3.50 \(\int \frac{1}{(a \cos ^3(x))^{5/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt{a \cos ^3(x)}} \]

[Out]

(-154*Cos[x]^(3/2)*EllipticE[x/2, 2])/(195*a^2*Sqrt[a*Cos[x]^3]) + (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Cos[x]^
3]) + (154*Tan[x])/(585*a^2*Sqrt[a*Cos[x]^3]) + (22*Sec[x]^2*Tan[x])/(117*a^2*Sqrt[a*Cos[x]^3]) + (2*Sec[x]^4*
Tan[x])/(13*a^2*Sqrt[a*Cos[x]^3])

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Rubi [A]  time = 0.0509484, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2636, 2639} \[ \frac{154 \sin (x) \cos (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec ^4(x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \tan (x) \sec ^2(x)}{117 a^2 \sqrt{a \cos ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[x]^3)^(-5/2),x]

[Out]

(-154*Cos[x]^(3/2)*EllipticE[x/2, 2])/(195*a^2*Sqrt[a*Cos[x]^3]) + (154*Cos[x]*Sin[x])/(195*a^2*Sqrt[a*Cos[x]^
3]) + (154*Tan[x])/(585*a^2*Sqrt[a*Cos[x]^3]) + (22*Sec[x]^2*Tan[x])/(117*a^2*Sqrt[a*Cos[x]^3]) + (2*Sec[x]^4*
Tan[x])/(13*a^2*Sqrt[a*Cos[x]^3])

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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}{\left (a \cos ^3(x)\right )^{5/2}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{1}{\cos ^{\frac{15}{2}}(x)} \, dx}{a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (11 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{11}{2}}(x)} \, dx}{13 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{7}{2}}(x)} \, dx}{117 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}+\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(x)} \, dx}{195 a^2 \sqrt{a \cos ^3(x)}}\\ &=\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}-\frac{\left (77 \cos ^{\frac{3}{2}}(x)\right ) \int \sqrt{\cos (x)} \, dx}{195 a^2 \sqrt{a \cos ^3(x)}}\\ &=-\frac{154 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \cos (x) \sin (x)}{195 a^2 \sqrt{a \cos ^3(x)}}+\frac{154 \tan (x)}{585 a^2 \sqrt{a \cos ^3(x)}}+\frac{22 \sec ^2(x) \tan (x)}{117 a^2 \sqrt{a \cos ^3(x)}}+\frac{2 \sec ^4(x) \tan (x)}{13 a^2 \sqrt{a \cos ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.104767, size = 57, normalized size = 0.49 \[ \frac{-462 \cos ^{\frac{3}{2}}(x) E\left (\left .\frac{x}{2}\right |2\right )+462 \sin (x) \cos (x)+2 \tan (x) \left (45 \sec ^4(x)+55 \sec ^2(x)+77\right )}{585 a^2 \sqrt{a \cos ^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[x]^3)^(-5/2),x]

[Out]

(-462*Cos[x]^(3/2)*EllipticE[x/2, 2] + 462*Cos[x]*Sin[x] + 2*(77 + 55*Sec[x]^2 + 45*Sec[x]^4)*Tan[x])/(585*a^2
*Sqrt[a*Cos[x]^3])

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Maple [C]  time = 0.423, size = 223, normalized size = 1.9 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) ^{2}\cos \left ( x \right ) }{585\, \left ( \sin \left ( x \right ) \right ) ^{5}} \left ( 231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) +231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) -231\,i \left ( \cos \left ( x \right ) \right ) ^{6}\sin \left ( x \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) +231\, \left ( \cos \left ( x \right ) \right ) ^{7}-154\, \left ( \cos \left ( x \right ) \right ) ^{6}-22\, \left ( \cos \left ( x \right ) \right ) ^{4}-10\, \left ( \cos \left ( x \right ) \right ) ^{2}-45 \right ) \left ( a \left ( \cos \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cos(x)^3)^(5/2),x)

[Out]

-2/585*(cos(x)+1)^2*(-1+cos(x))^2*(231*I*cos(x)^7*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*Ellipt
icF(I*(-1+cos(x))/sin(x),I)-231*I*cos(x)^7*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(
-1+cos(x))/sin(x),I)+231*I*cos(x)^6*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticF(I*(-1+cos(
x))/sin(x),I)-231*I*cos(x)^6*sin(x)*(1/(cos(x)+1))^(1/2)*(cos(x)/(cos(x)+1))^(1/2)*EllipticE(I*(-1+cos(x))/sin
(x),I)+231*cos(x)^7-154*cos(x)^6-22*cos(x)^4-10*cos(x)^2-45)*cos(x)/sin(x)^5/(a*cos(x)^3)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cos \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*cos(x)^3)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \cos \left (x\right )^{3}}}{a^{3} \cos \left (x\right )^{9}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)^3)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*cos(x)^3)/(a^3*cos(x)^9), 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/(a*cos(x)**3)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \cos \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*cos(x)^3)^(-5/2), x)