3.1.19 \(\int (c \sec (a+b x))^{3/2} \, dx\) [19]

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

[Out]

-2*c^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/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

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Rubi [A]
time = 0.03, 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 = {3853, 3856, 2719} \begin {gather*} \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)}} \end {gather*}

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 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]

Rule 3853

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 3856

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*}

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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.73 \begin {gather*} \frac {2 c \sqrt {c \sec (a+b x)} \left (-\sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+\sin (a+b x)\right )}{b} \end {gather*}

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] Result contains complex when optimal does not.
time = 42.37, size = 320, normalized size = 4.85

method	result	size
default	\(-\frac {2 \left (\cos \left (b x +a \right )+1\right )^{2} \left (-1+\cos \left (b x +a \right )\right )^{2} \left (i \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )-i \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+i \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-i \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+\cos \left (b x +a \right )-1\right ) \cos \left (b x +a \right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}}{b \sin \left (b x +a \right )^{5}}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/b*(cos(b*x+a)+1)^2*(-1+cos(b*x+a))^2*(I*sin(b*x+a)*cos(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*EllipticE(I*(-1+cos(b*x+a))/sin(b*x+a),I)*sin(b*x+a)
*cos(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)-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)+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 = 0.00, 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((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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.83, size = 84, normalized size = 1.27 \begin {gather*} \frac {-i \, \sqrt {2} c^{\frac {3}{2}} {\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} c^{\frac {3}{2}} {\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 \, c \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sin \left (b x + a\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-I*sqrt(2)*c^(3/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) + I*sin(b*x + a))) + I*sqrt
(2)*c^(3/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(b*x + a) - I*sin(b*x + a))) + 2*c*sqrt(c/cos
(b*x + a))*sin(b*x + a))/b

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

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 = 0.00, 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((c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((c*sec(b*x + a))^(3/2), x)

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

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)

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