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

Optimal. Leaf size=80 \[ -\frac {b (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {x (b B-a C)}{a^2+b^2}+\frac {B \log (\sin (c+d x))}{a d} \]

[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

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Rubi [A]  time = 0.20, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3632, 3611, 3530, 3475} \[ -\frac {b (b B-a C) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac {x (b B-a C)}{a^2+b^2}+\frac {B \log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[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 3475

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), 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 3611

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 3632

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*} \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 &=\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*}

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Mathematica [C]  time = 0.37, size = 113, normalized size = 1.41 \[ -\frac {\frac {2 b (b B-a C) \log (a+b \tan (c+d x))}{a \left (a^2+b^2\right )}+\frac {(B+i C) \log (-\tan (c+d x)+i)}{a+i b}+\frac {(B-i C) \log (\tan (c+d x)+i)}{a-i b}-\frac {2 B \log (\tan (c+d x))}{a}}{2 d} \]

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.67, size = 118, normalized size = 1.48 \[ \frac {2 \, {\left (C a^{2} - B a b\right )} d x + {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (C a b - B b^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 4.11, size = 113, normalized size = 1.41 \[ \frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b^{2} - B b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}} + \frac {2 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.94, size = 174, normalized size = 2.18 \[ -\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d a \left (a^{2}+b^{2}\right )}+\frac {b \ln \left (a +b \tan \left (d x +c \right )\right ) C}{d \left (a^{2}+b^{2}\right )}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{d a}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a B}{2 d \left (a^{2}+b^{2}\right )}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C b}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b}{d \left (a^{2}+b^{2}\right )}+\frac {C \arctan \left (\tan \left (d x +c \right )\right ) a}{d \left (a^{2}+b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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

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maxima [A]  time = 0.46, size = 107, normalized size = 1.34 \[ \frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (C a b - B b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, B \log \left (\tan \left (d x + c\right )\right )}{a}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 9.46, size = 115, normalized size = 1.44 \[ \frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b-C\,a\right )}{a\,d\,\left (a^2+b^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 5.75, size = 966, normalized size = 12.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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*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)), (I*B*d*x*tan(c + d*x)/(2*
I*b*d*tan(c + d*x) + 2*b*d) + B*d*x/(2*I*b*d*tan(c + d*x) + 2*b*d) + B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(
2*I*b*d*tan(c + d*x) + 2*b*d) - I*B*log(tan(c + d*x)**2 + 1)/(2*I*b*d*tan(c + d*x) + 2*b*d) - 2*B*log(tan(c +
d*x))*tan(c + d*x)/(2*I*b*d*tan(c + d*x) + 2*b*d) + 2*I*B*log(tan(c + d*x))/(2*I*b*d*tan(c + d*x) + 2*b*d) + I
*B/(2*I*b*d*tan(c + d*x) + 2*b*d) - C*d*x*tan(c + d*x)/(2*I*b*d*tan(c + d*x) + 2*b*d) + I*C*d*x/(2*I*b*d*tan(c
+ d*x) + 2*b*d) - C/(2*I*b*d*tan(c + d*x) + 2*b*d), Eq(a, -I*b)), (-I*B*d*x*tan(c + d*x)/(-2*I*b*d*tan(c + d*
x) + 2*b*d) + B*d*x/(-2*I*b*d*tan(c + d*x) + 2*b*d) + B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(-2*I*b*d*tan(c
+ d*x) + 2*b*d) + I*B*log(tan(c + d*x)**2 + 1)/(-2*I*b*d*tan(c + d*x) + 2*b*d) - 2*B*log(tan(c + d*x))*tan(c +
d*x)/(-2*I*b*d*tan(c + d*x) + 2*b*d) - 2*I*B*log(tan(c + d*x))/(-2*I*b*d*tan(c + d*x) + 2*b*d) - I*B/(-2*I*b*
d*tan(c + d*x) + 2*b*d) - C*d*x*tan(c + d*x)/(-2*I*b*d*tan(c + d*x) + 2*b*d) - I*C*d*x/(-2*I*b*d*tan(c + d*x)
+ 2*b*d) - C/(-2*I*b*d*tan(c + d*x) + 2*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*log(tan(c + d*x)**2 + 1)/(2*d) + B*log(tan(c + d*x))/d + C*x)/a, Eq(b, 0)), (-B*a**2*log(ta
n(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))

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