\(\int (c (d \tan (e+f x))^p)^n (a+i a \tan (e+f x))^2 \, dx\) [1319]

Optimal result

Integrand size = 30, antiderivative size = 93 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {2 a^2 \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x)) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)} \]

[Out]

-a^2*tan(f*x+e)*(c*(d*tan(f*x+e))^p)^n/f/(n*p+1)+2*a^2*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 = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1600, 1970, 81, 66} \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=-\frac {a^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)}+\frac {2 a^2 \tan (e+f x) \operatorname {Hypergeometric2F1}(1,n p+1,n p+2,i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]

[In]

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

[Out]

-((a^2*Tan[e + f*x]*(c*(d*Tan[e + f*x])^p)^n)/(f*(1 + n*p))) + (2*a^2*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 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 81

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

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 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 {\text {Subst}\left (\int \frac {\left (c (d x)^p\right )^n (a+i a x)^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {\left (c (d x)^p\right )^n (a+i a x)}{\frac {1}{a}-\frac {i x}{a}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (\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} (a+i a x)}{\frac {1}{a}-\frac {i x}{a}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {\left (2 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}}{\frac {1}{a}-\frac {i x}{a}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a^2 \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f (1+n p)}+\frac {2 a^2 \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 = 0.87 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.62 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=\frac {a^2 (-1+2 \operatorname {Hypergeometric2F1}(1,1+n p,2+n p,i \tan (e+f x))) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n}{f+f n p} \]

[In]

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

[Out]

(a^2*(-1 + 2*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
+ f*n*p)

Maple [F]

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

[In]

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x)

[Out]

int((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x)

Fricas [F]

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

[In]

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(4*a^2*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)) + 4*I*f*x + n*log(c) + 4
*I*e)/(e^(4*I*f*x + 4*I*e) + 2*e^(2*I*f*x + 2*I*e) + 1), x)

Sympy [F]

\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=- a^{2} \left (\int \left (- \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n}\right )\, dx + \int \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx\right ) \]

[In]

integrate((c*(d*tan(f*x+e))**p)**n*(a+I*a*tan(f*x+e))**2,x)

[Out]

-a**2*(Integral(-(c*(d*tan(e + f*x))**p)**n, x) + Integral((c*(d*tan(e + f*x))**p)**n*tan(e + f*x)**2, x) + In
tegral(-2*I*(c*(d*tan(e + f*x))**p)**n*tan(e + f*x), x))

Maxima [F]

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

[In]

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((I*a*tan(f*x + e) + a)^2*((d*tan(f*x + e))^p*c)^n, x)

Giac [F]

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

[In]

integrate((c*(d*tan(f*x+e))^p)^n*(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^2*((d*tan(f*x + e))^p*c)^n, x)

Mupad [F(-1)]

Timed out. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^2 \, dx=\int {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]

[In]

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^2,x)

[Out]

int((c*(d*tan(e + f*x))^p)^n*(a + a*tan(e + f*x)*1i)^2, x)