3.5.64 \(\int \frac {\cot ^2(c+d x)}{a+b \tan (c+d x)} \, dx\) [464]

Optimal. Leaf size=81 \[ -\frac {a x}{a^2+b^2}-\frac {\cot (c+d x)}{a d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right ) d} \]

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3650, 3732, 3611, 3556} \begin {gather*} -\frac {a x}{a^2+b^2}+\frac {b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )}-\frac {b \log (\sin (c+d x))}{a^2 d}-\frac {\cot (c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.50, size = 96, normalized size = 1.19 \begin {gather*} -\frac {\frac {\cot (c+d x)}{a}-\frac {\log (i-\cot (c+d x))}{2 (i a+b)}+\frac {\log (i+\cot (c+d x))}{2 (i a-b)}-\frac {b^3 \log (b+a \cot (c+d x))}{a^2 \left (a^2+b^2\right )}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.26, size = 94, normalized size = 1.16

method	result	size
derivativedivides	\(\frac {-\frac {1}{a \tan \left (d x +c \right )}-\frac {b \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{2}}+\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(94\)
default	\(\frac {-\frac {1}{a \tan \left (d x +c \right )}-\frac {b \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a^{2}}+\frac {\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-a \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(94\)
norman	\(\frac {-\frac {1}{d a}-\frac {a x \tan \left (d x +c \right )}{a^{2}+b^{2}}}{\tan \left (d x +c \right )}+\frac {b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} d \left (a^{2}+b^{2}\right )}-\frac {b \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}\)	\(111\)
risch	\(\frac {x}{i b -a}+\frac {2 i b x}{a^{2}}+\frac {2 i b c}{a^{2} d}-\frac {2 i b^{3} x}{a^{2} \left (a^{2}+b^{2}\right )}-\frac {2 i b^{3} c}{a^{2} d \left (a^{2}+b^{2}\right )}-\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{2} d \left (a^{2}+b^{2}\right )}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.59, size = 100, normalized size = 1.23 \begin {gather*} \frac {\frac {2 \, b^{3} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + a^{2} b^{2}} - \frac {2 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b \log \left (\tan \left (d x + c\right )\right )}{a^{2}} - \frac {2}{a \tan \left (d x + c\right )}}{2 \, 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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.92, size = 140, normalized size = 1.73 \begin {gather*} -\frac {2 \, a^{3} d x \tan \left (d x + c\right ) - b^{3} \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 ) \tan \left (d x + c\right ) + 2 \, a^{3} + 2 \, a b^{2} + {\left (a^{2} b + b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\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)),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.51, size = 1080, normalized size = 13.33 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge c = 0 \wedge d = 0 \\\frac {- x - \frac {\cot {\left (c + d x \right )}}{d}}{a} & \text {for}\: b = 0 \\\frac {\frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {\log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {1}{2 d \tan ^{2}{\left (c + d x \right )}}}{b} & \text {for}\: a = 0 \\- \frac {3 i d x \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} - \frac {3 d x \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} + \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} + \frac {2 \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} - \frac {2 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} - \frac {3 i \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} - \frac {2}{2 b d \tan ^{2}{\left (c + d x \right )} - 2 i b d \tan {\left (c + d x \right )}} & \text {for}\: a = - i b \\\frac {3 i d x \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} - \frac {3 d x \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} - \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} - \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} + \frac {2 \log {\left (\tan {\left (c + d x \right )} \right )} \tan ^{2}{\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} + \frac {2 i \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} + \frac {3 i \tan {\left (c + d x \right )}}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} - \frac {2}{2 b d \tan ^{2}{\left (c + d x \right )} + 2 i b d \tan {\left (c + d x \right )}} & \text {for}\: a = i b \\\frac {\tilde {\infty } x}{a} & \text {for}\: c = - d x \\\frac {x \cot ^{2}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {2 a^{3} d x \tan {\left (c + d x \right )}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} - \frac {2 a^{3}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} + \frac {a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} - \frac {2 a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} - \frac {2 a b^{2}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} + \frac {2 b^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} - \frac {2 b^{3} \log {\left (\tan {\left (c + d x \right )} \right )} \tan {\left (c + d x \right )}}{2 a^{4} d \tan {\left (c + d x \right )} + 2 a^{2} b^{2} d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \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)),x)

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((-x - cot(c + d*x)/d)/a, Eq(b, 0)), ((log(tan(c
+ d*x)**2 + 1)/(2*d) - log(tan(c + d*x))/d - 1/(2*d*tan(c + d*x)**2))/b, Eq(a, 0)), (-3*I*d*x*tan(c + d*x)**2
/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c + d*x)) - 3*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c +
d*x)) - log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c + d*x)) + I*log(tan(c
+ d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c + d*x)) + 2*log(tan(c + d*x))*tan(c + d*x)*
*2/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c + d*x)) - 2*I*log(tan(c + d*x))*tan(c + d*x)/(2*b*d*tan(c + d*x)**2
- 2*I*b*d*tan(c + d*x)) - 3*I*tan(c + d*x)/(2*b*d*tan(c + d*x)**2 - 2*I*b*d*tan(c + d*x)) - 2/(2*b*d*tan(c + d
*x)**2 - 2*I*b*d*tan(c + d*x)), Eq(a, -I*b)), (3*I*d*x*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c
+ d*x)) - 3*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c + d*x)) - log(tan(c + d*x)**2 + 1)*tan(c +
d*x)**2/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c + d*x)) - I*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c
+ d*x)**2 + 2*I*b*d*tan(c + d*x)) + 2*log(tan(c + d*x))*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(
c + d*x)) + 2*I*log(tan(c + d*x))*tan(c + d*x)/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c + d*x)) + 3*I*tan(c + d*
x)/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c + d*x)) - 2/(2*b*d*tan(c + d*x)**2 + 2*I*b*d*tan(c + d*x)), Eq(a, I*
b)), (zoo*x/a, Eq(c, -d*x)), (x*cot(c)**2/(a + b*tan(c)), Eq(d, 0)), (-2*a**3*d*x*tan(c + d*x)/(2*a**4*d*tan(c
+ d*x) + 2*a**2*b**2*d*tan(c + d*x)) - 2*a**3/(2*a**4*d*tan(c + d*x) + 2*a**2*b**2*d*tan(c + d*x)) + a**2*b*l
og(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**4*d*tan(c + d*x) + 2*a**2*b**2*d*tan(c + d*x)) - 2*a**2*b*log(tan(c
+ d*x))*tan(c + d*x)/(2*a**4*d*tan(c + d*x) + 2*a**2*b**2*d*tan(c + d*x)) - 2*a*b**2/(2*a**4*d*tan(c + d*x) +
2*a**2*b**2*d*tan(c + d*x)) + 2*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**4*d*tan(c + d*x) + 2*a**2*b**
2*d*tan(c + d*x)) - 2*b**3*log(tan(c + d*x))*tan(c + d*x)/(2*a**4*d*tan(c + d*x) + 2*a**2*b**2*d*tan(c + d*x))
, True))

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Giac [A]
time = 0.72, size = 116, normalized size = 1.43 \begin {gather*} \frac {\frac {2 \, b^{4} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + a^{2} b^{3}} - \frac {2 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {2 \, {\left (b \tan \left (d x + c\right ) - a\right )}}{a^{2} \tan \left (d x + c\right )}}{2 \, 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)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.16, size = 108, normalized size = 1.33 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {\mathrm {cot}\left (c+d\,x\right )}{a\,d}-\frac {b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}+\frac {b^3\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\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)),x)

[Out]

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

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