3.14 \(\int \frac {1}{\cos ^{\frac {3}{2}}(a+b x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.02, 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 = {2636, 2639} \[ \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]

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 2636

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 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, 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 = 0.06, size = 38, normalized size = 1.00 \[ \frac {2 \sin (a+b x)}{b \sqrt {\cos (a+b x)}}-\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]

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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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cos(b*x + a)^(-3/2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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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maple [A]  time = 0.08, size = 101, normalized size = 2.66 \[ -\frac {2 \left (\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{\sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))-2*sin(
1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a))/sin(1/2*b*x+1/2*a)/(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 \[ \int \frac {1}{\cos \left (b x + a\right )^{\frac {3}{2}}}\,{d x} \]

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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mupad [B]  time = 0.25, size = 42, normalized size = 1.11 \[ \frac {2\,\sin \left (a+b\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{b\,\sqrt {\cos \left (a+b\,x\right )}\,\sqrt {{\sin \left (a+b\,x\right )}^2}} \]

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

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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