\(\int \cot ^3(e+f x) (a+b \sec ^2(e+f x))^2 \, dx\) [328]

Optimal result

Integrand size = 23, antiderivative size = 57 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac {b^2 \log (\cos (e+f x))}{f}-\frac {\left (a^2-b^2\right ) \log (\sin (e+f x))}{f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 457, 90} \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {\left (a^2-b^2\right ) \log (\sin (e+f x))}{f}-\frac {(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac {b^2 \log (\cos (e+f x))}{f} \]

[In]

Int[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]

[Out]

-1/2*((a + b)^2*Csc[e + f*x]^2)/f - (b^2*Log[Cos[e + f*x]])/f - ((a^2 - b^2)*Log[Sin[e + f*x]])/f

Rule 90

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

Rule 457

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

Rule 4223

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.42 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {2 \left (b+a \cos ^2(e+f x)\right )^2 \left ((a+b)^2 \csc ^2(e+f x)+2 b^2 \log (\cos (e+f x))+2 \left (a^2-b^2\right ) \log (\sin (e+f x))\right )}{f (a+2 b+a \cos (2 (e+f x)))^2} \]

[In]

Integrate[Cot[e + f*x]^3*(a + b*Sec[e + f*x]^2)^2,x]

[Out]

(-2*(b + a*Cos[e + f*x]^2)^2*((a + b)^2*Csc[e + f*x]^2 + 2*b^2*Log[Cos[e + f*x]] + 2*(a^2 - b^2)*Log[Sin[e + f
*x]]))/(f*(a + 2*b + a*Cos[2*(e + f*x)])^2)

Maple [A] (verified)

Time = 2.62 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cot \left (f x +e \right )^{2}}{2}-\ln \left (\sin \left (f x +e \right )\right )\right )-\frac {a b}{\sin \left (f x +e \right )^{2}}+b^{2} \left (-\frac {1}{2 \sin \left (f x +e \right )^{2}}+\ln \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(64\)
default	\(\frac {a^{2} \left (-\frac {\cot \left (f x +e \right )^{2}}{2}-\ln \left (\sin \left (f x +e \right )\right )\right )-\frac {a b}{\sin \left (f x +e \right )^{2}}+b^{2} \left (-\frac {1}{2 \sin \left (f x +e \right )^{2}}+\ln \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(64\)
risch	\(i a^{2} x +\frac {2 i a^{2} e}{f}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) {\mathrm e}^{2 i \left (f x +e \right )}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) a^{2}}{f}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) b^{2}}{f}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}\)	\(116\)

[In]

int(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(a^2*(-1/2*cot(f*x+e)^2-ln(sin(f*x+e)))-a*b/sin(f*x+e)^2+b^2*(-1/2/sin(f*x+e)^2+ln(tan(f*x+e))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.75 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {a^{2} + 2 \, a b + b^{2} - {\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \log \left (\cos \left (f x + e\right )^{2}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} + b^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (f x + e\right )^{2} + \frac {1}{4}\right )}{2 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \]

[In]

integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \cot ^{3}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**3*(a+b*sec(f*x+e)**2)**2,x)

[Out]

Integral((a + b*sec(e + f*x)**2)**2*cot(e + f*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=-\frac {b^{2} \log \left (\sin \left (f x + e\right )^{2} - 1\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\sin \left (f x + e\right )^{2}\right ) + \frac {a^{2} + 2 \, a b + b^{2}}{\sin \left (f x + e\right )^{2}}}{2 \, f} \]

[In]

integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

-1/2*(b^2*log(sin(f*x + e)^2 - 1) + (a^2 - b^2)*log(sin(f*x + e)^2) + (a^2 + 2*a*b + b^2)/sin(f*x + e)^2)/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (55) = 110\).

Time = 0.33 (sec) , antiderivative size = 233, normalized size of antiderivative = 4.09 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {a^{2} {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + 2 \, a b {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + b^{2} {\left (\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + 4 \, a^{2} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2 \right |}\right ) - 4 \, b^{2} \log \left ({\left | -\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 2 \right |}\right )}{8 \, f} \]

[In]

integrate(cot(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")

[Out]

1/8*(a^2*((cos(f*x + e) + 1)/(cos(f*x + e) - 1) + (cos(f*x + e) - 1)/(cos(f*x + e) + 1)) + 2*a*b*((cos(f*x + e
) + 1)/(cos(f*x + e) - 1) + (cos(f*x + e) - 1)/(cos(f*x + e) + 1)) + b^2*((cos(f*x + e) + 1)/(cos(f*x + e) - 1
) + (cos(f*x + e) - 1)/(cos(f*x + e) + 1)) + 4*a^2*log(abs(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1) - (cos(f*x +
e) - 1)/(cos(f*x + e) + 1) + 2)) - 4*b^2*log(abs(-(cos(f*x + e) + 1)/(cos(f*x + e) - 1) - (cos(f*x + e) - 1)/
(cos(f*x + e) + 1) - 2)))/f

Mupad [B] (verification not implemented)

Time = 20.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.19 \[ \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {a^2\,\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^2-b^2\right )}{f}-\frac {{\mathrm {cot}\left (e+f\,x\right )}^2\,\left (\frac {a^2}{2}+a\,b+\frac {b^2}{2}\right )}{f} \]

[In]

int(cot(e + f*x)^3*(a + b/cos(e + f*x)^2)^2,x)

[Out]

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