x])/(7*b)

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Rubi [A]  time = 0.0586644, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 2642, 2641} \[ \frac{10 c^3 \sin (a+b x) \sqrt{c \cos (a+b x)}}{21 b}+\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{2 c \sin (a+b x) (c \cos (a+b x))^{5/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(7/2),x]

[Out]

(10*c^4*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(21*b*Sqrt[c*Cos[a + b*x]]) + (10*c^3*Sqrt[c*Cos[a + b*x
]]*Sin[a + b*x])/(21*b) + (2*c*(c*Cos[a + b*x])^(5/2)*Sin[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 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 (c \cos (a+b x))^{7/2} \, dx &=\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{1}{7} \left (5 c^2\right ) \int (c \cos (a+b x))^{3/2} \, dx\\ &=\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{1}{21} \left (5 c^4\right ) \int \frac{1}{\sqrt{c \cos (a+b x)}} \, dx\\ &=\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}+\frac{\left (5 c^4 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{21 \sqrt{c \cos (a+b x)}}\\ &=\frac{10 c^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b \sqrt{c \cos (a+b x)}}+\frac{10 c^3 \sqrt{c \cos (a+b x)} \sin (a+b x)}{21 b}+\frac{2 c (c \cos (a+b x))^{5/2} \sin (a+b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0920171, size = 76, normalized size = 0.78 \[ \frac{c^3 \sqrt{c \cos (a+b x)} \left (20 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )+(23 \sin (a+b x)+3 \sin (3 (a+b x))) \sqrt{\cos (a+b x)}\right )}{42 b \sqrt{\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(7/2),x]

[Out]

(c^3*Sqrt[c*Cos[a + b*x]]*(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*Sqrt[Cos[a + b*x]])

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Maple [A]  time = 2.047, size = 210, normalized size = 2.1 \begin{align*} -{\frac{2\,{c}^{4}}{21\,b}\sqrt{c \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{-c \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{c \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(c*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*c^4*(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))/(-c*(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)/(c*(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 \left (c \cos \left (b x + a\right )\right )^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*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 (\sqrt{c \cos \left (b x + a\right )} c^{3} \cos \left (b x + a\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*cos(b*x + a))*c^3*cos(b*x + a)^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((c*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((c*cos(b*x+a))^(7/2),x, algorithm="giac")

[Out]

Timed out