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

Optimal. Leaf size=42 \[ \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt {a \cos ^2(x)}} \]

[Out]

1/2*arctanh(sin(x))*cos(x)/a/(a*cos(x)^2)^(1/2)+1/2*tan(x)/a/(a*cos(x)^2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3204, 3207, 3770} \[ \frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt {a \cos ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTanh[Sin[x]]*Cos[x])/(2*a*Sqrt[a*Cos[x]^2]) + Tan[x]/(2*a*Sqrt[a*Cos[x]^2])

Rule 3204

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1))/(b*f*(
2*p + 1)), x] + Dist[(2*(p + 1))/(b*(2*p + 1)), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx &=\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx}{2 a}\\ &=\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\cos (x) \int \sec (x) \, dx}{2 a \sqrt {a \cos ^2(x)}}\\ &=\frac {\tanh ^{-1}(\sin (x)) \cos (x)}{2 a \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{2 a \sqrt {a \cos ^2(x)}}\\ \end {align*}

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Mathematica [B]  time = 0.06, size = 91, normalized size = 2.17 \[ -\frac {\cos (x) \left (-2 \sin (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\cos (2 x) \left (\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right )}{4 \left (a \cos ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(Cos[x]*(Log[Cos[x/2] - Sin[x/2]] + Cos[2*x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) - Log[
Cos[x/2] + Sin[x/2]] - 2*Sin[x]))/(a*Cos[x]^2)^(3/2)

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fricas [A]  time = 0.42, size = 40, normalized size = 0.95 \[ -\frac {\sqrt {a \cos \relax (x)^{2}} {\left (\cos \relax (x)^{2} \log \left (-\frac {\sin \relax (x) - 1}{\sin \relax (x) + 1}\right ) - 2 \, \sin \relax (x)\right )}}{4 \, a^{2} \cos \relax (x)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(a*cos(x)^2)*(cos(x)^2*log(-(sin(x) - 1)/(sin(x) + 1)) - 2*sin(x))/(a^2*cos(x)^3)

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giac [A]  time = 0.42, size = 47, normalized size = 1.12 \[ -\frac {\frac {\log \left ({\left | -\sqrt {a} \tan \relax (x) + \sqrt {a \tan \relax (x)^{2} + a} \right |}\right )}{\sqrt {a}} - \frac {\sqrt {a \tan \relax (x)^{2} + a} \tan \relax (x)}{a}}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(abs(-sqrt(a)*tan(x) + sqrt(a*tan(x)^2 + a)))/sqrt(a) - sqrt(a*tan(x)^2 + a)*tan(x)/a)/a

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maple [B]  time = 0.09, size = 70, normalized size = 1.67 \[ \frac {\sqrt {a \left (\sin ^{2}\relax (x )\right )}\, \left (\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\relax (x )\right )}+2 a}{\cos \relax (x )}\right ) a \left (\cos ^{2}\relax (x )\right )+\sqrt {a}\, \sqrt {a \left (\sin ^{2}\relax (x )\right )}\right )}{2 a^{\frac {5}{2}} \cos \relax (x ) \sin \relax (x ) \sqrt {a \left (\cos ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^(5/2)/cos(x)*(a*sin(x)^2)^(1/2)*(ln(2*(a^(1/2)*(a*sin(x)^2)^(1/2)+a)/cos(x))*a*cos(x)^2+a^(1/2)*(a*sin(x
)^2)^(1/2))/sin(x)/(a*cos(x)^2)^(1/2)

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maxima [B]  time = 0.68, size = 304, normalized size = 7.24 \[ \frac {4 \, {\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) - 4 \, {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) + 4 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) - 8 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 8 \, \cos \relax (x) \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \sin \relax (x) - 4 \, \sin \relax (x)}{4 \, {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(sin(3*x) - sin(x))*cos(4*x) + (2*(2*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 4*cos(2*x)^2 + sin(4*x)^2 +
4*sin(4*x)*sin(2*x) + 4*sin(2*x)^2 + 4*cos(2*x) + 1)*log(cos(x)^2 + sin(x)^2 + 2*sin(x) + 1) - (2*(2*cos(2*x)
+ 1)*cos(4*x) + cos(4*x)^2 + 4*cos(2*x)^2 + sin(4*x)^2 + 4*sin(4*x)*sin(2*x) + 4*sin(2*x)^2 + 4*cos(2*x) + 1)*
log(cos(x)^2 + sin(x)^2 - 2*sin(x) + 1) - 4*(cos(3*x) - cos(x))*sin(4*x) + 4*(2*cos(2*x) + 1)*sin(3*x) - 8*cos
(3*x)*sin(2*x) + 8*cos(x)*sin(2*x) - 8*cos(2*x)*sin(x) - 4*sin(x))/((a*cos(4*x)^2 + 4*a*cos(2*x)^2 + a*sin(4*x
)^2 + 4*a*sin(4*x)*sin(2*x) + 4*a*sin(2*x)^2 + 2*(2*a*cos(2*x) + a)*cos(4*x) + 4*a*cos(2*x) + a)*sqrt(a))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (a\,{\cos \relax (x)}^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a*cos(x)^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cos ^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cos(x)**2)**(3/2),x)

[Out]

Integral((a*cos(x)**2)**(-3/2), x)

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