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

Optimal. Leaf size=71 \[ \frac{10 \sin (x)}{21 a \sqrt{a \cos ^3(x)}}+\frac{10 \cos ^{\frac{3}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right )}{21 a \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec (x)}{7 a \sqrt{a \cos ^3(x)}} \]

[Out]

(10*Cos[x]^(3/2)*EllipticF[x/2, 2])/(21*a*Sqrt[a*Cos[x]^3]) + (10*Sin[x])/(21*a*Sqrt[a*Cos[x]^3]) + (2*Sec[x]*
Tan[x])/(7*a*Sqrt[a*Cos[x]^3])

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Rubi [A]  time = 0.0303169, antiderivative size = 71, 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 = {3207, 2636, 2641} \[ \frac{10 \sin (x)}{21 a \sqrt{a \cos ^3(x)}}+\frac{10 \cos ^{\frac{3}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right )}{21 a \sqrt{a \cos ^3(x)}}+\frac{2 \tan (x) \sec (x)}{7 a \sqrt{a \cos ^3(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Cos[x]^(3/2)*EllipticF[x/2, 2])/(21*a*Sqrt[a*Cos[x]^3]) + (10*Sin[x])/(21*a*Sqrt[a*Cos[x]^3]) + (2*Sec[x]*
Tan[x])/(7*a*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 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}{\left (a \cos ^3(x)\right )^{3/2}} \, dx &=\frac{\cos ^{\frac{3}{2}}(x) \int \frac{1}{\cos ^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \cos ^3(x)}}\\ &=\frac{2 \sec (x) \tan (x)}{7 a \sqrt{a \cos ^3(x)}}+\frac{\left (5 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(x)} \, dx}{7 a \sqrt{a \cos ^3(x)}}\\ &=\frac{10 \sin (x)}{21 a \sqrt{a \cos ^3(x)}}+\frac{2 \sec (x) \tan (x)}{7 a \sqrt{a \cos ^3(x)}}+\frac{\left (5 \cos ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\cos (x)}} \, dx}{21 a \sqrt{a \cos ^3(x)}}\\ &=\frac{10 \cos ^{\frac{3}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right )}{21 a \sqrt{a \cos ^3(x)}}+\frac{10 \sin (x)}{21 a \sqrt{a \cos ^3(x)}}+\frac{2 \sec (x) \tan (x)}{7 a \sqrt{a \cos ^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0592409, size = 44, normalized size = 0.62 \[ \frac{2 \cos ^2(x) \left (3 \tan (x)+5 \cos ^{\frac{5}{2}}(x) F\left (\left .\frac{x}{2}\right |2\right )+5 \sin (x) \cos (x)\right )}{21 \left (a \cos ^3(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[x]^2*(5*Cos[x]^(5/2)*EllipticF[x/2, 2] + 5*Cos[x]*Sin[x] + 3*Tan[x]))/(21*(a*Cos[x]^3)^(3/2))

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Maple [C]  time = 0.289, size = 87, normalized size = 1.2 \begin{align*} -{\frac{2\, \left ( \cos \left ( x \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( x \right ) \right ) \cos \left ( x \right ) }{21\, \left ( \sin \left ( x \right ) \right ) ^{3}} \left ( 5\,i \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( x \right ) \right ) }{\sin \left ( x \right ) }},i \right ) \sqrt{ \left ( \cos \left ( x \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}-5\, \left ( \cos \left ( x \right ) \right ) ^{3}+5\, \left ( \cos \left ( x \right ) \right ) ^{2}-3\,\cos \left ( x \right ) +3 \right ) \left ( a \left ( \cos \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(cos(x)+1)^2*(-1+cos(x))*(5*I*cos(x)^3*sin(x)*EllipticF(I*(-1+cos(x))/sin(x),I)*(1/(cos(x)+1))^(1/2)*(co
s(x)/(cos(x)+1))^(1/2)-5*cos(x)^3+5*cos(x)^2-3*cos(x)+3)*cos(x)/sin(x)^3/(a*cos(x)^3)^(3/2)

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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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cos(x)^3)^(-3/2), 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^{2} \cos \left (x\right )^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*cos(x)^3)/(a^2*cos(x)^6), 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)**(3/2),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{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*cos(x)^3)^(-3/2), x)