3.485 \(\int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=211 \[ -\frac{b \left (6 a^2 b^2+a^4+3 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b^3 \left (9 a^2 b^2+10 a^4+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac{3 b \log (\sin (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]

[Out]

-((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (3*b*Log[Sin[c + d*x]])/(a^4*d) + (b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log
[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2 + 3*b^2))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan
[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a^3*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.602136, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3569, 3649, 3651, 3530, 3475} \[ -\frac{b \left (6 a^2 b^2+a^4+3 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b^3 \left (9 a^2 b^2+10 a^4+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac{a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac{3 b \log (\sin (c+d x))}{a^4 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3) - (3*b*Log[Sin[c + d*x]])/(a^4*d) + (b^3*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log
[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^3*d) - (b*(2*a^2 + 3*b^2))/(2*a^2*(a^2 + b^2)*d*(a + b*Tan
[c + d*x])^2) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^2) - (b*(a^4 + 6*a^2*b^2 + 3*b^4))/(a^3*(a^2 + b^2)^2*d
*(a + b*Tan[c + d*x]))

Rule 3569

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) &&  !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3651

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[((a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d
))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac{\int \frac{\cot (c+d x) \left (3 b+a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{a}\\ &=-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac{\int \frac{\cot (c+d x) \left (6 b \left (a^2+b^2\right )+2 a^3 \tan (c+d x)+2 b \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2 \left (a^2+b^2\right )}\\ &=-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac{b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (6 b \left (a^2+b^2\right )^2+2 a^3 \left (a^2-b^2\right ) \tan (c+d x)+2 b \left (a^4+6 a^2 b^2+3 b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac{b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{(3 b) \int \cot (c+d x) \, dx}{a^4}+\frac{\left (b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^3}\\ &=-\frac{a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{3 b \log (\sin (c+d x))}{a^4 d}+\frac{b^3 \left (10 a^4+9 a^2 b^2+3 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac{b \left (2 a^2+3 b^2\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac{b \left (a^4+6 a^2 b^2+3 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.82974, size = 178, normalized size = 0.84 \[ -\frac{\frac{b^5}{a^4 \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac{2 b^4 \left (5 a^2+3 b^2\right )}{a^4 \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}-\frac{2 b^3 \left (9 a^2 b^2+10 a^4+3 b^4\right ) \log (a \cot (c+d x)+b)}{a^4 \left (a^2+b^2\right )^3}+\frac{2 \cot (c+d x)}{a^3}+\frac{\log (-\cot (c+d x)+i)}{(b+i a)^3}+\frac{\log (\cot (c+d x)+i)}{(b-i a)^3}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*Cot[c + d*x])/a^3 + b^5/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])^2) - (2*b^4*(5*a^2 + 3*b^2))/(a^4*(a^2 + b^
2)^2*(b + a*Cot[c + d*x])) + Log[I - Cot[c + d*x]]/(I*a + b)^3 + Log[I + Cot[c + d*x]]/((-I)*a + b)^3 - (2*b^3
*(10*a^4 + 9*a^2*b^2 + 3*b^4)*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)^3))/(2*d)

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Maple [A]  time = 0.099, size = 326, normalized size = 1.6 \begin{align*}{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) b{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{1}{d{a}^{3}\tan \left ( dx+c \right ) }}-3\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{4}d}}-{\frac{{b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ){a}^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+10\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+9\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}{a}^{2}}}+3\,{\frac{{b}^{7}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{3}{a}^{4}}}-4\,{\frac{{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}a \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{{b}^{5}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2/(a+b*tan(d*x+c))^3,x)

[Out]

3/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*b*a^2-1/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*b^3-1/d/(a^2+b^2)^3*arctan(tan
(d*x+c))*a^3+3/d/(a^2+b^2)^3*arctan(tan(d*x+c))*a*b^2-1/d/a^3/tan(d*x+c)-3*b*ln(tan(d*x+c))/a^4/d-1/2/d*b^3/(a
^2+b^2)/a^2/(a+b*tan(d*x+c))^2+10/d*b^3/(a^2+b^2)^3*ln(a+b*tan(d*x+c))+9/d*b^5/(a^2+b^2)^3/a^2*ln(a+b*tan(d*x+
c))+3/d*b^7/(a^2+b^2)^3/a^4*ln(a+b*tan(d*x+c))-4/d*b^3/(a^2+b^2)^2/a/(a+b*tan(d*x+c))-2/d*b^5/(a^2+b^2)^2/a^3/
(a+b*tan(d*x+c))

