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

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

[Out]

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

________________________________________________________________________________________

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 = {2636, 2642, 2641} \[ \frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(-5/2),x]

[Out]

(2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(3*b*c^2*Sqrt[c*Cos[a + b*x]]) + (2*Sin[a + b*x])/(3*b*c*(c*C
os[a + b*x])^(3/2))

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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{(c \cos (a+b x))^{5/2}} \, dx &=\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac {\int \frac {1}{\sqrt {c \cos (a+b x)}} \, dx}{3 c^2}\\ &=\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}+\frac {\sqrt {\cos (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{3 c^2 \sqrt {c \cos (a+b x)}}\\ &=\frac {2 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b c^2 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{3 b c (c \cos (a+b x))^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 51, normalized size = 0.71 \[ \frac {2 \left (\tan (a+b x)+\sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{3 b c^2 \sqrt {c \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(-5/2),x]

[Out]

(2*(Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + Tan[a + b*x]))/(3*b*c^2*Sqrt[c*Cos[a + b*x]])

________________________________________________________________________________________

fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \cos \left (b x + a\right )}}{c^{3} \cos \left (b x + a\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(-5/2), x)

________________________________________________________________________________________

maple [B]  time = 0.21, size = 241, normalized size = 3.35 \[ -\frac {2 \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{3 c^{2} \sqrt {-c \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(-2*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))*s
in(1/2*b*x+1/2*a)^2+(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticF(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))/c^2*(c*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(
1/2)/(-c*(2*sin(1/2*b*x+1/2*a)^4-sin(1/2*b*x+1/2*a)^2))^(1/2)/(2*cos(1/2*b*x+1/2*a)^2-1)/sin(1/2*b*x+1/2*a)/(c
*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*cos(b*x + a))^(-5/2), x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*cos(a + b*x))**(-5/2), x)

________________________________________________________________________________________