3.280 \(\int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=192 \[ -\frac{b \left (a^2 A-a b B+2 A b^2\right )}{a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{b^2 \left (4 a^2 A b-3 a^3 B-a b^2 B+2 A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2}-\frac{(2 A b-a B) \log (\sin (c+d x))}{a^3 d}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.541117, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3609, 3649, 3651, 3530, 3475} \[ -\frac{b \left (a^2 A-a b B+2 A b^2\right )}{a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{b^2 \left (4 a^2 A b-3 a^3 B-a b^2 B+2 A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2}-\frac{(2 A b-a B) \log (\sin (c+d x))}{a^3 d}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[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[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(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))}{(a+b \tan (c+d x))^2} \, dx &=-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (2 A b-a B+a A \tan (c+d x)+2 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{a}\\ &=-\frac{b \left (a^2 A+2 A b^2-a b B\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (\left (a^2+b^2\right ) (2 A b-a B)+a^2 (a A+b B) \tan (c+d x)+b \left (a^2 A+2 A b^2-a b B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}\\ &=-\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}-\frac{b \left (a^2 A+2 A b^2-a b B\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{(2 A b-a B) \int \cot (c+d x) \, dx}{a^3}+\frac{\left (b^2 \left (4 a^2 A b+2 A b^3-3 a^3 B-a b^2 B\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}-\frac{(2 A b-a B) \log (\sin (c+d x))}{a^3 d}+\frac{b^2 \left (4 a^2 A b+2 A b^3-3 a^3 B-a b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac{b \left (a^2 A+2 A b^2-a b B\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.27862, size = 193, normalized size = 1.01 \[ \frac{\frac{2 b^2 (a B-A b)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac{2 b^2 \left (-4 a^2 A b+3 a^3 B+a b^2 B-2 A b^3\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^2}+\frac{2 (a B-2 A b) \log (\tan (c+d x))}{a^3}-\frac{2 A \cot (c+d x)}{a^2}+\frac{i (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac{(B+i A) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*A*Cot[c + d*x])/a^2 + (I*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2 + (2*(-2*A*b + a*B)*Log[Tan[c + d*x
]])/a^3 - ((I*A + B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 - (2*b^2*(-4*a^2*A*b - 2*A*b^3 + 3*a^3*B + a*b^2*B)*Lo
g[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)^2) + (2*b^2*(-(A*b) + a*B))/(a^2*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2
*d)

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Maple [B]  time = 0.124, size = 399, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Aab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){a}^{2}B}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ){b}^{2}B}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-2\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) ab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A}{{a}^{2}d\tan \left ( dx+c \right ) }}-2\,{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) Ab}{{a}^{3}d}}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}+4\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{ad \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+2\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{{a}^{3}d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-3\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){b}^{2}B}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{{b}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{{a}^{2}d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A{b}^{3}}{{a}^{2}d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) }}+{\frac{{b}^{2}B}{ad \left ({a}^{2}+{b}^{2} \right ) \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))/(a+b*tan(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.6046, size = 354, normalized size = 1.84 \begin{align*} -\frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (3 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} + B a b^{4} - 2 \, A b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}} + \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (A a^{3} + A a b^{2} +{\left (A a^{2} b - B a b^{2} + 2 \, A b^{3}\right )} \tan \left (d x + c\right )\right )}}{{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{5} + a^{3} b^{2}\right )} \tan \left (d x + c\right )} - \frac{2 \,{\left (B a - 2 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{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))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/2*(2*(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*B*a^3*b^2 - 4*A*a^2*b^3 + B*a*b^4 -
2*A*b^5)*log(b*tan(d*x + c) + a)/(a^7 + 2*a^5*b^2 + a^3*b^4) + (B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x + c)^2 +
1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(A*a^3 + A*a*b^2 + (A*a^2*b - B*a*b^2 + 2*A*b^3)*tan(d*x + c))/((a^4*b + a^2*b
^3)*tan(d*x + c)^2 + (a^5 + a^3*b^2)*tan(d*x + c)) - 2*(B*a - 2*A*b)*log(tan(d*x + c))/a^3)/d

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Fricas [B]  time = 2.39417, size = 1017, normalized size = 5.3 \begin{align*} -\frac{2 \, A a^{6} + 4 \, A a^{4} b^{2} + 2 \, A a^{2} b^{4} + 2 \,{\left (B a^{3} b^{3} - A a^{2} b^{4} +{\left (A a^{5} b + 2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} -{\left ({\left (B a^{5} b - 2 \, A a^{4} b^{2} + 2 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (B a^{6} - 2 \, A a^{5} b + 2 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\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 (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (3 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3} + B a^{2} b^{4} - 2 \, A a b^{5}\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 ) + 2 \,{\left (A a^{5} b + 2 \, A a^{3} b^{3} - B a^{2} b^{4} + 2 \, A a b^{5} +{\left (A a^{6} + 2 \, B a^{5} b - A a^{4} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} d \tan \left (d x + c\right )^{2} +{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\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))/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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))/(a+b*tan(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.28134, size = 489, normalized size = 2.55 \begin{align*} -\frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (3 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4} + B a b^{5} - 2 \, A b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}} + \frac{B a^{4} b \tan \left (d x + c\right )^{2} - 2 \, A a^{3} b^{2} \tan \left (d x + c\right )^{2} - B a^{2} b^{3} \tan \left (d x + c\right )^{2} + B a^{5} \tan \left (d x + c\right ) - 3 \, B a^{3} b^{2} \tan \left (d x + c\right ) + 6 \, A a^{2} b^{3} \tan \left (d x + c\right ) - 2 \, B a b^{4} \tan \left (d x + c\right ) + 4 \, A b^{5} \tan \left (d x + c\right ) + 2 \, A a^{5} + 4 \, A a^{3} b^{2} + 2 \, A a b^{4}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )}{\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}} - \frac{2 \,{\left (B a - 2 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}}}{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))/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

-1/2*(2*(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + (B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x +
c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(3*B*a^3*b^3 - 4*A*a^2*b^4 + B*a*b^5 - 2*A*b^6)*log(abs(b*tan(d*x + c) +
a))/(a^7*b + 2*a^5*b^3 + a^3*b^5) + (B*a^4*b*tan(d*x + c)^2 - 2*A*a^3*b^2*tan(d*x + c)^2 - B*a^2*b^3*tan(d*x
+ c)^2 + B*a^5*tan(d*x + c) - 3*B*a^3*b^2*tan(d*x + c) + 6*A*a^2*b^3*tan(d*x + c) - 2*B*a*b^4*tan(d*x + c) + 4
*A*b^5*tan(d*x + c) + 2*A*a^5 + 4*A*a^3*b^2 + 2*A*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(d*x + c)^2 + a*ta
n(d*x + c))) - 2*(B*a - 2*A*b)*log(abs(tan(d*x + c)))/a^3)/d