3.1.22 \(\int \frac {1}{(c \cos (a+b x))^{3/2}} \, dx\) [22]

Optimal. Leaf size=68 \[ -\frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b c^2 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}} \]

[Out]

2*sin(b*x+a)/b/c/(c*cos(b*x+a))^(1/2)-2*(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^2/cos(b*x+a)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{b c^2 \sqrt {\cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(-3/2),x]

[Out]

(-2*Sqrt[c*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(b*c^2*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(b*c*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))^{3/2}} \, dx &=\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\int \sqrt {c \cos (a+b x)} \, dx}{c^2}\\ &=\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}-\frac {\sqrt {c \cos (a+b x)} \int \sqrt {\cos (a+b x)} \, dx}{c^2 \sqrt {\cos (a+b x)}}\\ &=-\frac {2 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b c^2 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{b c \sqrt {c \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 50, normalized size = 0.74 \begin {gather*} \frac {2 \left (-\sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sin (a+b x)\right )}{b c \sqrt {c \cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(-3/2),x]

[Out]

(2*(-(Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]) + Sin[a + b*x]))/(b*c*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. \(197\) vs. \(2(88)=176\).
time = 0.04, size = 198, normalized size = 2.91

method	result	size
default	\(-\frac {2 \left (-2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{c \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cos(b*x+a))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/c*(-2*cos(1/2*b*x+1/2*a)*(-2*sin(1/2*b*x+1/2*a)^4*c+c*sin(1/2*b*x+1/2*a)^2)^(1/2)*sin(1/2*b*x+1/2*a)^2+(sin
(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(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)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2)))/(-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(1/(c*cos(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((c*cos(b*x + a))^(-3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.09, size = 104, normalized size = 1.53 \begin {gather*} \frac {-i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right ) {\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 ) + i \, \sqrt {2} \sqrt {c} \cos \left (b x + a\right ) {\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 )} \sin \left (b x + a\right )}{b c^{2} \cos \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))**(3/2),x)

[Out]

Integral((c*cos(a + b*x))**(-3/2), x)

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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))^(3/2),x, algorithm="giac")

[Out]

integrate((c*cos(b*x + a))^(-3/2), 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 )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cos(a + b*x))^(3/2),x)

[Out]

int(1/(c*cos(a + b*x))^(3/2), x)

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