3.1.17 \(\int (c \cos (a+b x))^{7/2} \, dx\) [17]

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

[Out]

2/7*c*(c*cos(b*x+a))^(5/2)*sin(b*x+a)/b+10/21*c^4*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+1/2*b*x)*EllipticF(si
n(1/2*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)/b/(c*cos(b*x+a))^(1/2)+10/21*c^3*sin(b*x+a)*(c*cos(b*x+a))^(1/2)/b

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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \begin {gather*} \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 \sin (a+b x) \sqrt {c \cos (a+b x)}}{21 b}+\frac {2 c \sin (a+b x) (c \cos (a+b x))^{5/2}}{7 b} \end {gather*}

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 2715

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] && Integ
erQ[2*n]

Rule 2720

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

Rule 2721

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

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*}

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Mathematica [A]
time = 0.06, size = 76, normalized size = 0.78 \begin {gather*} \frac {c^3 \sqrt {c \cos (a+b x)} \left (20 F\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sqrt {\cos (a+b x)} (23 \sin (a+b x)+3 \sin (3 (a+b x)))\right )}{42 b \sqrt {\cos (a+b x)}} \end {gather*}

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 = 0.05, size = 210, normalized size = 2.14

method	result	size
default	\(-\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, c^{4} \left (48 \left (\cos ^{9}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-120 \left (\cos ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+128 \left (\cos ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-72 \left (\cos ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{21 \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(210\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 95, normalized size = 0.97 \begin {gather*} \frac {-5 i \, \sqrt {2} c^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} c^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + 2 \, {\left (3 \, c^{3} \cos \left (b x + a\right )^{2} + 5 \, c^{3}\right )} \sqrt {c \cos \left (b x + a\right )} \sin \left (b x + a\right )}{21 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*(-5*I*sqrt(2)*c^(7/2)*weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a)) + 5*I*sqrt(2)*c^(7/2)*wei
erstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a)) + 2*(3*c^3*cos(b*x + a)^2 + 5*c^3)*sqrt(c*cos(b*x + a))
*sin(b*x + a))/b

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c\,\cos \left (a+b\,x\right )\right )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*cos(a + b*x))^(7/2),x)

[Out]

int((c*cos(a + b*x))^(7/2), x)

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