3.1.24 \(\int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx\) [24]

Optimal. Leaf size=100 \[ -\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]

[Out]

2/5*sin(b*x+a)/b/c/(c*cos(b*x+a))^(5/2)+6/5*sin(b*x+a)/b/c^3/(c*cos(b*x+a))^(1/2)-6/5*(cos(1/2*a+1/2*b*x)^2)^(
1/2)/cos(1/2*a+1/2*b*x)*EllipticE(sin(1/2*a+1/2*b*x),2^(1/2))*(c*cos(b*x+a))^(1/2)/b/c^4/cos(b*x+a)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 100, 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 = {2716, 2721, 2719} \begin {gather*} -\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(-7/2),x]

[Out]

(-6*Sqrt[c*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(5*b*c^4*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(5*b*c*(c*
Cos[a + b*x])^(5/2)) + (6*Sin[a + b*x])/(5*b*c^3*Sqrt[c*Cos[a + b*x]])

Rule 2716

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

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 \frac {1}{(c \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {3 \int \frac {1}{(c \cos (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {3 \int \sqrt {c \cos (a+b x)} \, dx}{5 c^4}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {\left (3 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 c^4 \sqrt {\cos (a+b x)}}\\ &=-\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 68, normalized size = 0.68 \begin {gather*} \frac {-6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+6 \sin (a+b x)+2 \sec (a+b x) \tan (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(-7/2),x]

[Out]

(-6*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 6*Sin[a + b*x] + 2*Sec[a + b*x]*Tan[a + b*x])/(5*b*c^3*Sqrt
[c*Cos[a + b*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(112)=224\).
time = 0.06, size = 366, normalized size = 3.66

method	result	size
default	\(-\frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{5 c^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cos(b*x+a))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(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/sin(1/2*b*x+1/2*a)^3/(8*sin(1/2*b*x+1/2*a)^
6-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(24*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^6-12*(2*sin(1/2*
b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+1/2*a)^4-
24*sin(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2*a)+12*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*El
lipticE(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+1/2*a)^2+8*sin(1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a)-3*(sin(1/2*
b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2)))*(-2*sin(1/2*b*x+1/
2*a)^4*c+c*sin(1/2*b*x+1/2*a)^2)^(1/2)/(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(1/(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 = 121, normalized size = 1.21 \begin {gather*} \frac {-3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + 2 \, \sqrt {c \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{5 \, b c^{4} \cos \left (b x + a\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

1/5*(-3*I*sqrt(2)*sqrt(c)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*si
n(b*x + a))) + 3*I*sqrt(2)*sqrt(c)*cos(b*x + a)^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x +
a) - I*sin(b*x + a))) + 2*sqrt(c*cos(b*x + a))*(3*cos(b*x + a)^2 + 1)*sin(b*x + a))/(b*c^4*cos(b*x + a)^3)

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3880 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(1/(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 \frac {1}{{\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(1/(c*cos(a + b*x))^(7/2),x)

[Out]

int(1/(c*cos(a + b*x))^(7/2), x)

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