3.197 \(\int \cot ^6(e+f x) (a+b \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=61 \[ \frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{(a-b) \cot (e+f x)}{f}-x (a-b)-\frac{a \cot ^5(e+f x)}{5 f} \]

[Out]

-((a - b)*x) - ((a - b)*Cot[e + f*x])/f + ((a - b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0456984, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3629, 12, 3473, 8} \[ \frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{(a-b) \cot (e+f x)}{f}-x (a-b)-\frac{a \cot ^5(e+f x)}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]

[Out]

-((a - b)*x) - ((a - b)*Cot[e + f*x])/f + ((a - b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f)

Rule 3629

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[
((A*b^2 + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*
Tan[e + f*x])^(m + 1)*Simp[a*(A - C) - (A*b - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] &&
NeQ[A*b^2 + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cot ^6(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=-\frac{a \cot ^5(e+f x)}{5 f}-\int (a-b) \cot ^4(e+f x) \, dx\\ &=-\frac{a \cot ^5(e+f x)}{5 f}-(a-b) \int \cot ^4(e+f x) \, dx\\ &=\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{a \cot ^5(e+f x)}{5 f}-(-a+b) \int \cot ^2(e+f x) \, dx\\ &=-\frac{(a-b) \cot (e+f x)}{f}+\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{a \cot ^5(e+f x)}{5 f}-(a-b) \int 1 \, dx\\ &=-(a-b) x-\frac{(a-b) \cot (e+f x)}{f}+\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{a \cot ^5(e+f x)}{5 f}\\ \end{align*}

Mathematica [C]  time = 0.0503916, size = 69, normalized size = 1.13 \[ -\frac{a \cot ^5(e+f x) \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-\tan ^2(e+f x)\right )}{5 f}-\frac{b \cot ^3(e+f x) \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-\tan ^2(e+f x)\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]

[Out]

-(a*Cot[e + f*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, -Tan[e + f*x]^2])/(5*f) - (b*Cot[e + f*x]^3*Hypergeometric
2F1[-3/2, 1, -1/2, -Tan[e + f*x]^2])/(3*f)

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 67, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ( b \left ( -{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{3}}{3}}+\cot \left ( fx+e \right ) +fx+e \right ) +a \left ( -{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{5}}{5}}+{\frac{ \left ( \cot \left ( fx+e \right ) \right ) ^{3}}{3}}-\cot \left ( fx+e \right ) -fx-e \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^6*(a+b*tan(f*x+e)^2),x)

[Out]

1/f*(b*(-1/3*cot(f*x+e)^3+cot(f*x+e)+f*x+e)+a*(-1/5*cot(f*x+e)^5+1/3*cot(f*x+e)^3-cot(f*x+e)-f*x-e))

________________________________________________________________________________________

Maxima [A]  time = 1.67998, size = 82, normalized size = 1.34 \begin{align*} -\frac{15 \,{\left (f x + e\right )}{\left (a - b\right )} + \frac{15 \,{\left (a - b\right )} \tan \left (f x + e\right )^{4} - 5 \,{\left (a - b\right )} \tan \left (f x + e\right )^{2} + 3 \, a}{\tan \left (f x + e\right )^{5}}}{15 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

-1/15*(15*(f*x + e)*(a - b) + (15*(a - b)*tan(f*x + e)^4 - 5*(a - b)*tan(f*x + e)^2 + 3*a)/tan(f*x + e)^5)/f

________________________________________________________________________________________

Fricas [A]  time = 1.08593, size = 161, normalized size = 2.64 \begin{align*} -\frac{15 \,{\left (a - b\right )} f x \tan \left (f x + e\right )^{5} + 15 \,{\left (a - b\right )} \tan \left (f x + e\right )^{4} - 5 \,{\left (a - b\right )} \tan \left (f x + e\right )^{2} + 3 \, a}{15 \, f \tan \left (f x + e\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

-1/15*(15*(a - b)*f*x*tan(f*x + e)^5 + 15*(a - b)*tan(f*x + e)^4 - 5*(a - b)*tan(f*x + e)^2 + 3*a)/(f*tan(f*x
+ e)^5)

________________________________________________________________________________________

Sympy [A]  time = 18.2639, size = 97, normalized size = 1.59 \begin{align*} \begin{cases} \tilde{\infty } a x & \text{for}\: \left (e = 0 \vee e = - f x\right ) \wedge \left (e = - f x \vee f = 0\right ) \\x \left (a + b \tan ^{2}{\left (e \right )}\right ) \cot ^{6}{\left (e \right )} & \text{for}\: f = 0 \\- a x - \frac{a}{f \tan{\left (e + f x \right )}} + \frac{a}{3 f \tan ^{3}{\left (e + f x \right )}} - \frac{a}{5 f \tan ^{5}{\left (e + f x \right )}} + b x + \frac{b}{f \tan{\left (e + f x \right )}} - \frac{b}{3 f \tan ^{3}{\left (e + f x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**6*(a+b*tan(f*x+e)**2),x)

[Out]

Piecewise((zoo*a*x, (Eq(e, 0) | Eq(e, -f*x)) & (Eq(f, 0) | Eq(e, -f*x))), (x*(a + b*tan(e)**2)*cot(e)**6, Eq(f
, 0)), (-a*x - a/(f*tan(e + f*x)) + a/(3*f*tan(e + f*x)**3) - a/(5*f*tan(e + f*x)**5) + b*x + b/(f*tan(e + f*x
)) - b/(3*f*tan(e + f*x)**3), True))

________________________________________________________________________________________

Giac [B]  time = 1.34912, size = 227, normalized size = 3.72 \begin{align*} \frac{3 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 35 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 20 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 480 \,{\left (f x + e\right )}{\left (a - b\right )} + 330 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 300 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \frac{330 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 300 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 35 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 20 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, a}{\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5}}}{480 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(a+b*tan(f*x+e)^2),x, algorithm="giac")

[Out]

1/480*(3*a*tan(1/2*f*x + 1/2*e)^5 - 35*a*tan(1/2*f*x + 1/2*e)^3 + 20*b*tan(1/2*f*x + 1/2*e)^3 - 480*(f*x + e)*
(a - b) + 330*a*tan(1/2*f*x + 1/2*e) - 300*b*tan(1/2*f*x + 1/2*e) - (330*a*tan(1/2*f*x + 1/2*e)^4 - 300*b*tan(
1/2*f*x + 1/2*e)^4 - 35*a*tan(1/2*f*x + 1/2*e)^2 + 20*b*tan(1/2*f*x + 1/2*e)^2 + 3*a)/tan(1/2*f*x + 1/2*e)^5)/
f