\(\int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx\) [347]

Optimal result

Integrand size = 23, antiderivative size = 266 \[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\frac {d \operatorname {AppellF1}\left (1-n,\frac {1-n}{2},\frac {1-n}{2},2-n,\frac {a+b}{a+b \sec (e+f x)},\frac {a-b}{a+b \sec (e+f x)}\right ) \left (-\frac {b (1-\sec (e+f x))}{a+b \sec (e+f x)}\right )^{\frac {1-n}{2}} \left (\frac {b (1+\sec (e+f x))}{a+b \sec (e+f x)}\right )^{\frac {1-n}{2}} (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}+\frac {1}{2} (-1+n)}}{a f (1-n)}-\frac {d \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{-1+n} \left (-\tan ^2(e+f x)\right )^{\frac {1-n}{2}+\frac {1+n}{2}}}{a f (1+n)} \]

[Out]

d*AppellF1(1-n,1/2-1/2*n,1/2-1/2*n,2-n,(a-b)/(a+b*sec(f*x+e)),(a+b)/(a+b*sec(f*x+e)))*(-b*(1-sec(f*x+e))/(a+b*
sec(f*x+e)))^(1/2-1/2*n)*(b*(1+sec(f*x+e))/(a+b*sec(f*x+e)))^(1/2-1/2*n)*(d*tan(f*x+e))^(-1+n)/a/f/(1-n)+d*hyp
ergeom([1, 1/2+1/2*n],[3/2+1/2*n],-tan(f*x+e)^2)*(d*tan(f*x+e))^(-1+n)*tan(f*x+e)^2/a/f/(1+n)

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(786\) vs. \(2(266)=532\).

Time = 4.22 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.95 \[ \int \frac {(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx=\frac {2 \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \tan \left (\frac {1}{2} (e+f x)\right ) (d \tan (e+f x))^n}{f (a+b \sec (e+f x)) \left (\left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )-16 n \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \cos \left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x) \sec (e+f x) \sin ^5\left (\frac {1}{2} (e+f x)\right )+2 n \left ((a+b) \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \csc (e+f x) \sec (e+f x) \tan \left (\frac {1}{2} (e+f x)\right )-\frac {2 (1+n) \left ((a-b) b \operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )+(a+b)^2 \left (\operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )+b (a+b) n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),\frac {(a-b) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )}{(a+b) (3+n)}\right )} \]

[In]

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

[Out]

(2*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)
/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]*(d*Tan[e + f*
x])^n)/(f*(a + b*Sec[e + f*x])*(((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*
x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*S
ec[(e + f*x)/2]^2 - 16*n*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2
] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Cos[(e +
f*x)/2]*Csc[e + f*x]^3*Sec[e + f*x]*Sin[(e + f*x)/2]^5 + 2*n*((a + b)*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Ta
n[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - b*AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*T
an[(e + f*x)/2]^2)/(a + b)])*Csc[e + f*x]*Sec[e + f*x]*Tan[(e + f*x)/2] - (2*(1 + n)*((a - b)*b*AppellF1[(3 +
n)/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + (a + b)^2*(AppellF1[(3 + n)
/2, n, 2, (5 + n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan
[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]) + b*(a + b)*n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(e + f*x)/2]
^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2)/((a + b)*(3 + n))))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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