3.885 \(\int (d \cot (e+f x))^n (a+b \tan (e+f x))^m \, dx\)

Optimal. Leaf size=193 \[ \frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right )}{2 f (1-n)}+\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right )}{2 f (1-n)} \]

[Out]

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

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Rubi [A]  time = 0.254485, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4241, 3575, 912, 135, 133} \[ \frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},-i \tan (e+f x)\right )}{2 f (1-n)}+\frac{\tan (e+f x) (d \cot (e+f x))^n (a+b \tan (e+f x))^m \left (\frac{b \tan (e+f x)}{a}+1\right )^{-m} F_1\left (1-n;-m,1;2-n;-\frac{b \tan (e+f x)}{a},i \tan (e+f x)\right )}{2 f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4241

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ
[u, x]

Rule 3575

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Wit
h[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((a + b*ff*x)^m*(c + d*ff*x)^n)/(1 + ff^2*x^2), x]
, x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rule 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 135

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Dist[(c^IntPart[n]*(c +
d*x)^FracPart[n])/(1 + (d*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d
, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

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

Mathematica [F]  time = 2.90663, size = 0, normalized size = 0. \[ \int (d \cot (e+f x))^n (a+b \tan (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.304, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))**n*(a+b*tan(f*x+e))**m,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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