\(\int (c (d \tan (e+f x))^p)^n (a+i a \tan (e+f x)) \, dx\) [1320]

Optimal result

Integrand size = 28, antiderivative size = 54 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \]

[Out]

a*hypergeom([1, n*p+1],[n*p+2],I*tan(f*x+e))*tan(f*x+e)*(c*(d*tan(f*x+e))^p)^n/f/(n*p+1)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 1970, 66} \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \tan (e+f x) \operatorname {Hypergeometric2F1}(1,n p+1,n p+2,i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 1970

Int[(u_.)*((c_.)*((d_)*((a_.) + (b_.)*(x_)))^(q_))^(p_), x_Symbol] :> Dist[(c*(d*(a + b*x))^q)^p/(a + b*x)^(p*
q), Int[u*(a + b*x)^(p*q), x], x] /; FreeQ[{a, b, c, d, q, p}, x] &&  !IntegerQ[q] &&  !IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e)),x)

[Out]

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e)),x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((c*(d*tan(f*x+e))**p)**n*(a+I*a*tan(f*x+e)),x)

[Out]

I*a*(Integral(-I*(c*(d*tan(e + f*x))**p)**n, x) + Integral((c*(d*tan(e + f*x))**p)**n*tan(e + f*x), x))

Maxima [F]

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

[In]

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

[Out]

integrate((I*a*tan(f*x + e) + a)*((d*tan(f*x + e))^p*c)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((I*a*tan(f*x + e) + a)*((d*tan(f*x + e))^p*c)^n, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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