3.283 \(\int \frac{1}{\sqrt{a+a \tan ^2(c+d x)}} \, dx\)

Optimal. Leaf size=24 \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]

[Out]

Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])

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Rubi [A]  time = 0.0292873, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3657, 4122, 191} \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + a*Tan[c + d*x]^2],x]

[Out]

Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])

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 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0407648, size = 24, normalized size = 1. \[ \frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + a*Tan[c + d*x]^2],x]

[Out]

Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])

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Maple [A]  time = 0.031, size = 25, normalized size = 1. \begin{align*}{\frac{\tan \left ( dx+c \right ) }{d}{\frac{1}{\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.80395, size = 18, normalized size = 0.75 \begin{align*} \frac{\sin \left (d x + c\right )}{\sqrt{a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(d*x + c)/(sqrt(a)*d)

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Fricas [A]  time = 1.59211, size = 92, normalized size = 3.83 \begin{align*} \frac{\sqrt{a \tan \left (d x + c\right )^{2} + a} \tan \left (d x + c\right )}{a d \tan \left (d x + c\right )^{2} + a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*tan(d*x + c)^2 + a), x)