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

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Rubi [A]  time = 0.0177816, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}

Mathematica [A]  time = 0.0525389, size = 38, normalized size = 1. \[ \frac{2 \sin (a+b x)}{b \sqrt{\cos (a+b x)}}-\frac{2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b} \]

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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Maple [A]  time = 1.684, size = 101, normalized size = 2.7 \begin{align*} -2\,{\frac{\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) }{\sin \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((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))-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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\cos \left (b x + a\right )^{\frac{3}{2}}}, x\right ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)