\(\int \cot ^4(e+f x) (a+b \sec ^2(e+f x)) \, dx\) [322]

Optimal result

Integrand size = 21, antiderivative size = 33 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=a x+\frac {a \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4226, 1816, 209} \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {a \cot (e+f x)}{f}+a x \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {b \cot ^3(e+f x)}{3 f}-\frac {a \cot ^3(e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2(e+f x)\right )}{3 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cot \left (f x +e \right )^{3}}{3}+\cot \left (f x +e \right )+f x +e \right )-\frac {b \cos \left (f x +e \right )^{3}}{3 \sin \left (f x +e \right )^{3}}}{f}\)	\(48\)
default	\(\frac {a \left (-\frac {\cot \left (f x +e \right )^{3}}{3}+\cot \left (f x +e \right )+f x +e \right )-\frac {b \cos \left (f x +e \right )^{3}}{3 \sin \left (f x +e \right )^{3}}}{f}\)	\(48\)
risch	\(a x +\frac {2 i \left (6 a \,{\mathrm e}^{4 i \left (f x +e \right )}+3 b \,{\mathrm e}^{4 i \left (f x +e \right )}-6 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 a +b \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}\)	\(65\)

[In]

int(cot(f*x+e)^4*(a+b*sec(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.30 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {{\left (4 \, a + b\right )} \cos \left (f x + e\right )^{3} - 3 \, a \cos \left (f x + e\right ) + 3 \, {\left (a f x \cos \left (f x + e\right )^{2} - a f x\right )} \sin \left (f x + e\right )}{3 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]

[In]

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

[Out]

1/3*((4*a + b)*cos(f*x + e)^3 - 3*a*cos(f*x + e) + 3*(a*f*x*cos(f*x + e)^2 - a*f*x)*sin(f*x + e))/((f*cos(f*x
+ e)^2 - f)*sin(f*x + e))

Sympy [F]

\[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot ^{4}{\left (e + f x \right )}\, dx \]

[In]

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

[Out]

Integral((a + b*sec(e + f*x)**2)*cot(e + f*x)**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, {\left (f x + e\right )} a + \frac {3 \, a \tan \left (f x + e\right )^{2} - a - b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (31) = 62\).

Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.36 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, {\left (f x + e\right )} a - 15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]

[In]

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

[Out]

1/24*(a*tan(1/2*f*x + 1/2*e)^3 + b*tan(1/2*f*x + 1/2*e)^3 + 24*(f*x + e)*a - 15*a*tan(1/2*f*x + 1/2*e) - 3*b*t
an(1/2*f*x + 1/2*e) + (15*a*tan(1/2*f*x + 1/2*e)^2 + 3*b*tan(1/2*f*x + 1/2*e)^2 - a - b)/tan(1/2*f*x + 1/2*e)^
3)/f

Mupad [B] (verification not implemented)

Time = 19.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=a\,x-\frac {-a\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{3}+\frac {b}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3} \]

[In]

int(cot(e + f*x)^4*(a + b/cos(e + f*x)^2),x)

[Out]

a*x - (a/3 + b/3 - a*tan(e + f*x)^2)/(f*tan(e + f*x)^3)