3.4.72 \(\int \cot ^4(e+f x) (a+b \tan ^2(e+f x))^p \, dx\) [372]

Optimal. Leaf size=83 \[ -\frac {F_1\left (-\frac {3}{2};1,-p;-\frac {1}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{3 f} \]

[Out]

-1/3*AppellF1(-3/2,1,-p,-1/2,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*cot(f*x+e)^3*(a+b*tan(f*x+e)^2)^p/f/((1+b*tan(f*
x+e)^2/a)^p)

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Rubi [A]
time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 525, 524} \begin {gather*} -\frac {\cot ^3(e+f x) \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 {3}{2};1,-p;-\frac {1}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

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

Rule 524

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)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 3751

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]))

Rubi steps

\begin {align*} \int \cot ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^4 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^4 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {F_1\left (-\frac {3}{2};1,-p;-\frac {1}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \cot ^3(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}}{3 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1887\) vs. \(2(83)=166\).
time = 6.53, size = 1887, normalized size = 22.73 \begin {gather*} \frac {2 \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f}+\frac {\cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (-a-b \tan ^2(e+f x)-(3 a+b (-1+2 p)) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \tan ^2(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right )}{3 a f}+\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos (e+f x) \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^{2 p}}{f \left (3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)\right ) \left (\frac {6 a b p F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \tan ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{-1+p}}{3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}+\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p}{3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}-\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sin ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^p}{3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}+\frac {3 a \cos (e+f x) \sin (e+f x) \left (\frac {2 b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac {2}{3} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right ) \left (a+b \tan ^2(e+f x)\right )^p}{3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)}-\frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos (e+f x) \sin (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (4 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \sec ^2(e+f x) \tan (e+f x)+3 a \left (\frac {2 b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{3 a}-\frac {2}{3} F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right )+2 \tan ^2(e+f x) \left (b p \left (-\frac {6}{5} F_1\left (\frac {5}{2};1-p,2;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)-\frac {6 b (1-p) F_1\left (\frac {5}{2};2-p,1;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{5 a}\right )-a \left (\frac {6 b p F_1\left (\frac {5}{2};1-p,2;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)}{5 a}-\frac {12}{5} F_1\left (\frac {5}{2};-p,3;\frac {7}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sec ^2(e+f x) \tan (e+f x)\right )\right )\right )}{\left (3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)\right ){}^2}\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]

[Out]

(2*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b
*Tan[e + f*x]^2)/a)^p) + (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p*(-a - b*Tan[e + f*x]^2 - ((3*a + b*(-1 + 2*p
))*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^2)/(1 + (b*Tan[e + f*x]^2)/a)^p))/(3
*a*f) + (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a
+ b*Tan[e + f*x]^2)^(2*p))/(f*(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*
p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Ta
n[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)*((6*a*b*p*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a
), -Tan[e + f*x]^2]*Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(-1 + p))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e
+ f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]
- a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*AppellF1[1/2,
-p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[
1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e +
f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e +
f*x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sin[e + f*x]^2*(a + b*Tan[e
+ f*x]^2)^p)/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2,
1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a)
, -Tan[e + f*x]^2])*Tan[e + f*x]^2) + (3*a*Cos[e + f*x]*Sin[e + f*x]*((2*b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b
*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*AppellF1[3/2, -p, 2, 5/2, -((b*T
an[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3)*(a + b*Tan[e + f*x]^2)^p)/(3*a*AppellF1[1/
2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*
x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*
x]^2) - (3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Cos[e + f*x]*Sin[e + f*x]*(a
+ b*Tan[e + f*x]^2)^p*(4*(b*p*AppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*Appel
lF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2])*Sec[e + f*x]^2*Tan[e + f*x] + 3*a*((2*b*p*Appe
llF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/(3*a) - (2*App
ellF1[3/2, -p, 2, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e + f*x])/3) + 2*Tan[e + f
*x]^2*(b*p*((-6*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*Tan[e +
f*x])/5 - (6*b*(1 - p)*AppellF1[5/2, 2 - p, 1, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e + f*x]^2*T
an[e + f*x])/(5*a)) - a*((6*b*p*AppellF1[5/2, 1 - p, 2, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e +
f*x]^2*Tan[e + f*x])/(5*a) - (12*AppellF1[5/2, -p, 3, 7/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2]*Sec[e +
f*x]^2*Tan[e + f*x])/5))))/(3*a*AppellF1[1/2, -p, 1, 3/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*(b*p*A
ppellF1[3/2, 1 - p, 1, 5/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] - a*AppellF1[3/2, -p, 2, 5/2, -((b*Tan[e
+ f*x]^2)/a), -Tan[e + f*x]^2])*Tan[e + f*x]^2)^2))

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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \left (\cot ^{4}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)

[Out]

int(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm="maxima")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^4, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm="fricas")

[Out]

integral((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^4, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(a+b*tan(f*x+e)**2)**p,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(a+b*tan(f*x+e)^2)^p,x, algorithm="giac")

[Out]

integrate((b*tan(f*x + e)^2 + a)^p*cot(f*x + e)^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4*(a + b*tan(e + f*x)^2)^p,x)

[Out]

int(cot(e + f*x)^4*(a + b*tan(e + f*x)^2)^p, x)

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