3.258 \(\int \tan ^3(x) \sqrt{a+a \tan ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ \frac{\left (a \sec ^2(x)\right )^{3/2}}{3 a}-\sqrt{a \sec ^2(x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0870712, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3657, 4124, 43} \[ \frac{\left (a \sec ^2(x)\right )^{3/2}}{3 a}-\sqrt{a \sec ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]^3*Sqrt[a + a*Tan[x]^2],x]

[Out]

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

Rule 3657

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 4124

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]

Rule 43

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])

Rubi steps

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

Mathematica [A]  time = 0.0288181, size = 20, normalized size = 0.67 \[ \frac{1}{3} \left (\sec ^2(x)-3\right ) \sqrt{a \sec ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*Sec[x]^2]*(-3 + Sec[x]^2))/3

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 29, normalized size = 1. \begin{align*}{\frac{1}{3\,a} \left ( a+a \left ( \tan \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}}-\sqrt{a+a \left ( \tan \left ( x \right ) \right ) ^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.93065, size = 373, normalized size = 12.43 \begin{align*} -\frac{2 \,{\left ({\left (3 \, \cos \left (5 \, x\right ) + 2 \, \cos \left (3 \, x\right ) + 3 \, \cos \left (x\right )\right )} \cos \left (6 \, x\right ) + 3 \,{\left (3 \, \cos \left (4 \, x\right ) + 3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (5 \, x\right ) + 3 \,{\left (2 \, \cos \left (3 \, x\right ) + 3 \, \cos \left (x\right )\right )} \cos \left (4 \, x\right ) + 2 \,{\left (3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (3 \, x\right ) + 9 \, \cos \left (2 \, x\right ) \cos \left (x\right ) +{\left (3 \, \sin \left (5 \, x\right ) + 2 \, \sin \left (3 \, x\right ) + 3 \, \sin \left (x\right )\right )} \sin \left (6 \, x\right ) + 9 \,{\left (\sin \left (4 \, x\right ) + \sin \left (2 \, x\right )\right )} \sin \left (5 \, x\right ) + 3 \,{\left (2 \, \sin \left (3 \, x\right ) + 3 \, \sin \left (x\right )\right )} \sin \left (4 \, x\right ) + 6 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 3 \, \cos \left (x\right )\right )} \sqrt{a}}{3 \,{\left (2 \,{\left (3 \, \cos \left (4 \, x\right ) + 3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (6 \, x\right ) + \cos \left (6 \, x\right )^{2} + 6 \,{\left (3 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + 9 \, \cos \left (4 \, x\right )^{2} + 9 \, \cos \left (2 \, x\right )^{2} + 6 \,{\left (\sin \left (4 \, x\right ) + \sin \left (2 \, x\right )\right )} \sin \left (6 \, x\right ) + \sin \left (6 \, x\right )^{2} + 9 \, \sin \left (4 \, x\right )^{2} + 18 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 9 \, \sin \left (2 \, x\right )^{2} + 6 \, \cos \left (2 \, x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*((3*cos(5*x) + 2*cos(3*x) + 3*cos(x))*cos(6*x) + 3*(3*cos(4*x) + 3*cos(2*x) + 1)*cos(5*x) + 3*(2*cos(3*x)
+ 3*cos(x))*cos(4*x) + 2*(3*cos(2*x) + 1)*cos(3*x) + 9*cos(2*x)*cos(x) + (3*sin(5*x) + 2*sin(3*x) + 3*sin(x))
*sin(6*x) + 9*(sin(4*x) + sin(2*x))*sin(5*x) + 3*(2*sin(3*x) + 3*sin(x))*sin(4*x) + 6*sin(3*x)*sin(2*x) + 9*si
n(2*x)*sin(x) + 3*cos(x))*sqrt(a)/(2*(3*cos(4*x) + 3*cos(2*x) + 1)*cos(6*x) + cos(6*x)^2 + 6*(3*cos(2*x) + 1)*
cos(4*x) + 9*cos(4*x)^2 + 9*cos(2*x)^2 + 6*(sin(4*x) + sin(2*x))*sin(6*x) + sin(6*x)^2 + 9*sin(4*x)^2 + 18*sin
(4*x)*sin(2*x) + 9*sin(2*x)^2 + 6*cos(2*x) + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.35013, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{3} \, \sqrt{a \tan \left (x\right )^{2} + a}{\left (\tan \left (x\right )^{2} - 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\tan ^{2}{\left (x \right )} + 1\right )} \tan ^{3}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.15969, size = 39, normalized size = 1.3 \begin{align*} \frac{{\left (a \tan \left (x\right )^{2} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{a \tan \left (x\right )^{2} + a} a}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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