\(\int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx\) [273]

Optimal result

Integrand size = 17, antiderivative size = 25 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {1}{\sqrt {a \sec ^2(x)}}+\frac {\sqrt {a \sec ^2(x)}}{a} \]

[Out]

1/(a*sec(x)^2)^(1/2)+(a*sec(x)^2)^(1/2)/a

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3738, 4209, 45} \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {a \sec ^2(x)}}{a}+\frac {1}{\sqrt {a \sec ^2(x)}} \]

[In]

Int[Tan[x]^3/Sqrt[a + a*Tan[x]^2],x]

[Out]

1/Sqrt[a*Sec[x]^2] + Sqrt[a*Sec[x]^2]/a

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 ^3(x)}{\sqrt {a \sec ^2(x)}} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {-1+x}{(a x)^{3/2}} \, dx,x,\sec ^2(x)\right ) \\ & = \frac {1}{2} a \text {Subst}\left (\int \left (-\frac {1}{(a x)^{3/2}}+\frac {1}{a \sqrt {a x}}\right ) \, dx,x,\sec ^2(x)\right ) \\ & = \frac {1}{\sqrt {a \sec ^2(x)}}+\frac {\sqrt {a \sec ^2(x)}}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {(3+\cos (2 x)) \sqrt {a \sec ^2(x)}}{2 a} \]

[In]

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

[Out]

((3 + Cos[2*x])*Sqrt[a*Sec[x]^2])/(2*a)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04

method	result	size
derivativedivides	\(\frac {\sqrt {a +a \tan \left (x \right )^{2}}}{a}+\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\)	\(26\)
default	\(\frac {\sqrt {a +a \tan \left (x \right )^{2}}}{a}+\frac {1}{\sqrt {a +a \tan \left (x \right )^{2}}}\)	\(26\)
risch	\(\frac {{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}+1}{2 \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}\)	\(44\)

[In]

int(tan(x)^3/(a+a*tan(x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\tan \left (x\right )^{2} + 2}{\sqrt {a \tan \left (x\right )^{2} + a}} \]

[In]

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

[Out]

(tan(x)^2 + 2)/sqrt(a*tan(x)^2 + a)

Sympy [F]

\[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\int \frac {\tan ^{3}{\left (x \right )}}{\sqrt {a \left (\tan ^{2}{\left (x \right )} + 1\right )}}\, dx \]

[In]

integrate(tan(x)**3/(a+a*tan(x)**2)**(1/2),x)

[Out]

Integral(tan(x)**3/sqrt(a*(tan(x)**2 + 1)), x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {{\left (\sin \left (x\right )^{2} - 2\right )} \sqrt {\sin \left (x\right ) + 1} \sqrt {-\sin \left (x\right ) + 1}}{\sqrt {a} \sin \left (x\right )^{2} - \sqrt {a}} \]

[In]

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

[Out]

(sin(x)^2 - 2)*sqrt(sin(x) + 1)*sqrt(-sin(x) + 1)/(sqrt(a)*sin(x)^2 - sqrt(a))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {a \tan \left (x\right )^{2} + a} + \frac {a}{\sqrt {a \tan \left (x\right )^{2} + a}}}{a} \]

[In]

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

[Out]

(sqrt(a*tan(x)^2 + a) + a/sqrt(a*tan(x)^2 + a))/a

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\tan ^3(x)}{\sqrt {a+a \tan ^2(x)}} \, dx=\frac {\sqrt {2}\,\left (\cos \left (2\,x\right )+3\right )}{2\,\sqrt {a}\,\sqrt {\cos \left (2\,x\right )+1}} \]

[In]

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

[Out]

(2^(1/2)*(cos(2*x) + 3))/(2*a^(1/2)*(cos(2*x) + 1)^(1/2))