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

Optimal. Leaf size=69 \[ -\frac{b^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac{b B x}{a^2+b^2}+\frac{B \log (\sin (c+d x))}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0854284, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {21, 3571, 3530, 3475} \[ -\frac{b^2 B \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}-\frac{b B x}{a^2+b^2}+\frac{B \log (\sin (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 3571

Int[1/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[((a*
c - b*d)*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[b^2/((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^2/((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}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

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 3475

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

Rubi steps

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

Mathematica [C]  time = 0.113865, size = 79, normalized size = 1.14 \[ -\frac{B \left (2 b^2 \log (a \cot (c+d x)+b)+a (a+i b) \log (-\cot (c+d x)+i)+a (a-i b) \log (\cot (c+d x)+i)\right )}{2 a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.089, size = 99, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) aB}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) b}{d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ) }{ad}}-{\frac{{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{ad \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.52219, size = 119, normalized size = 1.72 \begin{align*} -\frac{\frac{2 \, B b^{2} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} + \frac{2 \,{\left (d x + c\right )} B b}{a^{2} + b^{2}} + \frac{B a \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.78834, size = 242, normalized size = 3.51 \begin{align*} -\frac{2 \, B a b d x + B b^{2} \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 ) -{\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 )}{2 \,{\left (a^{3} + a b^{2}\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.24427, size = 124, normalized size = 1.8 \begin{align*} -\frac{\frac{2 \, B b^{3} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}} + \frac{2 \,{\left (d x + c\right )} B b}{a^{2} + b^{2}} + \frac{B a \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 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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