3.3.28 \(\int \frac {\cot ^3(e+f x)}{(a+b \tan ^2(e+f x))^2} \, dx\) [228]

Optimal. Leaf size=132 \[ -\frac {\cot ^2(e+f x)}{2 a^2 f}-\frac {\log (\cos (e+f x))}{(a-b)^2 f}-\frac {(a+2 b) \log (\tan (e+f x))}{a^3 f}-\frac {(3 a-2 b) b^2 \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 (a-b)^2 f}+\frac {b^2}{2 a^2 (a-b) f \left (a+b \tan ^2(e+f x)\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 90} \begin {gather*} -\frac {b^2 (3 a-2 b) \log \left (a+b \tan ^2(e+f x)\right )}{2 a^3 f (a-b)^2}-\frac {(a+2 b) \log (\tan (e+f x))}{a^3 f}+\frac {b^2}{2 a^2 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {\cot ^2(e+f x)}{2 a^2 f}-\frac {\log (\cos (e+f x))}{f (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.58, size = 98, normalized size = 0.74 \begin {gather*} -\frac {\frac {\cot ^2(e+f x)}{a^2}+\frac {b^3}{a^3 (a-b) \left (b+a \cot ^2(e+f x)\right )}+\frac {(3 a-2 b) b^2 \log \left (b+a \cot ^2(e+f x)\right )}{a^3 (a-b)^2}+\frac {2 \log (\sin (e+f x))}{(a-b)^2}}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Cot[e + f*x]^2/a^2 + b^3/(a^3*(a - b)*(b + a*Cot[e + f*x]^2)) + ((3*a - 2*b)*b^2*Log[b + a*Cot[e + f*x]^
2])/(a^3*(a - b)^2) + (2*Log[Sin[e + f*x]])/(a - b)^2)/f

