\(\int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx\) [86]

Optimal result

Integrand size = 21, antiderivative size = 48 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {b \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a^2 d (1+n)} \]

[Out]

b*hypergeom([2, 1+n],[2+n],1+b*tan(d*x+c)/a)*(a+b*tan(d*x+c))^(1+n)/a^2/d/(1+n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3597, 67} \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {b (a+b \tan (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {b \tan (c+d x)}{a}+1\right )}{a^2 d (n+1)} \]

[In]

Int[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]

[Out]

(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 3597

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

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

Mathematica [A] (verified)

Time = 6.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {b \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{a^2 d (1+n)} \]

[In]

Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]

[Out]

(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))

Maple [F]

\[\int \left (\csc ^{2}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}d x\]

[In]

int(csc(d*x+c)^2*(a+b*tan(d*x+c))^n,x)

[Out]

int(csc(d*x+c)^2*(a+b*tan(d*x+c))^n,x)

Fricas [F]

\[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(csc(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*tan(d*x + c) + a)^n*csc(d*x + c)^2, x)

Sympy [F]

\[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{n} \csc ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(csc(d*x+c)**2*(a+b*tan(d*x+c))**n,x)

[Out]

Integral((a + b*tan(c + d*x))**n*csc(c + d*x)**2, x)

Maxima [F]

\[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(csc(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^n*csc(d*x + c)^2, x)

Giac [F]

\[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{2} \,d x } \]

[In]

integrate(csc(d*x+c)^2*(a+b*tan(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c) + a)^n*csc(d*x + c)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{{\sin \left (c+d\,x\right )}^2} \,d x \]

[In]

int((a + b*tan(c + d*x))^n/sin(c + d*x)^2,x)

[Out]

int((a + b*tan(c + d*x))^n/sin(c + d*x)^2, x)