3.466 \(\int \frac{\cot ^4(c+d x)}{a+b \tan (c+d x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{\left (a^2-b^2\right ) \cot (c+d x)}{a^3 d}+\frac{b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )}+\frac{a x}{a^2+b^2}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d} \]

[Out]

(a*x)/(a^2 + b^2) + ((a^2 - b^2)*Cot[c + d*x])/(a^3*d) + (b*Cot[c + d*x]^2)/(2*a^2*d) - Cot[c + d*x]^3/(3*a*d)
+ (b*(a^2 - b^2)*Log[Sin[c + d*x]])/(a^4*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)*d)

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Rubi [A]  time = 0.490472, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3569, 3649, 3650, 3651, 3530, 3475} \[ \frac{\left (a^2-b^2\right ) \cot (c+d x)}{a^3 d}+\frac{b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )}+\frac{a x}{a^2+b^2}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4/(a + b*Tan[c + d*x]),x]

[Out]

(a*x)/(a^2 + b^2) + ((a^2 - b^2)*Cot[c + d*x])/(a^3*d) + (b*Cot[c + d*x]^2)/(2*a^2*d) - Cot[c + d*x]^3/(3*a*d)
+ (b*(a^2 - b^2)*Log[Sin[c + d*x]])/(a^4*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)*d)

Rule 3569

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 3649

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

Rule 3650

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

Rule 3651

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]

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 ^4(c+d x)}{a+b \tan (c+d x)} \, dx &=-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\int \frac{\cot ^3(c+d x) \left (3 b+3 a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{3 a}\\ &=\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\int \frac{\cot ^2(c+d x) \left (-6 \left (a^2-b^2\right )+6 b^2 \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^2}\\ &=\frac{\left (a^2-b^2\right ) \cot (c+d x)}{a^3 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}-\frac{\int \frac{\cot (c+d x) \left (-6 b \left (a^2-b^2\right )-6 a^3 \tan (c+d x)-6 b \left (a^2-b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3}\\ &=\frac{a x}{a^2+b^2}+\frac{\left (a^2-b^2\right ) \cot (c+d x)}{a^3 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{\left (b \left (a^2-b^2\right )\right ) \int \cot (c+d x) \, dx}{a^4}+\frac{b^5 \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )}\\ &=\frac{a x}{a^2+b^2}+\frac{\left (a^2-b^2\right ) \cot (c+d x)}{a^3 d}+\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{\cot ^3(c+d x)}{3 a d}+\frac{b \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right ) d}\\ \end{align*}

Mathematica [C]  time = 1.42871, size = 131, normalized size = 0.98 \[ -\frac{-\frac{6 \left (a^2-b^2\right ) \cot (c+d x)}{a^3}-\frac{6 b^5 \log (a \cot (c+d x)+b)}{a^4 \left (a^2+b^2\right )}-\frac{3 b \cot ^2(c+d x)}{a^2}+\frac{3 \log (-\cot (c+d x)+i)}{b+i a}+\frac{3 i \log (\cot (c+d x)+i)}{a+i b}+\frac{2 \cot ^3(c+d x)}{a}}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4/(a + b*Tan[c + d*x]),x]

[Out]

-((-6*(a^2 - b^2)*Cot[c + d*x])/a^3 - (3*b*Cot[c + d*x]^2)/a^2 + (2*Cot[c + d*x]^3)/a + (3*Log[I - Cot[c + d*x
]])/(I*a + b) + ((3*I)*Log[I + Cot[c + d*x]])/(a + I*b) - (6*b^5*Log[b + a*Cot[c + d*x]])/(a^4*(a^2 + b^2)))/(
6*d)

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Maple [A]  time = 0.07, size = 179, normalized size = 1.4 \begin{align*} -{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{a\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{1}{3\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{1}{ad\tan \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\tan \left ( dx+c \right ) }}+{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{{a}^{2}d}}-{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}+{\frac{b}{2\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d \left ({a}^{2}+{b}^{2} \right ){a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4/(a+b*tan(d*x+c)),x)

[Out]

-1/2/d/(a^2+b^2)*b*ln(1+tan(d*x+c)^2)+1/d/(a^2+b^2)*a*arctan(tan(d*x+c))-1/3/d/a/tan(d*x+c)^3+1/d/a/tan(d*x+c)
-1/d/a^3/tan(d*x+c)*b^2+b*ln(tan(d*x+c))/a^2/d-1/d/a^4*b^3*ln(tan(d*x+c))+1/2/d*b/a^2/tan(d*x+c)^2+1/d*b^5/(a^
2+b^2)/a^4*ln(a+b*tan(d*x+c))

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Maxima [A]  time = 1.64293, size = 196, normalized size = 1.47 \begin{align*} \frac{\frac{6 \, b^{5} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + a^{4} b^{2}} + \frac{6 \,{\left (d x + c\right )} a}{a^{2} + b^{2}} - \frac{3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{6 \,{\left (a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}} + \frac{3 \, a b \tan \left (d x + c\right ) + 6 \,{\left (a^{2} - b^{2}\right )} \tan \left (d x + c\right )^{2} - 2 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.16867, size = 470, normalized size = 3.53 \begin{align*} \frac{3 \, b^{5} \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 ) \tan \left (d x + c\right )^{3} - 2 \, a^{5} - 2 \, a^{3} b^{2} + 3 \,{\left (a^{4} b - b^{5}\right )} \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 )^{3} + 3 \,{\left (2 \, a^{5} d x + a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (a^{5} - a b^{4}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{4} b + a^{2} b^{3}\right )} \tan \left (d x + c\right )}{6 \,{\left (a^{6} + a^{4} b^{2}\right )} d \tan \left (d x + c\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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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)**4/(a+b*tan(d*x+c)),x)

[Out]

Timed out

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Giac [A]  time = 1.37585, size = 252, normalized size = 1.89 \begin{align*} \frac{\frac{6 \, b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + a^{4} b^{3}} + \frac{6 \,{\left (d x + c\right )} a}{a^{2} + b^{2}} - \frac{3 \, b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{6 \,{\left (a^{2} b - b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{11 \, a^{2} b \tan \left (d x + c\right )^{3} - 11 \, b^{3} \tan \left (d x + c\right )^{3} - 6 \, a^{3} \tan \left (d x + c\right )^{2} + 6 \, a b^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{2} b \tan \left (d x + c\right ) + 2 \, a^{3}}{a^{4} \tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*b^6*log(abs(b*tan(d*x + c) + a))/(a^6*b + a^4*b^3) + 6*(d*x + c)*a/(a^2 + b^2) - 3*b*log(tan(d*x + c)^2
+ 1)/(a^2 + b^2) + 6*(a^2*b - b^3)*log(abs(tan(d*x + c)))/a^4 - (11*a^2*b*tan(d*x + c)^3 - 11*b^3*tan(d*x + c
)^3 - 6*a^3*tan(d*x + c)^2 + 6*a*b^2*tan(d*x + c)^2 - 3*a^2*b*tan(d*x + c) + 2*a^3)/(a^4*tan(d*x + c)^3))/d