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

Optimal. Leaf size=26 \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0435077, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3506, 32} \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.178291, size = 26, normalized size = 1. \[ \frac{(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 27, normalized size = 1. \begin{align*}{\frac{ \left ( a+b\tan \left ( dx+c \right ) \right ) ^{1+n}}{bd \left ( 1+n \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.55939, size = 155, normalized size = 5.96 \begin{align*} \frac{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )} \left (\frac{a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{n}}{{\left (b d n + b d\right )} \cos \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.96348, size = 35, normalized size = 1.35 \begin{align*} \frac{{\left (b \tan \left (d x + c\right ) + a\right )}^{n + 1}}{b d{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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