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(b*x+a)^2)^(1+1/2*n)*hypergeom([1+1/2*n, 1/2+1/2*n],[3/2+1/2*n],sin(b*x+a)^2)*sec(b*x+a)*(d*tan(b*x+a))^(1
+n)/b/d/(1+n)

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Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right ), x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{d x} \]

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)

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maple [F]  time = 0.69, size = 0, normalized size = 0.00 \[ \int \sec \left (b x +a \right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx \]

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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )\,{d x} \]

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n}{\cos \left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec {\left (a + b x \right )}\, dx \]

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)

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