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

Optimal. Leaf size=76 \[ \frac{\sec (a+b x) \cos ^2(a+b x)^{\frac{n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]

[Out]

((Cos[a + b*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]*
(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))

________________________________________________________________________________________

Rubi [A]  time = 0.0239285, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2617} \[ \frac{\sec (a+b x) \cos ^2(a+b x)^{\frac{n+2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{n+2}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[a + b*x]^2)^((2 + n)/2)*Hypergeometric2F1[(1 + n)/2, (2 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]*
(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))

Rule 2617

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1)*(Cos[e + f*x]^2)^((m + n + 1)/2)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps

\begin{align*} \int \sec (a+b x) (d \tan (a+b x))^n \, dx &=\frac{\cos ^2(a+b x)^{\frac{2+n}{2}} \, _2F_1\left (\frac{1+n}{2},\frac{2+n}{2};\frac{3+n}{2};\sin ^2(a+b x)\right ) \sec (a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0665986, size = 64, normalized size = 0.84 \[ \frac{\csc (a+b x) \left (-\tan ^2(a+b x)\right )^{\frac{1-n}{2}} (d \tan (a+b x))^n \, _2F_1\left (\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\sec ^2(a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[a + b*x]*Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Sec[a + b*x]^2]*(d*Tan[a + b*x])^n*(-Tan[a + b*x]^2)^((1
- n)/2))/b

________________________________________________________________________________________

Maple [F]  time = 0.223, size = 0, normalized size = 0. \begin{align*} \int \sec \left ( bx+a \right ) \left ( d\tan \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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