\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx\) [430]

Optimal result

Integrand size = 21, antiderivative size = 41 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=-\left (\left (a^2-b^2\right ) x\right )-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3623, 3612, 3556} \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=-x \left (a^2-b^2\right )-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d} \]

[In]

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

[Out]

-((a^2 - b^2)*x) - (a^2*Cot[c + d*x])/d + (2*a*b*Log[Sin[c + d*x]])/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 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && NeQ[a*c + b*d, 0]

Rule 3623

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Ta
n[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Tan[e + f*x], x], x], x] /; FreeQ
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.00 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {-2 a^2 \cot (c+d x)+i \left ((a+i b)^2 \log (i-\tan (c+d x))-4 i a b \log (\tan (c+d x))-(a-i b)^2 \log (i+\tan (c+d x))\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 a b \ln \left (\sin \left (d x +c \right )\right )+b^{2} \left (d x +c \right )}{d}\)	\(46\)
default	\(\frac {a^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+2 a b \ln \left (\sin \left (d x +c \right )\right )+b^{2} \left (d x +c \right )}{d}\)	\(46\)
parallelrisch	\(\frac {-a b \ln \left (\sec ^{2}\left (d x +c \right )\right )+2 a b \ln \left (\tan \left (d x +c \right )\right )-\cot \left (d x +c \right ) a^{2}-d x \left (a -b \right ) \left (a +b \right )}{d}\)	\(53\)
norman	\(\frac {\left (-a^{2}+b^{2}\right ) x \tan \left (d x +c \right )-\frac {a^{2}}{d}}{\tan \left (d x +c \right )}+\frac {2 a b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\)	\(69\)
risch	\(-2 i a b x -a^{2} x +b^{2} x -\frac {4 i a b c}{d}-\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(69\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.63 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {{\left (a^{2} - b^{2}\right )} d x \tan \left (d x + c\right ) - a b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + a^{2}}{d \tan \left (d x + c\right )} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (34) = 68\).

Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.02 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=\begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \cot ^{2}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\- a^{2} x - \frac {a^{2}}{d \tan {\left (c + d x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + b^{2} x & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**2*(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*a**2*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**2*cot(c)**2, Eq(d, 0)), (zoo*a**2*x, Eq(c, -d*
x)), (-a**2*x - a**2/(d*tan(c + d*x)) - a*b*log(tan(c + d*x)**2 + 1)/d + 2*a*b*log(tan(c + d*x))/d + b**2*x, T
rue))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, a b \log \left (\tan \left (d x + c\right )\right ) + {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {a^{2}}{\tan \left (d x + c\right )}}{d} \]

[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) - 2*a*b*log(tan(d*x + c)) + (a^2 - b^2)*(d*x + c) + a^2/tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (41) = 82\).

Time = 0.62 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.39 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=-\frac {4 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.93 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx=\frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {a^2\,\mathrm {cot}\left (c+d\,x\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d} \]

[In]

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

[Out]

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