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

Optimal. Leaf size=24 \[ \frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]

[Out]

(d*Tan[a + b*x])^(1 + n)/(b*d*(1 + n))

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Rubi [A]  time = 0.0385258, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 32} \[ \frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Tan[a + b*x])^(1 + n)/(b*d*(1 + n))

Rule 2607

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

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

Mathematica [A]  time = 0.020364, size = 25, normalized size = 1.04 \[ \frac{\tan (a+b x) (d \tan (a+b x))^n}{b (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Tan[a + b*x]*(d*Tan[a + b*x])^n)/(b*(1 + n))

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Maple [A]  time = 0.017, size = 25, normalized size = 1. \begin{align*}{\frac{ \left ( d\tan \left ( bx+a \right ) \right ) ^{1+n}}{bd \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65202, size = 96, normalized size = 4. \begin{align*} \frac{\left (\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right )^{n} \sin \left (b x + a\right )}{{\left (b n + b\right )} \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError