Integrand size = 12, antiderivative size = 59 \[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+2 n),\frac {1}{2} (3+2 n),-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^2(c+d x)\right )^n}{d (1+2 n)} \] Output:
hypergeom([1, 1/2+n],[3/2+n],-tan(d*x+c)^2)*tan(d*x+c)*(b*tan(d*x+c)^2)^n/ d/(1+2*n)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+n,\frac {3}{2}+n,-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^2(c+d x)\right )^n}{d (1+2 n)} \] Input:
Integrate[(b*Tan[c + d*x]^2)^n,x]
Output:
(Hypergeometric2F1[1, 1/2 + n, 3/2 + n, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*T an[c + d*x]^2)^n)/(d*(1 + 2*n))
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 4141, 3042, 3957, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (b \tan ^2(c+d x)\right )^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (b \tan (c+d x)^2\right )^ndx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \tan ^{-2 n}(c+d x) \left (b \tan ^2(c+d x)\right )^n \int \tan ^{2 n}(c+d x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan ^{-2 n}(c+d x) \left (b \tan ^2(c+d x)\right )^n \int \tan (c+d x)^{2 n}dx\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\tan ^{-2 n}(c+d x) \left (b \tan ^2(c+d x)\right )^n \int \frac {\tan ^{2 n}(c+d x)}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\tan (c+d x) \left (b \tan ^2(c+d x)\right )^n \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2 n+1),\frac {1}{2} (2 n+3),-\tan ^2(c+d x)\right )}{d (2 n+1)}\) |
Input:
Int[(b*Tan[c + d*x]^2)^n,x]
Output:
(Hypergeometric2F1[1, (1 + 2*n)/2, (3 + 2*n)/2, -Tan[c + d*x]^2]*Tan[c + d *x]*(b*Tan[c + d*x]^2)^n)/(d*(1 + 2*n))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \left (b \tan \left (d x +c \right )^{2}\right )^{n}d x\]
Input:
int((b*tan(d*x+c)^2)^n,x)
Output:
int((b*tan(d*x+c)^2)^n,x)
\[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\int { \left (b \tan \left (d x + c\right )^{2}\right )^{n} \,d x } \] Input:
integrate((b*tan(d*x+c)^2)^n,x, algorithm="fricas")
Output:
integral((b*tan(d*x + c)^2)^n, x)
\[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\int \left (b \tan ^{2}{\left (c + d x \right )}\right )^{n}\, dx \] Input:
integrate((b*tan(d*x+c)**2)**n,x)
Output:
Integral((b*tan(c + d*x)**2)**n, x)
\[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\int { \left (b \tan \left (d x + c\right )^{2}\right )^{n} \,d x } \] Input:
integrate((b*tan(d*x+c)^2)^n,x, algorithm="maxima")
Output:
integrate((b*tan(d*x + c)^2)^n, x)
\[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\int { \left (b \tan \left (d x + c\right )^{2}\right )^{n} \,d x } \] Input:
integrate((b*tan(d*x+c)^2)^n,x, algorithm="giac")
Output:
integrate((b*tan(d*x + c)^2)^n, x)
Timed out. \[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}^n \,d x \] Input:
int((b*tan(c + d*x)^2)^n,x)
Output:
int((b*tan(c + d*x)^2)^n, x)
\[ \int \left (b \tan ^2(c+d x)\right )^n \, dx=b^{n} \left (\int \tan \left (d x +c \right )^{2 n}d x \right ) \] Input:
int((b*tan(d*x+c)^2)^n,x)
Output:
b**n*int(tan(c + d*x)**(2*n),x)