3.9 \(\int \sec ^{\frac {7}{2}}(a+b x) \, dx\)

Optimal. Leaf size=85 \[ \frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {6 \sin (a+b x) \sqrt {\sec (a+b x)}}{5 b}-\frac {6 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3768, 3771, 2639} \[ \frac {2 \sin (a+b x) \sec ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {6 \sin (a+b x) \sqrt {\sec (a+b x)}}{5 b}-\frac {6 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^(7/2),x]

[Out]

(-6*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(5*b) + (6*Sqrt[Sec[a + b*x]]*Sin[a + b*x
])/(5*b) + (2*Sec[a + b*x]^(5/2)*Sin[a + b*x])/(5*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 \sec ^{\frac {7}{2}}(a+b x) \, dx &=\frac {2 \sec ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{5 b}+\frac {3}{5} \int \sec ^{\frac {3}{2}}(a+b x) \, dx\\ &=\frac {6 \sqrt {\sec (a+b x)} \sin (a+b x)}{5 b}+\frac {2 \sec ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{5 b}-\frac {3}{5} \int \frac {1}{\sqrt {\sec (a+b x)}} \, dx\\ &=\frac {6 \sqrt {\sec (a+b x)} \sin (a+b x)}{5 b}+\frac {2 \sec ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{5 b}-\frac {1}{5} \left (3 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx\\ &=-\frac {6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {\sec (a+b x)}}{5 b}+\frac {6 \sqrt {\sec (a+b x)} \sin (a+b x)}{5 b}+\frac {2 \sec ^{\frac {5}{2}}(a+b x) \sin (a+b x)}{5 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 59, normalized size = 0.69 \[ \frac {\sec ^{\frac {5}{2}}(a+b x) \left (7 \sin (a+b x)+3 \sin (3 (a+b x))-12 \cos ^{\frac {5}{2}}(a+b x) E\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{10 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^(7/2),x]

[Out]

(Sec[a + b*x]^(5/2)*(-12*Cos[a + b*x]^(5/2)*EllipticE[(a + b*x)/2, 2] + 7*Sin[a + b*x] + 3*Sin[3*(a + b*x)]))/
(10*b)

________________________________________________________________________________________

fricas [F]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sec \left (b x + a\right )^{\frac {7}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sec(b*x + a)^(7/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^(7/2), x)

________________________________________________________________________________________

maple [B]  time = 5.09, size = 358, normalized size = 4.21 \[ \frac {2 \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}}{5 \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x+a)^(7/2),x)

[Out]

2/5*(-(-2*cos(1/2*b*x+1/2*a)^2+1)*sin(1/2*b*x+1/2*a)^2)^(1/2)/(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)/sin(1/2*b*x+1/2*a)^3*(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)^6-12*El
lipticE(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+sin(1/2*b*x+1/2*a)^2)^(1/2)/(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 \[ \int \sec \left (b x + a\right )^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^(7/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cos(a + b*x))^(7/2),x)

[Out]

int((1/cos(a + b*x))^(7/2), x)

________________________________________________________________________________________

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(sec(b*x+a)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________