________________________________________________________________________________________

Maple [A]
time = 0.33, size = 161, normalized size = 1.22

method	result	size
derivativedivides	\(\frac {-\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a -2 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{2 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (-a -2 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{2 a^{3}}-\frac {b^{2} \left (\frac {a b}{\left (a -b \right )^{2} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}+\frac {\left (3 a -2 b \right ) \ln \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}{\left (a -b \right )^{2}}\right )}{2 a^{3}}}{f}\)	\(161\)
default	\(\frac {-\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-a -2 b \right ) \ln \left (\cos \left (f x +e \right )+1\right )}{2 a^{3}}+\frac {1}{4 a^{2} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (-a -2 b \right ) \ln \left (\cos \left (f x +e \right )-1\right )}{2 a^{3}}-\frac {b^{2} \left (\frac {a b}{\left (a -b \right )^{2} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}+\frac {\left (3 a -2 b \right ) \ln \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )}{\left (a -b \right )^{2}}\right )}{2 a^{3}}}{f}\)	\(161\)
norman	\(\frac {-\frac {1}{2 a f}+\frac {\left (-a \,b^{2}+2 b^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 a^{2} f b \left (a -b \right )}}{\tan \left (f x +e \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (a +2 b \right ) \ln \left (\tan \left (f x +e \right )\right )}{a^{3} f}-\frac {b^{2} \left (3 a -2 b \right ) \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 a^{3} f \left (a^{2}-2 a b +b^{2}\right )}\)	\(165\)
risch	\(-\frac {i x}{a^{2}-2 a b +b^{2}}+\frac {2 i x}{a^{2}}+\frac {2 i e}{a^{2} f}+\frac {4 i b x}{a^{3}}+\frac {4 i b e}{a^{3} f}+\frac {6 i b^{2} x}{a^{2} \left (a^{2}-2 a b +b^{2}\right )}+\frac {6 i b^{2} e}{a^{2} f \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i b^{3} x}{a^{3} \left (a^{2}-2 a b +b^{2}\right )}-\frac {4 i b^{3} e}{a^{3} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {2 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-6 a^{2} b \,{\mathrm e}^{6 i \left (f x +e \right )}+6 a \,b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+4 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}-4 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-4 a \,b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+8 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+2 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-6 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+6 a \,b^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{a^{2} f \left (a -b \right )^{2} \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}-b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a -b \right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{a^{2} f}-\frac {2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right ) b}{a^{3} f}-\frac {3 b^{2} \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a -b}+1\right )}{2 a^{2} f \left (a^{2}-2 a b +b^{2}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a -b}+1\right )}{a^{3} f \left (a^{2}-2 a b +b^{2}\right )}\)	\(562\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 192, normalized size = 1.45 \begin {gather*} -\frac {\frac {{\left (3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}} - \frac {a^{3} - 2 \, a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - 2 \, b^{3}\right )} \sin \left (f x + e\right )^{2}}{{\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \sin \left (f x + e\right )^{4} - {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \sin \left (f x + e\right )^{2}} + \frac {{\left (a + 2 \, b\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{3}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (131) = 262\).
time = 1.48, size = 304, normalized size = 2.30 \begin {gather*} -\frac {{\left (a^{3} b - 2 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{4} + a^{4} - 2 \, a^{3} b + a^{2} b^{2} + {\left (a^{4} - a^{3} b - a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2} + {\left ({\left (a^{3} b - 3 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (a^{4} - 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {\tan \left (f x + e\right )^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left ({\left (3 \, a b^{3} - 2 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{2} b^{2} - 2 \, a b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left ({\left (a^{5} b - 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} f \tan \left (f x + e\right )^{4} + {\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \tan \left (f x + e\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a^3*b - 2*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^4 + a^4 - 2*a^3*b + a^2*b^2 + (a^4 - a^3*b - a^2*b^2 + 2*a*b^
3)*tan(f*x + e)^2 + ((a^3*b - 3*a*b^3 + 2*b^4)*tan(f*x + e)^4 + (a^4 - 3*a^2*b^2 + 2*a*b^3)*tan(f*x + e)^2)*lo
g(tan(f*x + e)^2/(tan(f*x + e)^2 + 1)) + ((3*a*b^3 - 2*b^4)*tan(f*x + e)^4 + (3*a^2*b^2 - 2*a*b^3)*tan(f*x + e
)^2)*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)))/((a^5*b - 2*a^4*b^2 + a^3*b^3)*f*tan(f*x + e)^4 + (a^6
- 2*a^5*b + a^4*b^2)*f*tan(f*x + e)^2)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3526 vs. \(2 (107) = 214\).
time = 129.01, size = 3526, normalized size = 26.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(e, 0) & Eq(f, 0)), ((log(tan(e + f*x)**2 + 1)/(2*f) - log(tan(e + f
*x))/f - 1/(2*f*tan(e + f*x)**2))/a**2, Eq(b, 0)), ((log(tan(e + f*x)**2 + 1)/(2*f) - log(tan(e + f*x))/f - 1/
(2*f*tan(e + f*x)**2) + 1/(4*f*tan(e + f*x)**4) - 1/(6*f*tan(e + f*x)**6))/b**2, Eq(a, 0)), (6*log(tan(e + f*x
)**2 + 1)*tan(e + f*x)**6/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*tan(e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) + 1
2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**4/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*tan(e + f*x)**4 + 4*b**2*f*tan
(e + f*x)**2) + 6*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*tan(e + f*x)**
4 + 4*b**2*f*tan(e + f*x)**2) - 12*log(tan(e + f*x))*tan(e + f*x)**6/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*tan(
e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) - 24*log(tan(e + f*x))*tan(e + f*x)**4/(4*b**2*f*tan(e + f*x)**6 + 8*b
