3.19 \(\int (c \sec (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 c \sin (a+b x) \sqrt{c \sec (a+b x)}}{b}-\frac{2 c^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0395353, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3771, 2639} \[ \frac{2 c \sin (a+b x) \sqrt{c \sec (a+b x)}}{b}-\frac{2 c^2 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b \sqrt{\cos (a+b x)} \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sec[a + b*x])^(3/2),x]

[Out]

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

Rule 3768

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

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

Mathematica [A]  time = 0.0391133, size = 48, normalized size = 0.73 \[ \frac{2 c \sqrt{c \sec (a+b x)} \left (\sin (a+b x)-\sqrt{\cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sec[a + b*x])^(3/2),x]

[Out]

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

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Maple [C]  time = 0.225, size = 322, normalized size = 4.9 \begin{align*} 2\,{\frac{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) }{b \left ( \sin \left ( bx+a \right ) \right ) ^{5}} \left ( i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}-i\sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) +i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \sin \left ( bx+a \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}-i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( bx+a \right ) \right ) }{\sin \left ( bx+a \right ) }},i \right ) \sin \left ( bx+a \right ) \sqrt{ \left ( \cos \left ( bx+a \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( bx+a \right ) }{\cos \left ( bx+a \right ) +1}}}-\cos \left ( bx+a \right ) +1 \right ) \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{3/2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sec(b*x+a))^(3/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sec(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((c*sec(b*x + a))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{c \sec \left (b x + a\right )} c \sec \left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*sec(b*x + a))*c*sec(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*sec(a + b*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*sec(b*x + a))^(3/2), x)