\(\int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx\) [345]

Optimal result

Integrand size = 23, antiderivative size = 160 \[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\frac {b^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {a^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3971, 3557, 371, 2697, 2687, 32} \[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\frac {a^2 (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{d f (n+1)}+\frac {2 a b \sec (e+f x) \cos ^2(e+f x)^{\frac {n+2}{2}} (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+2}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1)}+\frac {b^2 (d \tan (e+f x))^{n+1}}{d f (n+1)} \]

[In]

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

[Out]

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

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]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2687

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 2697

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)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (d \tan (e+f x))^n+2 a b \sec (e+f x) (d \tan (e+f x))^n+b^2 \sec ^2(e+f x) (d \tan (e+f x))^n\right ) \, dx \\ & = a^2 \int (d \tan (e+f x))^n \, dx+(2 a b) \int \sec (e+f x) (d \tan (e+f x))^n \, dx+b^2 \int \sec ^2(e+f x) (d \tan (e+f x))^n \, dx \\ & = \frac {2 a b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b^2 \text {Subst}\left (\int (d x)^n \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (a^2 d\right ) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f} \\ & = \frac {b^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {a^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.11 \[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\frac {d (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{-n/2} \left (2 a b (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3}{2},\sec ^2(e+f x)\right ) \sec (e+f x) \sqrt {-\tan ^2(e+f x)}-a^2 \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{\frac {2+n}{2}}+b^2 \left (\sqrt {-\tan ^2(e+f x)}-\left (-\tan ^2(e+f x)\right )^{\frac {2+n}{2}}\right )\right )}{f (1+n)} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +b \sec \left (f x +e \right )\right )^{2} \left (d \tan \left (f x +e \right )\right )^{n}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{2} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

integral((b^2*sec(f*x + e)^2 + 2*a*b*sec(f*x + e) + a^2)*(d*tan(f*x + e))^n, x)

Sympy [F]

\[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\int \left (d \tan {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{2} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{2} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a+b \sec (e+f x))^2 (d \tan (e+f x))^n \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^2 \,d x \]

[In]

int((d*tan(e + f*x))^n*(a + b/cos(e + f*x))^2,x)

[Out]

int((d*tan(e + f*x))^n*(a + b/cos(e + f*x))^2, x)