\(\int \frac {\cot ^2(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{a+b \tan (c+d x)} \, dx\) [29]

Optimal result

Integrand size = 40, antiderivative size = 80 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {(b B-a C) x}{a^2+b^2}+\frac {B \log (\sin (c+d x))}{a d}-\frac {b (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3713, 3692, 3611, 3556} \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {b (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {x (b B-a C)}{a^2+b^2}+\frac {B \log (\sin (c+d x))}{a d} \]

[In]

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

[Out]

-(((b*B - a*C)*x)/(a^2 + b^2)) + (B*Log[Sin[c + d*x]])/(a*d) - (b*(b*B - a*C)*Log[a*Cos[c + d*x] + b*Sin[c + d
*x]])/(a*(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 3692

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

Rule 3713

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=-\frac {\frac {(B+i C) \log (i-\tan (c+d x))}{a+i b}-\frac {2 B \log (\tan (c+d x))}{a}+\frac {(B-i C) \log (i+\tan (c+d x))}{a-i b}+\frac {2 b (b B-a C) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19

method	result	size
parallelrisch	\(\frac {\left (-2 B \,b^{2}+2 C a b \right ) \ln \left (a +b \tan \left (d x +c \right )\right )+\left (-B \,a^{2}-C a b \right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 B \left (a^{2}+b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-2 a d x \left (B b -C a \right )}{2 \left (a^{2}+b^{2}\right ) a d}\)	\(95\)
derivativedivides	\(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a}+\frac {\frac {\left (-B a -C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B b +C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (B b -C a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a}}{d}\)	\(101\)
default	\(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a}+\frac {\frac {\left (-B a -C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B b +C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {\left (B b -C a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a}}{d}\)	\(101\)
norman	\(-\frac {\left (B b -C a \right ) x}{a^{2}+b^{2}}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a d}-\frac {\left (B a +C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {\left (B b -C a \right ) b \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right ) a d}\)	\(106\)
risch	\(-\frac {i x B}{i b -a}-\frac {x C}{i b -a}+\frac {2 i b^{2} B x}{\left (a^{2}+b^{2}\right ) a}+\frac {2 i b^{2} B c}{\left (a^{2}+b^{2}\right ) a d}-\frac {2 i b C x}{a^{2}+b^{2}}-\frac {2 i b C c}{\left (a^{2}+b^{2}\right ) d}-\frac {2 i B x}{a}-\frac {2 i B c}{a d}-\frac {b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{2}+b^{2}\right ) a d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{2}+b^{2}\right ) d}+\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a d}\)	\(240\)

[In]

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

[Out]

1/2*((-2*B*b^2+2*C*a*b)*ln(a+b*tan(d*x+c))+(-B*a^2-C*a*b)*ln(sec(d*x+c)^2)+2*B*(a^2+b^2)*ln(tan(d*x+c))-2*a*d*
x*(B*b-C*a))/(a^2+b^2)/a/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {2 \, {\left (C a^{2} - B a b\right )} d x + {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (C a b - B b^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.17 (sec) , antiderivative size = 966, normalized size of antiderivative = 12.08 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*x*(B*tan(c) + C*tan(c)**2)*cot(c)**2/tan(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-B*log(tan(c +
d*x)**2 + 1)/(2*d) + B*log(tan(c + d*x))/d + C*x)/a, Eq(b, 0)), ((-B*x - B/(d*tan(c + d*x)) - C*log(tan(c + d*
x)**2 + 1)/(2*d) + C*log(tan(c + d*x))/d)/b, Eq(a, 0)), (B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) - I
*B*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) - I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*
d) - B*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 2*I*B*log(tan(c + d*x))*tan(c + d*x)/(2*b*d*t
an(c + d*x) - 2*I*b*d) + 2*B*log(tan(c + d*x))/(2*b*d*tan(c + d*x) - 2*I*b*d) + B/(2*b*d*tan(c + d*x) - 2*I*b*
d) + I*C*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + C*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) + I*C/(2*b*d*t
an(c + d*x) - 2*I*b*d), Eq(a, -I*b)), (B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + I*B*d*x/(2*b*d*tan(
c + d*x) + 2*I*b*d) + I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - B*log(tan(c +
d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 2*I*B*log(tan(c + d*x))*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*
b*d) + 2*B*log(tan(c + d*x))/(2*b*d*tan(c + d*x) + 2*I*b*d) + B/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*C*d*x*tan(c
+ d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + C*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*C/(2*b*d*tan(c + d*x) + 2*I*
b*d), Eq(a, I*b)), (x*(B*tan(c) + C*tan(c)**2)*cot(c)**2/(a + b*tan(c)), Eq(d, 0)), (-B*a**2*log(tan(c + d*x)*
*2 + 1)/(2*a**3*d + 2*a*b**2*d) + 2*B*a**2*log(tan(c + d*x))/(2*a**3*d + 2*a*b**2*d) - 2*B*a*b*d*x/(2*a**3*d +
2*a*b**2*d) - 2*B*b**2*log(a/b + tan(c + d*x))/(2*a**3*d + 2*a*b**2*d) + 2*B*b**2*log(tan(c + d*x))/(2*a**3*d
+ 2*a*b**2*d) + 2*C*a**2*d*x/(2*a**3*d + 2*a*b**2*d) + 2*C*a*b*log(a/b + tan(c + d*x))/(2*a**3*d + 2*a*b**2*d
) - C*a*b*log(tan(c + d*x)**2 + 1)/(2*a**3*d + 2*a*b**2*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.34 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b - B b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, B \log \left (\tan \left (d x + c\right )\right )}{a}}{2 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.80 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b^{2} - B b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}} + \frac {2 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a}}{2 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.44 \[ \int \frac {\cot ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx=\frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b-C\,a\right )}{a\,d\,\left (a^2+b^2\right )} \]

[In]

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

[Out]

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