3.24 \(\int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]

[Out]

2/5*sin(b*x+a)/b/c/(c*cos(b*x+a))^(5/2)+6/5*sin(b*x+a)/b/c^3/(c*cos(b*x+a))^(1/2)-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))*(c*cos(b*x+a))^(1/2)/b/c^4/cos(b*x+a)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2636, 2640, 2639} \[ \frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*Cos[a + b*x])^(-7/2),x]

[Out]

(-6*Sqrt[c*Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2])/(5*b*c^4*Sqrt[Cos[a + b*x]]) + (2*Sin[a + b*x])/(5*b*c*(c*
Cos[a + b*x])^(5/2)) + (6*Sin[a + b*x])/(5*b*c^3*Sqrt[c*Cos[a + b*x]])

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 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 2640

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

Rubi steps

\begin {align*} \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {3 \int \frac {1}{(c \cos (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {3 \int \sqrt {c \cos (a+b x)} \, dx}{5 c^4}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {\left (3 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 c^4 \sqrt {\cos (a+b x)}}\\ &=-\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 68, normalized size = 0.68 \[ \frac {6 \sin (a+b x)-6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+2 \tan (a+b x) \sec (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*Cos[a + b*x])^(-7/2),x]

[Out]

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

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \cos \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(c*cos(b*x + a))/(c^4*cos(b*x + a)^4), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))^(7/2),x, algorithm="giac")

[Out]

integrate((c*cos(b*x + a))^(-7/2), x)

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maple [B]  time = 0.20, size = 366, normalized size = 3.66 \[ \frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{5 c^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cos(b*x+a))^(7/2),x)

[Out]

2/5*(c*(2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/c^4/sin(1/2*b*x+1/2*a)^3/(8*sin(1/2*b*x+1/2*a)^6
-12*sin(1/2*b*x+1/2*a)^4+6*sin(1/2*b*x+1/2*a)^2-1)*(12*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin(1/2*b*x+1/2*
a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*b*x+1/2*a)^4-24*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^6-1
2*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*sin(1/2*
b*x+1/2*a)^2+24*sin(1/2*b*x+1/2*a)^4*cos(1/2*b*x+1/2*a)+3*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(2*sin(1/2*b*x+1/2*a)^2
-1)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))-8*sin(1/2*b*x+1/2*a)^2*cos(1/2*b*x+1/2*a))*(-2*sin(1/2*b*x+1/2
*a)^4*c+c*sin(1/2*b*x+1/2*a)^2)^(1/2)/(c*(2*cos(1/2*b*x+1/2*a)^2-1))^(1/2)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))^(7/2),x, algorithm="maxima")

[Out]

integrate((c*cos(b*x + a))^(-7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*cos(a + b*x))^(7/2),x)

[Out]

int(1/(c*cos(a + b*x))^(7/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(b*x+a))**(7/2),x)

[Out]

Timed out

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