3.693 \(\int \sec ^2(x) (a+b \tan (x))^n \, dx\)

Optimal. Leaf size=19 \[ \frac{(a+b \tan (x))^{n+1}}{b (n+1)} \]

[Out]

(a + b*Tan[x])^(1 + n)/(b*(1 + n))

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Rubi [A]  time = 0.0354784, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3506, 32} \[ \frac{(a+b \tan (x))^{n+1}}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^2*(a + b*Tan[x])^n,x]

[Out]

(a + b*Tan[x])^(1 + n)/(b*(1 + n))

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/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]

Rubi steps

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

Mathematica [A]  time = 0.17782, size = 18, normalized size = 0.95 \[ \frac{(a+b \tan (x))^{n+1}}{b n+b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^2*(a + b*Tan[x])^n,x]

[Out]

(a + b*Tan[x])^(1 + n)/(b + b*n)

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Maple [A]  time = 0.021, size = 20, normalized size = 1.1 \begin{align*}{\frac{ \left ( a+b\tan \left ( x \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^2*(a+b*tan(x))^n,x)

[Out]

(a+b*tan(x))^(1+n)/b/(1+n)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(a+b*tan(x))^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.10435, size = 101, normalized size = 5.32 \begin{align*} \frac{{\left (a \cos \left (x\right ) + b \sin \left (x\right )\right )} \left (\frac{a \cos \left (x\right ) + b \sin \left (x\right )}{\cos \left (x\right )}\right )^{n}}{{\left (b n + b\right )} \cos \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(a+b*tan(x))^n,x, algorithm="fricas")

[Out]

(a*cos(x) + b*sin(x))*((a*cos(x) + b*sin(x))/cos(x))^n/((b*n + b)*cos(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**2*(a+b*tan(x))**n,x)

[Out]

Integral((a + b*tan(x))**n*sec(x)**2, x)

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Giac [A]  time = 1.14073, size = 26, normalized size = 1.37 \begin{align*} \frac{{\left (b \tan \left (x\right ) + a\right )}^{n + 1}}{b{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^2*(a+b*tan(x))^n,x, algorithm="giac")

[Out]

(b*tan(x) + a)^(n + 1)/(b*(n + 1))