\(\int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx\) [319]

Optimal result

Integrand size = 26, antiderivative size = 158 \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\frac {(1-n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{2 a d f (1+n)}-\frac {i n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{2 a d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))} \]

[Out]

1/2*(1-n)*hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],-tan(f*x+e)^2)*(d*tan(f*x+e))^(1+n)/a/d/f/(1+n)-1/2*I*n*hyperge
om([1, 1+1/2*n],[2+1/2*n],-tan(f*x+e)^2)*(d*tan(f*x+e))^(2+n)/a/d^2/f/(2+n)+1/2*(d*tan(f*x+e))^(1+n)/d/f/(a-I*
a*tan(f*x+e))

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3633, 3619, 3557, 371} \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=-\frac {i n (d \tan (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{2},\frac {n+4}{2},-\tan ^2(e+f x)\right )}{2 a d^2 f (n+2)}+\frac {(1-n) (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{2 a d f (n+1)}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))} \]

[In]

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

[Out]

((1 - n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(1 + n))/(2*a*d*f*(1 + n
)) - ((I/2)*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(a*d^2*f*(
2 + n)) + (d*Tan[e + f*x])^(1 + n)/(2*d*f*(a - I*a*Tan[e + f*x]))

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 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 3619

Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*T
an[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ
[c^2 + d^2, 0] &&  !IntegerQ[2*m]

Rule 3633

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-a
)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x]))), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c
+ d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x
] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}-\frac {\int (d \tan (e+f x))^n (-a d (1-n)+i a d n \tan (e+f x)) \, dx}{2 a^2 d} \\ & = \frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}+\frac {(1-n) \int (d \tan (e+f x))^n \, dx}{2 a}-\frac {(i n) \int (d \tan (e+f x))^{1+n} \, dx}{2 a d} \\ & = \frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))}+\frac {(d (1-n)) \text {Subst}\left (\int \frac {x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{2 a f}-\frac {(i n) \text {Subst}\left (\int \frac {x^{1+n}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{2 a f} \\ & = \frac {(1-n) \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}}{2 a d f (1+n)}-\frac {i n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{2 a d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\frac {\tan (e+f x) (d \tan (e+f x))^n \left (-\frac {(-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right )}{a (1+n)}-\frac {i n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)}{a (2+n)}+\frac {1}{a-i a \tan (e+f x)}\right )}{2 f} \]

[In]

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

[Out]

(Tan[e + f*x]*(d*Tan[e + f*x])^n*(-(((-1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2])/(a*
(1 + n))) - (I*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(a*(2 + n)) + (a -
I*a*Tan[e + f*x])^(-1)))/(2*f)

Maple [F]

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

[In]

int((d*tan(f*x+e))^n/(a-I*a*tan(f*x+e)),x)

[Out]

int((d*tan(f*x+e))^n/(a-I*a*tan(f*x+e)),x)

Fricas [F]

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

[In]

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

[Out]

integral(1/2*((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))^n*(e^(2*I*f*x + 2*I*e) + 1)/a, x)

Sympy [F]

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

[In]

integrate((d*tan(f*x+e))**n/(a-I*a*tan(f*x+e)),x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

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

[In]

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

[Out]

integrate((d*tan(f*x + e))^n/(-I*a*tan(f*x + e) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{a-a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]

[In]

int((d*tan(e + f*x))^n/(a - a*tan(e + f*x)*1i),x)

[Out]

int((d*tan(e + f*x))^n/(a - a*tan(e + f*x)*1i), x)