3.5.75 \(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [475]

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

[Out]

-(a^2-b^2)*x/(a^2+b^2)^2-2*b*ln(sin(d*x+c))/a^3/d+2*b^3*(2*a^2+b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^3/(a^2+b^2
)^2/d-b*(a^2+2*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))-cot(d*x+c)/a/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.26, antiderivative size = 150, normalized size of antiderivative = 1.00, 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 = {3650, 3730, 3732, 3611, 3556} \begin {gather*} -\frac {2 b \log (\sin (c+d x))}{a^3 d}-\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 {x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}+\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 {\cot (c+d x)}{a d (a+b \tan (c+d x))} \end {gather*}

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 3556

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

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], 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 3650

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 3730

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*Ta
n[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 3732

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]

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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.78, size = 136, normalized size = 0.91 \begin {gather*} -\frac {\frac {\cot (c+d x)}{a^2}-\frac {b^4}{a^3 \left (a^2+b^2\right ) (b+a \cot (c+d x))}+\frac {i \log (i-\cot (c+d x))}{2 (a-i b)^2}-\frac {i \log (i+\cot (c+d x))}{2 (a+i b)^2}-\frac {2 b^3 \left (2 a^2+b^2\right ) \log (b+a \cot (c+d x))}{a^3 \left (a^2+b^2\right )^2}}{d} \end {gather*}

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 = 0.30, size = 140, normalized size = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {1}{a^{2} \tan \left (d x +c \right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}-\frac {b^{3}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(140\)
default	\(\frac {-\frac {1}{a^{2} \tan \left (d x +c \right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}-\frac {b^{3}}{\left (a^{2}+b^{2}\right ) a^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )+\left (-a^{2}+b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(140\)
norman	\(\frac {\frac {\left (-a^{2} b^{2}-2 b^{4}\right ) \tan \left (d x +c \right )}{a^{2} b d \left (a^{2}+b^{2}\right )}-\frac {1}{d a}-\frac {b \left (a^{2}-b^{2}\right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (a^{2}-b^{2}\right ) a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{\tan \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {2 b^{3} \left (2 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(243\)
risch	\(\frac {x}{2 i a b -a^{2}+b^{2}}+\frac {4 i b x}{a^{3}}+\frac {4 i b c}{a^{3} d}-\frac {8 i b^{3} x}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {8 i b^{3} c}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 i b^{5} x}{a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {4 i b^{5} c}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i \left (a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+a^{4}+2 a^{2} b^{2}+2 b^{4}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) a^{2} d}-\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}+\frac {4 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{3} d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(434\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.53, size = 200, normalized size = 1.33 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (150) = 300\).
time = 1.25, size = 323, normalized size = 2.15 \begin {gather*} -\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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 1.97, size = 4070, normalized size = 27.13 \begin {gather*} \text {Too large to display} \end {gather*}

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]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((x + 1/(d*tan(c + d*x)) - 1/(3*d*tan(c + d*x)**
3))/b**2, Eq(a, 0)), (9*d*x*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*
tan(c + d*x)) - 18*I*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan
(c + d*x)) - 9*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)
) - 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b*
*2*d*tan(c + d*x)) - 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c +
d*x)**2 - 4*b**2*d*tan(c + d*x)) + 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**3 - 8*I*
b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 8*I*log(tan(c + d*x))*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)
**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 16*log(tan(c + d*x))*tan(c + d*x)**2/(4*b**2*d*tan
(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 8*I*log(tan(c + d*x))*tan(c + d*x)/(4*b**
2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 9*tan(c + d*x)**2/(4*b**2*d*tan(c
+ d*x)**3 - 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 14*I*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**3
- 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 4/(4*b**2*d*tan(c + d*x)**3 - 8*I*b**2*d*tan(c + d*x)*
*2 - 4*b**2*d*tan(c + d*x)), Eq(a, -I*b)), (9*d*x*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c
+ d*x)**2 - 4*b**2*d*tan(c + d*x)) + 18*I*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c +
d*x)**2 - 4*b**2*d*tan(c + d*x)) - 9*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 -
4*b**2*d*tan(c + d*x)) + 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*
tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 8*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**
3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 4*I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*
tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 8*I*log(tan(c + d*x))*tan(c + d*x)**3/
(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 16*log(tan(c + d*x))*tan(c +
d*x)**2/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 8*I*log(tan(c + d*x
))*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 9*tan(c + d*
x)**2/(4*b**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) + 14*I*tan(c + d*x)/(4*b
**2*d*tan(c + d*x)**3 + 8*I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)) - 4/(4*b**2*d*tan(c + d*x)**3 + 8*
I*b**2*d*tan(c + d*x)**2 - 4*b**2*d*tan(c + d*x)), Eq(a, I*b)), (zoo*x/a**2, Eq(c, -d*x)), (x*cot(c)**2/(a + b
*tan(c))**2, Eq(d, 0)), ((-x - cot(c + d*x)/d)/a**2, Eq(b, 0)), (-a**6*d*x*tan(c + d*x)/(a**8*d*tan(c + d*x) +
a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d
*x) + a**3*b**5*d*tan(c + d*x)**2) - a**6/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(
c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) - a**5*b*d*
x*tan(c + d*x)**2/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d
*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) + a**5*b*log(tan(c + d*x)**2 + 1)*t
an(c + d*x)/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c
+ d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) - 2*a**5*b*log(tan(c + d*x))*tan(c + d*x)
/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2
+ a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) - a**5*b*tan(c + d*x)/(a**8*d*tan(c + d*x) + a**7*b*
d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a*
*3*b**5*d*tan(c + d*x)**2) + a**4*b**2*d*x*tan(c + d*x)/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a*
*6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**
2) + a**4*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a*
*6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**
2) - 2*a**4*b**2*log(tan(c + d*x))*tan(c + d*x)**2/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b*
*2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c + d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) -
2*a**4*b**2/(a**8*d*tan(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6*b**2*d*tan(c + d*x) + 2*a**5*b**3*d*tan(c
+ d*x)**2 + a**4*b**4*d*tan(c + d*x) + a**3*b**5*d*tan(c + d*x)**2) + a**3*b**3*d*x*tan(c + d*x)**2/(a**8*d*t
an(c + d*x) + a**7*b*d*tan(c + d*x)**2 + 2*a**6...

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Giac [A]
time = 0.88, size = 235, normalized size = 1.57 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 4.35, size = 183, normalized size = 1.22 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {\frac {1}{a}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^2\,b+2\,b^3\right )}{a^2\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {2\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {2\,b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (2\,a^2+b^2\right )}{a^3\,d\,{\left (a^2+b^2\right )}^2}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(tan(c + d*x) + 1i)*1i)/(2*d*(a*b*2i - a^2 + b^2)) + log(tan(c + d*x) - 1i)/(2*d*(2*a*b - a^2*1i + b^2*1i)
) - (1/a + (tan(c + d*x)*(a^2*b + 2*b^3))/(a^2*(a^2 + b^2)))/(d*(a*tan(c + d*x) + b*tan(c + d*x)^2)) - (2*b*lo
g(tan(c + d*x)))/(a^3*d) + (2*b^3*log(a + b*tan(c + d*x))*(2*a^2 + b^2))/(a^3*d*(a^2 + b^2)^2)

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