3.1.87 \(\int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx\) [87]

Optimal. Leaf size=38 \[ -\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}+\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2716, 2719} \begin {gather*} \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(-3/2),x]

[Out]

(-2*EllipticE[(a + b*x)/2, 2])/b + (2*Sin[a + b*x])/(b*Sqrt[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]

Rubi steps

\begin {align*} \int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx &=\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\int \sqrt {\cos (a+b x)} \, dx\\ &=-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}+\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 38, normalized size = 1.00 \begin {gather*} -\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b}+\frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(-3/2),x]

[Out]

(-2*EllipticE[(a + b*x)/2, 2])/b + (2*Sin[a + b*x])/(b*Sqrt[Cos[a + b*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(181\) vs. \(2(62)=124\).
time = 0.00, size = 182, normalized size = 4.79

method	result	size
default	\(-\frac {2 \left (-2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b}\)	\(182\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(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 = 0.12, size = 93, normalized size = 2.45 \begin {gather*} \frac {-i \, \sqrt {2} \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} \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 {\cos \left (b x + a\right )} \sin \left (b x + a\right )}{b \cos \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(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/cos(b*x+a)^(3/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(-3/2), x)

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Mupad [B]
time = 0.44, size = 42, normalized size = 1.11 \begin {gather*} \frac {2\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,\sqrt {\cos \left (a+b\,x\right )}\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/cos(a + b*x)^(3/2),x)

[Out]

(2*sin(a + b*x)*hypergeom([-1/4, 1/2], 3/4, cos(a + b*x)^2))/(b*cos(a + b*x)^(1/2)*(sin(a + b*x)^2)^(1/2))

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