3.422 \(\int (d \cot (e+f x))^m (a+b \tan ^2(e+f x))^p \, dx\)

Optimal. Leaf size=107 \[ \frac{\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]

[Out]

(AppellF1[(1 - m)/2, 1, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cot[e + f*x])^m*Tan[e + f*
x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 - m)*(1 + (b*Tan[e + f*x]^2)/a)^p)

________________________________________________________________________________________

Rubi [A]  time = 0.198803, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3674, 3670, 511, 510} \[ \frac{\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac{b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(AppellF1[(1 - m)/2, 1, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cot[e + f*x])^m*Tan[e + f*
x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 - m)*(1 + (b*Tan[e + f*x]^2)/a)^p)

Rule 3674

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

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\left ((d \cot (e+f x))^m \left (\frac{\tan (e+f x)}{d}\right )^m\right ) \int \left (\frac{\tan (e+f x)}{d}\right )^{-m} \left (a+b \tan ^2(e+f x)\right )^p \, dx\\ &=\frac{\left ((d \cot (e+f x))^m \left (\frac{\tan (e+f x)}{d}\right )^m\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{x}{d}\right )^{-m} \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left ((d \cot (e+f x))^m \left (\frac{\tan (e+f x)}{d}\right )^m \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac{b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{x}{d}\right )^{-m} \left (1+\frac{b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{F_1\left (\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac{b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)}\\ \end{align*}

Mathematica [B]  time = 2.33114, size = 265, normalized size = 2.48 \[ -\frac{a (m-3) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p F_1\left (\frac{1-m}{2};-p,1;\frac{3-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )}{f (m-1) \left (-2 b p F_1\left (\frac{3-m}{2};1-p,1;\frac{5-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 a F_1\left (\frac{3-m}{2};-p,2;\frac{5-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a (m-3) \cot ^2(e+f x) F_1\left (\frac{1-m}{2};-p,1;\frac{3-m}{2};-\frac{b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*C
ot[e + f*x]*(d*Cot[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p)/(f*(-1 + m)*(-2*b*p*AppellF1[(3 - m)/2, 1 - p, 1, (5
- m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*a*AppellF1[(3 - m)/2, -p, 2, (5 - m)/2, -((b*Tan[e + f*x
]^2)/a), -Tan[e + f*x]^2] + a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e +
f*x]^2]*Cot[e + f*x]^2)))

________________________________________________________________________________________

Maple [F]  time = 0.522, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{m} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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