**2*f*tan(e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) - 12*log(tan(e + f*x))*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x
)**6 + 8*b**2*f*tan(e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) - 6*tan(e + f*x)**4/(4*b**2*f*tan(e + f*x)**6 + 8*
b**2*f*tan(e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) - 9*tan(e + f*x)**2/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*ta
n(e + f*x)**4 + 4*b**2*f*tan(e + f*x)**2) - 2/(4*b**2*f*tan(e + f*x)**6 + 8*b**2*f*tan(e + f*x)**4 + 4*b**2*f*
tan(e + f*x)**2), Eq(a, b)), (zoo*x/a**2, Eq(e, -f*x)), (x*cot(e)**3/(a + b*tan(e)**2)**2, Eq(f, 0)), (a**4*lo
g(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan
(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4)
- 2*a**4*log(tan(e + f*x))*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f
*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)*
*4) - a**4/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f
*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) + a**3*b*log(tan(e + f*x)**2
+ 1)*tan(e + f*x)**4/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*
a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - 2*a**3*b*log(ta
n(e + f*x))*tan(e + f*x)**4/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**
2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - a**3*b*ta
n(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**
2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) + 2*a**3*b/(2*a**6*f*tan(
e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4
*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - 3*a**2*b**2*log(-sqrt(-a/b) + tan(e + f*x))*tan(e +
f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f*t
an(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - 3*a**2*b**2*log(sqrt(-a/b) +
tan(e + f*x))*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x
)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) + 6*a**2
*b**2*log(tan(e + f*x))*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*ta
n(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4)
+ 3*a**2*b**2*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x
)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - a**2*b
**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e
+ f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) - 3*a*b**3*log(-sqrt(-a/b) + tan(e
+ f*x))*tan(e + f*x)**4/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 -
4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) + 2*a*b**3*log(
-sqrt(-a/b) + tan(e + f*x))*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*
f*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)
**4) - 3*a*b**3*log(sqrt(-a/b) + tan(e + f*x))*tan(e + f*x)**4/(2*a**6*f*tan(e + f*x)**2 + 2*a**5*b*f*tan(e +
f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e + f*x)**2 + 2*a**3*
b**3*f*tan(e + f*x)**4) + 2*a*b**3*log(sqrt(-a/b) + tan(e + f*x))*tan(e + f*x)**2/(2*a**6*f*tan(e + f*x)**2 +
2*a**5*b*f*tan(e + f*x)**4 - 4*a**5*b*f*tan(e + f*x)**2 - 4*a**4*b**2*f*tan(e + f*x)**4 + 2*a**4*b**2*f*tan(e
+ f*x)**2 + 2*a**3*b**3*f*tan(e + f*x)**4) + 6*...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (131) = 262\).
time = 1.11, size = 684, normalized size = 5.18 \begin {gather*} -\frac {\frac {12 \, {\left (3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (a + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}} - \frac {24 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, a^{4} - 6 \, a^{3} b + 3 \, a^{2} b^{2} + \frac {10 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {24 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {42 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {20 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {11 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {22 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {27 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {16 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {16 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {12 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {8 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (\frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}} + \frac {12 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{3}} - \frac {3 \, {\left (\cos \left (f x + e\right ) - 1\right )}}{a^{2} {\left (\cos \left (f x + e\right ) + 1\right )}}}{24 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(12*(3*a*b^2 - 2*b^3)*log(a + 2*a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 4*b*(cos(f*x + e) - 1)/(cos(f*
x + e) + 1) + a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/(a^5 - 2*a^4*b + a^3*b^2) - 24*log(abs(-(cos(f*x +
e) - 1)/(cos(f*x + e) + 1) + 1))/(a^2 - 2*a*b + b^2) - (3*a^4 - 6*a^3*b + 3*a^2*b^2 + 10*a^4*(cos(f*x + e) - 1
)/(cos(f*x + e) + 1) - 24*a^3*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 42*a^2*b^2*(cos(f*x + e) - 1)/(cos(f*x
+ e) + 1) - 20*a*b^3*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 11*a^4*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2
- 22*a^3*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 27*a^2*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 -
16*a*b^3*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 16*b^4*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 4*a^4
*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 + 12*a^2*b^2*(cos(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3 - 8*a*b^3*(c
os(f*x + e) - 1)^3/(cos(f*x + e) + 1)^3)/((a^5 - 2*a^4*b + a^3*b^2)*(a*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) +
2*a*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 4*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + a*(cos(f*x +
e) - 1)^3/(cos(f*x + e) + 1)^3)) + 12*(a + 2*b)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x + e) + 1))/a^3 - 3*(cos
(f*x + e) - 1)/(a^2*(cos(f*x + e) + 1)))/f

________________________________________________________________________________________

Mupad [B]
time = 11.88, size = 144, normalized size = 1.09 \begin {gather*} \frac {\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )\,\left (\frac {b}{a^3}+\frac {1}{2\,a^2}-\frac {1}{2\,{\left (a-b\right )}^2}\right )}{f}-\frac {\frac {1}{2\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a\,b-2\,b^2\right )}{2\,a^2\,\left (a-b\right )}}{f\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^4+a\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}+\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,{\left (a-b\right )}^2}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a+2\,b\right )}{a^3\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________