Integrand size = 12, antiderivative size = 59 \[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1+2 p),\frac {1}{2} (3+2 p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \] Output:
hypergeom([1, 1/2+p],[3/2+p],-tan(f*x+e)^2)*tan(f*x+e)*(b*tan(f*x+e)^2)^p/ f/(1+2*p)
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \] Input:
Integrate[(b*Tan[e + f*x]^2)^p,x]
Output:
(Hypergeometric2F1[1, 1/2 + p, 3/2 + p, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*T an[e + f*x]^2)^p)/(f*(1 + 2*p))
Time = 0.47 (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.
\(\displaystyle \int \left (b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (b \tan (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int \tan ^{2 p}(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int \tan (e+f x)^{2 p}dx\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int \frac {\tan ^{2 p}(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (2 p+1),\frac {1}{2} (2 p+3),-\tan ^2(e+f x)\right )}{f (2 p+1)}\) |
Input:
Int[(b*Tan[e + f*x]^2)^p,x]
Output:
(Hypergeometric2F1[1, (1 + 2*p)/2, (3 + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f *x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))
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]])
\[\int \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
Input:
int((b*tan(f*x+e)^2)^p,x)
Output:
int((b*tan(f*x+e)^2)^p,x)
\[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \,d x } \] Input:
integrate((b*tan(f*x+e)^2)^p,x, algorithm="fricas")
Output:
integral((b*tan(f*x + e)^2)^p, x)
\[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, dx \] Input:
integrate((b*tan(f*x+e)**2)**p,x)
Output:
Integral((b*tan(e + f*x)**2)**p, x)
\[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \,d x } \] Input:
integrate((b*tan(f*x+e)^2)^p,x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e)^2)^p, x)
\[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \,d x } \] Input:
integrate((b*tan(f*x+e)^2)^p,x, algorithm="giac")
Output:
integrate((b*tan(f*x + e)^2)^p, x)
Timed out. \[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \] Input:
int((b*tan(e + f*x)^2)^p,x)
Output:
int((b*tan(e + f*x)^2)^p, x)
\[ \int \left (b \tan ^2(e+f x)\right )^p \, dx=b^{p} \left (\int \tan \left (f x +e \right )^{2 p}d x \right ) \] Input:
int((b*tan(f*x+e)^2)^p,x)
Output:
b**p*int(tan(e + f*x)**(2*p),x)