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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 66, 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 = {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 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]

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]

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 = 0.04, 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

________________________________________________________________________________________

fricas [F]  time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c \sec \left (b x + a\right )} c \sec \left (b x + a\right ), x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [C]  time = 0.98, size = 322, normalized size = 4.88 \[ \frac {2 \left (\cos \left (b x +a \right )+1\right )^{2} \left (-1+\cos \left (b x +a \right )\right )^{2} \left (i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )+i \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-i \sin \left (b x +a \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-\cos \left (b x +a \right )+1\right ) \cos \left (b x +a \right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}{b \sin \left (b x +a \right )^{5}} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/cos(a + b*x))^(3/2),x)

[Out]

int((c/cos(a + b*x))^(3/2), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________