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Maxima [A]  time = 1.67137, size = 471, normalized size = 2.23 \begin{align*} -\frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (10 \, a^{4} b^{3} + 9 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \, a^{6} + 4 \, a^{4} b^{2} + 2 \, a^{2} b^{4} + 2 \,{\left (a^{4} b^{2} + 6 \, a^{2} b^{4} + 3 \, b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (4 \, a^{5} b + 17 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} + \frac{6 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(10*a^4*b^3 + 9*a^2*b^5 + 3*b^7)*log
(b*tan(d*x + c) + a)/(a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)/(a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6) + (2*a^6 + 4*a^4*b^2 + 2*a^2*b^4 + 2*(a^4*b^2 + 6*a^2*b^4 + 3*b^6)*tan(d*x + c)^
2 + (4*a^5*b + 17*a^3*b^3 + 9*a*b^5)*tan(d*x + c))/((a^7*b^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^3 + 2*(a^8*b
+ 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^2 + (a^9 + 2*a^7*b^2 + a^5*b^4)*tan(d*x + c)) + 6*b*log(tan(d*x + c))/a^4)
/d

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Fricas [B]  time = 2.38101, size = 1269, normalized size = 6.01 \begin{align*} -\frac{2 \, a^{9} + 6 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 2 \, a^{3} b^{6} -{\left (9 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - 2 \,{\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{7} b^{2} - 2 \, a^{5} b^{4} + 6 \, a^{3} b^{6} + 3 \, a b^{8} + 2 \,{\left (a^{8} b - 3 \, a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left ({\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) -{\left ({\left (10 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 3 \, b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (10 \, a^{5} b^{4} + 9 \, a^{3} b^{6} + 3 \, a b^{8}\right )} \tan \left (d x + c\right )^{2} +{\left (10 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + 3 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) +{\left (4 \, a^{8} b + 12 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + 9 \, a^{2} b^{7} + 2 \,{\left (a^{9} - 3 \, a^{7} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} +{\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^9 + 6*a^7*b^2 + 6*a^5*b^4 + 2*a^3*b^6 - (9*a^4*b^5 + 3*a^2*b^7 - 2*(a^7*b^2 - 3*a^5*b^4)*d*x)*tan(d*
x + c)^3 + 2*(a^7*b^2 - 2*a^5*b^4 + 6*a^3*b^6 + 3*a*b^8 + 2*(a^8*b - 3*a^6*b^3)*d*x)*tan(d*x + c)^2 + 3*((a^6*
b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 2*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2
+ (a^8*b + 3*a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - ((10*a^4
*b^5 + 9*a^2*b^7 + 3*b^9)*tan(d*x + c)^3 + 2*(10*a^5*b^4 + 9*a^3*b^6 + 3*a*b^8)*tan(d*x + c)^2 + (10*a^6*b^3 +
9*a^4*b^5 + 3*a^2*b^7)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)
) + (4*a^8*b + 12*a^6*b^3 + 23*a^4*b^5 + 9*a^2*b^7 + 2*(a^9 - 3*a^7*b^2)*d*x)*tan(d*x + c))/((a^10*b^2 + 3*a^8
*b^4 + 3*a^6*b^6 + a^4*b^8)*d*tan(d*x + c)^3 + 2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d*tan(d*x + c)^2 +
(a^12 + 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6)*d*tan(d*x + c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.37619, size = 482, normalized size = 2.28 \begin{align*} -\frac{\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (10 \, a^{4} b^{4} + 9 \, a^{2} b^{6} + 3 \, b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} + \frac{30 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} + 27 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 9 \, b^{9} \tan \left (d x + c\right )^{2} + 68 \, a^{5} b^{4} \tan \left (d x + c\right ) + 66 \, a^{3} b^{6} \tan \left (d x + c\right ) + 22 \, a b^{8} \tan \left (d x + c\right ) + 39 \, a^{6} b^{3} + 41 \, a^{4} b^{5} + 14 \, a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac{6 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{2 \,{\left (3 \, b \tan \left (d x + c\right ) - a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1
)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(10*a^4*b^4 + 9*a^2*b^6 + 3*b^8)*log(abs(b*tan(d*x + c) + a))/(a^10*
b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7) + (30*a^4*b^5*tan(d*x + c)^2 + 27*a^2*b^7*tan(d*x + c)^2 + 9*b^9*tan(d*x
+ c)^2 + 68*a^5*b^4*tan(d*x + c) + 66*a^3*b^6*tan(d*x + c) + 22*a*b^8*tan(d*x + c) + 39*a^6*b^3 + 41*a^4*b^5 +
14*a^2*b^7)/((a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*(b*tan(d*x + c) + a)^2) + 6*b*log(abs(tan(d*x + c)))/a^
4 - 2*(3*b*tan(d*x + c) - a)/(a^4*tan(d*x + c)))/d