3.5.74 \(\int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [474]

Optimal. Leaf size=107 \[ -\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b^2}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3650, 3732, 3611, 3556} \begin {gather*} \frac {b^2}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )^2}-\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {\log (\sin (c+d x))}{a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.72, size = 154, normalized size = 1.44 \begin {gather*} \frac {-\frac {a (a-i b) \log (i-\tan (c+d x))}{2 (a+i b)}+\frac {\left (a^2+b^2\right ) \log (\tan (c+d x))}{a}-\frac {a (a+i b) \log (i+\tan (c+d x))}{2 (a-i b)}-\frac {b^2 \left (3 a^2+b^2\right ) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}+\frac {b^2}{a+b \tan (c+d x)}}{a \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/2*(a*(a - I*b)*Log[I - Tan[c + d*x]])/(a + I*b) + ((a^2 + b^2)*Log[Tan[c + d*x]])/a - (a*(a + I*b)*Log[I +
Tan[c + d*x]])/(2*(a - I*b)) - (b^2*(3*a^2 + b^2)*Log[a + b*Tan[c + d*x]])/(a*(a^2 + b^2)) + b^2/(a + b*Tan[c
+ d*x]))/(a*(a^2 + b^2)*d)

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Maple [A]
time = 0.30, size = 126, normalized size = 1.18

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {b^{2}}{\left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}+\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(126\)
default	\(\frac {\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {b^{2}}{\left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2}}+\frac {\frac {\left (-a^{2}+b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}-2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(126\)
norman	\(\frac {-\frac {b^{3} \tan \left (d x +c \right )}{d \,a^{2} \left (a^{2}+b^{2}\right )}-\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 b^{2} a x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (3 a^{2}+b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}\)	\(200\)
risch	\(-\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {6 i b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {6 i b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i b^{4} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {2 i b^{4} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}-\frac {2 i x}{a^{2}}-\frac {2 i c}{a^{2} d}+\frac {2 i b^{3}}{\left (-i a +b \right ) d a \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 164, normalized size = 1.53 \begin {gather*} -\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, b^{2}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}} + \frac {{\left (a^{2} - 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 \, \log \left (\tan \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (107) = 214\).
time = 0.87, size = 235, normalized size = 2.20 \begin {gather*} -\frac {4 \, a^{4} b d x - 2 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + 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 (3 \, a^{3} b^{2} + a b^{4} + {\left (3 \, a^{2} b^{3} + 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 (2 \, a^{3} b^{2} d x + a^{2} b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.67, size = 2927, normalized size = 27.36 \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)/(a+b*tan(d*x+c))**2,x)

[Out]

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

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Giac [A]
time = 0.92, size = 206, normalized size = 1.93 \begin {gather*} -\frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - 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 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} - \frac {2 \, {\left (3 \, a^{2} b^{3} \tan \left (d x + c\right ) + b^{5} \tan \left (d x + c\right ) + 4 \, a^{3} b^{2} + 2 \, a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} - \frac {2 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.29, size = 152, normalized size = 1.42 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}+\frac {b^2}{a\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^2+b^2\right )}{a^2\,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\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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