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

Optimal. Leaf size=150 \[ -\frac{b \left (a^2+2 b^2\right )}{a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{2 b^3 \left (2 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.364979, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{b \left (a^2+2 b^2\right )}{a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{2 b^3 \left (2 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac{2 b \log (\sin (c+d x))}{a^3 d}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a^2 - b^2)*x)/(a^2 + b^2)^2) - (2*b*Log[Sin[c + d*x]])/(a^3*d) + (2*b^3*(2*a^2 + b^2)*Log[a*Cos[c + d*x] +
b*Sin[c + d*x]])/(a^3*(a^2 + b^2)^2*d) - (b*(a^2 + 2*b^2))/(a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) - Cot[c +
d*x]/(a*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))^2} \, dx &=-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (2 b+a \tan (c+d x)+2 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{a}\\ &=-\frac{b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{\int \frac{\cot (c+d x) \left (2 b \left (a^2+b^2\right )+a^3 \tan (c+d x)+b \left (a^2+2 b^2\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-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b \left (a^2+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))}-\frac{(2 b) \int \cot (c+d x) \, dx}{a^3}+\frac{\left (2 b^3 \left (2 a^2+b^2\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-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{2 b \log (\sin (c+d x))}{a^3 d}+\frac{2 b^3 \left (2 a^2+b^2\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+2 b^2\right )}{a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 0.683548, size = 136, normalized size = 0.91 \[ -\frac{-\frac{b^4}{a^3 \left (a^2+b^2\right ) (a \cot (c+d x)+b)}-\frac{2 b^3 \left (2 a^2+b^2\right ) \log (a \cot (c+d x)+b)}{a^3 \left (a^2+b^2\right )^2}+\frac{\cot (c+d x)}{a^2}+\frac{i \log (-\cot (c+d x)+i)}{2 (a-i b)^2}-\frac{i \log (\cot (c+d x)+i)}{2 (a+i b)^2}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.079, size = 201, normalized size = 1.3 \begin{align*} -{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{ab\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{1}{{a}^{2}d\tan \left ( dx+c \right ) }}-2\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ){a}^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+4\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}a}}+2\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}{a}^{3}}} \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))^2,x)

[Out]

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

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Maxima [A]  time = 1.53871, size = 270, normalized size = 1.8 \begin{align*} \frac{\frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (2 \, a^{2} b^{3} + 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{a^{3} + a b^{2} +{\left (a^{2} b + 2 \, b^{3}\right )} \tan \left (d x + c\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 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.33244, size = 701, normalized size = 4.67 \begin{align*} -\frac{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} -{\left (a^{2} b^{4} -{\left (a^{5} b - a^{3} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left ({\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{5} b + 2 \, a^{3} b^{3} + 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 (2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (2 \, a^{3} b^{3} + 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 ) +{\left (a^{5} b + 2 \, a^{3} b^{3} + 2 \, a b^{5} +{\left (a^{6} - a^{4} b^{2}\right )} d x\right )} \tan \left (d x + c\right )}{{\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 )} \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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.25648, size = 317, normalized size = 2.11 \begin{align*} \frac{\frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (2 \, a^{2} b^{4} + 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{a^{3} b^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{2} b^{3} \tan \left (d x + c\right ) - 2 \, b^{5} \tan \left (d x + c\right ) - a^{5} - 2 \, a^{3} b^{2} - 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 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}}}{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))^2,x, algorithm="giac")

[Out]

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