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

Optimal. Leaf size=65 \[ -\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b}+\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0292644, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2636, 2639} \[ -\frac{6 E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b}+\frac{2 \sin (a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}+\frac{6 \sin (a+b x)}{5 b \sqrt{\cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(-7/2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0932566, size = 59, normalized size = 0.91 \[ \frac{3 \sin (2 (a+b x))+2 \tan (a+b x)-6 \cos ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{5 b \cos ^{\frac{3}{2}}(a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(-7/2),x]

[Out]

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

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Maple [B]  time = 3.014, size = 358, normalized size = 5.5 \begin{align*}{\frac{2}{5\,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 ( 12\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-24\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) -12\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+24\,\cos \left ( 1/2\,bx+a/2 \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+3\,\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) \right ) \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 ( 8\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}-12\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+6\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-1} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-3}{\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/cos(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*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/
2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+1/2*a)^4-24*sin(1/2*b*x+1/2*a)^6*cos(1/2*b*x+1/2*a)-12*(2
*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^(1/2)*EllipticE(cos(1/2*b*x+1/2*a),2^(1/2))*sin(1/2*b*x+
1/2*a)^2+24*cos(1/2*b*x+1/2*a)*sin(1/2*b*x+1/2*a)^4+3*(2*sin(1/2*b*x+1/2*a)^2-1)^(1/2)*(sin(1/2*b*x+1/2*a)^2)^
(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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \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/cos(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(-7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\cos \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/cos(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

integral(cos(b*x + a)^(-7/2), x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\cos \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/cos(b*x+a)^(7/2),x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(-7/2), x)