3.16 \(\int \frac{1}{\sec ^{\frac{7}{2}}(a+b x)} \, dx\)

Optimal. Leaf size=85 \[ \frac{10 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{21 b}+\frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{10 \sin (a+b x)}{21 b \sqrt{\sec (a+b x)}} \]

[Out]

(10*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(21*b) + (2*Sin[a + b*x])/(7*b*Sec[a + b*
x]^(5/2)) + (10*Sin[a + b*x])/(21*b*Sqrt[Sec[a + b*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.0403677, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3769, 3771, 2641} \[ \frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{10 \sin (a+b x)}{21 b \sqrt{\sec (a+b x)}}+\frac{10 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{21 b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*x]^(-7/2),x]

[Out]

(10*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b*x]])/(21*b) + (2*Sin[a + b*x])/(7*b*Sec[a + b*
x]^(5/2)) + (10*Sin[a + b*x])/(21*b*Sqrt[Sec[a + b*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]

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]

Rubi steps

\begin{align*} \int \frac{1}{\sec ^{\frac{7}{2}}(a+b x)} \, dx &=\frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{5}{7} \int \frac{1}{\sec ^{\frac{3}{2}}(a+b x)} \, dx\\ &=\frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{10 \sin (a+b x)}{21 b \sqrt{\sec (a+b x)}}+\frac{5}{21} \int \sqrt{\sec (a+b x)} \, dx\\ &=\frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{10 \sin (a+b x)}{21 b \sqrt{\sec (a+b x)}}+\frac{1}{21} \left (5 \sqrt{\cos (a+b x)} \sqrt{\sec (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx\\ &=\frac{10 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{\sec (a+b x)}}{21 b}+\frac{2 \sin (a+b x)}{7 b \sec ^{\frac{5}{2}}(a+b x)}+\frac{10 \sin (a+b x)}{21 b \sqrt{\sec (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0986761, size = 61, normalized size = 0.72 \[ \frac{\sqrt{\sec (a+b x)} \left (40 \sqrt{\cos (a+b x)} \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )+26 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{84 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*x]^(-7/2),x]

[Out]

(Sqrt[Sec[a + b*x]]*(40*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 26*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)
]))/(84*b)

________________________________________________________________________________________

Maple [B]  time = 1.247, size = 199, normalized size = 2.3 \begin{align*} -{\frac{2}{21\,b}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 48\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-120\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+128\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-72\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +16\,\cos \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*((2*cos(1/2*b*x+1/2*a)^2-1)*sin(1/2*b*x+1/2*a)^2)^(1/2)*(48*cos(1/2*b*x+1/2*a)^9-120*cos(1/2*b*x+1/2*a)^
7+128*cos(1/2*b*x+1/2*a)^5-72*cos(1/2*b*x+1/2*a)^3+5*(sin(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cos(1/2*b*x+1/2*a)^2+1)^
(1/2)*EllipticF(cos(1/2*b*x+1/2*a),2^(1/2))+16*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)/sin(1/2*b*x+1/2*a)/(2*cos(1/2*b*x+1/2*a)^2-1)^(1/2)/b

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sec \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^(-7/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sec \left (b x + a\right )^{\frac{7}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sec(b*x + a)^(-7/2), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sec \left (b x + a\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*x + a)^(-7/2), x)