\(\int (d \cot (e+f x))^m (b \tan ^2(e+f x))^p \, dx\) [421]

Optimal result

Integrand size = 23, antiderivative size = 78 \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {(d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+2 p),\frac {1}{2} (3-m+2 p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)} \]

[Out]

(d*cot(f*x+e))^m*hypergeom([1, 1/2-1/2*m+p],[3/2-1/2*m+p],-tan(f*x+e)^2)*tan(f*x+e)*(b*tan(f*x+e)^2)^p/f/(1-m+
2*p)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3739, 2684, 3557, 371} \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+2 p+1),\frac {1}{2} (-m+2 p+3),-\tan ^2(e+f x)\right )}{f (-m+2 p+1)} \]

[In]

Int[(d*Cot[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]

[Out]

((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + 2*p)/2, (3 - m + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Ta
n[e + f*x]^2)^p)/(f*(1 - m + 2*p))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2684

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

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[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
]])

Rubi steps \begin{align*} \text {integral}& = \left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \cot (e+f x))^m \tan ^{2 p}(e+f x) \, dx \\ & = \left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \tan ^{-m+2 p}(e+f x) \, dx \\ & = \frac {\left ((d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \text {Subst}\left (\int \frac {x^{-m+2 p}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+2 p),\frac {1}{2} (3-m+2 p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=-\frac {d (d \cot (e+f x))^{-1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {m}{2}+p,\frac {3}{2}-\frac {m}{2}+p,-\tan ^2(e+f x)\right ) \left (b \tan ^2(e+f x)\right )^p}{f (-1+m-2 p)} \]

[In]

Integrate[(d*Cot[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]

[Out]

-((d*(d*Cot[e + f*x])^(-1 + m)*Hypergeometric2F1[1, 1/2 - m/2 + p, 3/2 - m/2 + p, -Tan[e + f*x]^2]*(b*Tan[e +
f*x]^2)^p)/(f*(-1 + m - 2*p)))

Maple [F]

\[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]

[In]

int((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x)

[Out]

int((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x)

Fricas [F]

\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^2)^p*(d*cot(f*x + e))^m, x)

Sympy [F]

\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cot {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

integrate((d*cot(f*x+e))**m*(b*tan(f*x+e)**2)**p,x)

[Out]

Integral((b*tan(e + f*x)**2)**p*(d*cot(e + f*x))**m, x)

Maxima [F]

\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^2)^p*(d*cot(f*x + e))^m, x)

Giac [F]

\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((d*cot(f*x+e))^m*(b*tan(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^2)^p*(d*cot(f*x + e))^m, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]

[In]

int((d*cot(e + f*x))^m*(b*tan(e + f*x)^2)^p,x)

[Out]

int((d*cot(e + f*x))^m*(b*tan(e + f*x)^2)^p, x)