\(\int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx\) [346]

Optimal result

Integrand size = 21, antiderivative size = 129 \[ \int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx=\frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {b \cos ^2(e+f x)^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {2+n}{2},\frac {3+n}{2},\sin ^2(e+f x)\right ) \sec (e+f x) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]

[Out]

a*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)+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 = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3969, 3557, 371, 2697} \[ \int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx=\frac {a (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{d f (n+1)}+\frac {b \sec (e+f x) \cos ^2(e+f x)^{\frac {n+2}{2}} (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+2}{2},\frac {n+3}{2},\sin ^2(e+f x)\right )}{d f (n+1)} \]

[In]

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

[Out]

(a*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(d*f*(1 + n)) + (b*(C
os[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 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 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 3969

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

Rubi steps \begin{align*} \text {integral}& = a \int (d \tan (e+f x))^n \, dx+b \int \sec (e+f x) (d \tan (e+f x))^n \, dx \\ & = \frac {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 {(a d) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{f} \\ & = \frac {a \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 {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 = 0.58 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int (a+b \sec (e+f x)) (d \tan (e+f x))^n \, dx=\frac {(d \tan (e+f x))^n \left (\frac {a \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)}{1+n}+b \csc (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1-n}{2},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}}\right )}{f} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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