\(\int \frac {\tan (x)}{(a+a \tan ^2(x))^{3/2}} \, dx\) [280]

Optimal result

Integrand size = 15, antiderivative size = 14 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

[Out]

-1/3/(a*sec(x)^2)^(3/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3738, 4209, 32} \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

[In]

Int[Tan[x]/(a + a*Tan[x]^2)^(3/2),x]

[Out]

-1/3*1/(a*Sec[x]^2)^(3/2)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3738

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4209

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> Dist[b/(2*f), Subst[In
t[(-1 + x)^((m - 1)/2)*(b*x)^(p - 1), x], x, Sec[e + f*x]^2], x] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p] &&
IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\tan (x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {1}{(a x)^{5/2}} \, dx,x,\sec ^2(x)\right ) \\ & = -\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

[In]

Integrate[Tan[x]/(a + a*Tan[x]^2)^(3/2),x]

[Out]

-1/3*1/(a*Sec[x]^2)^(3/2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(-\frac {1}{3 \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}\)	\(13\)
default	\(-\frac {1}{3 \left (a +a \tan \left (x \right )^{2}\right )^{\frac {3}{2}}}\)	\(13\)
risch	\(-\frac {{\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {{\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {1}{8 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right ) a}-\frac {{\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}\)	\(145\)

[In]

int(tan(x)/(a+a*tan(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(a+a*tan(x)^2)^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (10) = 20\).

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.50 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {\sqrt {a \tan \left (x\right )^{2} + a}}{3 \, {\left (a^{2} \tan \left (x\right )^{4} + 2 \, a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} \]

[In]

integrate(tan(x)/(a+a*tan(x)^2)^(3/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(a*tan(x)^2 + a)/(a^2*tan(x)^4 + 2*a^2*tan(x)^2 + a^2)

Sympy [A] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=- \frac {1}{3 \left (a \tan ^{2}{\left (x \right )} + a\right )^{\frac {3}{2}}} \]

[In]

integrate(tan(x)/(a+a*tan(x)**2)**(3/2),x)

[Out]

-1/(3*(a*tan(x)**2 + a)**(3/2))

Maxima [F]

\[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(tan(x)/(a+a*tan(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(tan(x)/(a*tan(x)^2 + a)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {1}{3 \, {\left (a \tan \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \]

[In]

integrate(tan(x)/(a+a*tan(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/3/(a*tan(x)^2 + a)^(3/2)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.64 \[ \int \frac {\tan (x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx=-\frac {\sqrt {a\,{\mathrm {tan}\left (x\right )}^2+a}}{3\,a^2\,{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^2} \]

[In]

int(tan(x)/(a + a*tan(x)^2)^(3/2),x)

[Out]

-(a + a*tan(x)^2)^(1/2)/(3*a^2*(tan(x)^2 + 1)^2)