3.23 \(\int \frac {1}{(c \sec (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}} \]

[Out]

2/5*sin(b*x+a)/b/c/(c*sec(b*x+a))^(3/2)+6/5*(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/c^2/cos(b*x+a)^(1/2)/(c*sec(b*x+a))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 72, 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 = {3769, 3771, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Sec[a + b*x])^(-5/2),x]

[Out]

(6*EllipticE[(a + b*x)/2, 2])/(5*b*c^2*Sqrt[Cos[a + b*x]]*Sqrt[c*Sec[a + b*x]]) + (2*Sin[a + b*x])/(5*b*c*(c*S
ec[a + b*x])^(3/2))

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 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 \frac {1}{(c \sec (a+b x))^{5/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {c \sec (a+b x)}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}+\frac {3 \int \sqrt {\cos (a+b x)} \, dx}{5 c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}\\ &=\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 0.83 \[ \frac {\sqrt {c \sec (a+b x)} \left (\sin (a+b x)+\sin (3 (a+b x))+12 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{10 b c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Sec[a + b*x])^(-5/2),x]

[Out]

(Sqrt[c*Sec[a + b*x]]*(12*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + Sin[a + b*x] + Sin[3*(a + b*x)]))/(10
*b*c^3)

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sec \left (b x + a\right )}}{c^{3} \sec \left (b x + a\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*sec(b*x + a))/(c^3*sec(b*x + a)^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((c*sec(b*x + a))^(-5/2), x)

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maple [C]  time = 0.97, size = 321, normalized size = 4.46 \[ -\frac {2 \left (-3 i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )+3 i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-3 i \sin \left (b x +a \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+3 i \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+\cos ^{4}\left (b x +a \right )+2 \left (\cos ^{2}\left (b x +a \right )\right )-3 \cos \left (b x +a \right )\right )}{5 b \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*sec(b*x+a))^(5/2),x)

[Out]

-2/5/b*(-3*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)+3*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)-3*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)+3*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)+cos(b*x+a)^4+2*cos(b*x+a)^2-3*cos(b*x+a))/(c/cos(b*x+a)
)^(5/2)/cos(b*x+a)^3/sin(b*x+a)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((c*sec(b*x + a))^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c/cos(a + b*x))^(5/2),x)

[Out]

int(1/(c/cos(a + b*x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*sec(b*x+a))**(5/2),x)

[Out]

Integral((c*sec(a + b*x))**(-5/2), x)

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