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

Optimal. Leaf size=25 \[ \frac {\sqrt {a \sec ^2(x)}}{a}+\frac {1}{\sqrt {a \sec ^2(x)}} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 25, normalized size of antiderivative = 1.00, 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} \[ \frac {\sqrt {a \sec ^2(x)}}{a}+\frac {1}{\sqrt {a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[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 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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.68 \[ \frac {\sec ^2(x)+1}{\sqrt {a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + Sec[x]^2)/Sqrt[a*Sec[x]^2]

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fricas [A]  time = 0.39, size = 17, normalized size = 0.68 \[ \frac {\tan \relax (x)^{2} + 2}{\sqrt {a \tan \relax (x)^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.39, size = 27, normalized size = 1.08 \[ \frac {\sqrt {a \tan \relax (x)^{2} + a} + \frac {a}{\sqrt {a \tan \relax (x)^{2} + a}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.22, size = 26, normalized size = 1.04 \[ \frac {\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}}{a}+\frac {1}{\sqrt {a +a \left (\tan ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.69, size = 37, normalized size = 1.48 \[ \frac {{\left (\sin \relax (x)^{2} - 2\right )} \sqrt {\sin \relax (x) + 1} \sqrt {-\sin \relax (x) + 1}}{\sqrt {a} \sin \relax (x)^{2} - \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.29, size = 22, normalized size = 0.88 \[ \frac {\sqrt {2}\,\left (\cos \left (2\,x\right )+3\right )}{2\,\sqrt {a}\,\sqrt {\cos \left (2\,x\right )+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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