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

Optimal. Leaf size=226 \[ \frac {\log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))} \]

[Out]

-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+ln(sin(d*x+c))/a^4/d-b^2*(10*a^6+5*a^4*b^2+4*a^2*b^4+b^6)*ln(a*cos(d*x+c)+b*sin
(d*x+c))/a^4/(a^2+b^2)^4/d+1/3*b^2/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+1/2*b^2*(3*a^2+b^2)/a^2/(a^2+b^2)^2/d/(a+b
*tan(d*x+c))^2+b^2*(6*a^4+3*a^2*b^2+b^4)/a^3/(a^2+b^2)^3/d/(a+b*tan(d*x+c))

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Rubi [A]  time = 0.65, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3569, 3649, 3651, 3530, 3475} \[ \frac {b^2 \left (3 a^2 b^2+6 a^4+b^4\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {b^2}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {b^2 \left (5 a^4 b^2+4 a^2 b^4+10 a^6+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}-\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}+\frac {\log (\sin (c+d x))}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 + Log[Sin[c + d*x]]/(a^4*d) - (b^2*(10*a^6 + 5*a^4*b^2 + 4*a^2*b^4 + b^6)
*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^4*d) + b^2/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3)
+ (b^2*(3*a^2 + b^2))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b^2*(6*a^4 + 3*a^2*b^2 + b^4))/(a^3*(a
^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

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 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 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]

Rubi steps

\begin {align*} \int \frac {\cot (c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) \left (3 \left (a^2+b^2\right )-3 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )}\\ &=\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^2-12 a^3 b \tan (c+d x)+6 b^2 \left (3 a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{6 a^2 \left (a^2+b^2\right )^2}\\ &=\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (6 \left (a^2+b^2\right )^3-6 a^3 b \left (3 a^2-b^2\right ) \tan (c+d x)+6 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{6 a^3 \left (a^2+b^2\right )^3}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \cot (c+d x) \, dx}{a^4}-\frac {\left (b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \left (a^2+b^2\right )^4}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\log (\sin (c+d x))}{a^4 d}-\frac {b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac {b^2}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {b^2 \left (3 a^2+b^2\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C]  time = 2.05, size = 243, normalized size = 1.08 \[ \frac {\frac {2 a b^2 \left (a^2+b^2\right )}{(a+b \tan (c+d x))^3}+\frac {3 \left (3 a^2 b^2+b^4\right )}{(a+b \tan (c+d x))^2}+\frac {6 \left (6 a^4 b^2+3 a^2 b^4+b^6\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {3 \left (-a^4 (a-i b)^4 \log (-\tan (c+d x)+i)-a^4 (a+i b)^4 \log (\tan (c+d x)+i)+2 \left (a^2+b^2\right )^4 \log (\tan (c+d x))-2 b^2 \left (10 a^6+5 a^4 b^2+4 a^2 b^4+b^6\right ) \log (a+b \tan (c+d x))\right )}{a^2 \left (a^2+b^2\right )^2}}{6 a^2 d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(-(a^4*(a - I*b)^4*Log[I - Tan[c + d*x]]) + 2*(a^2 + b^2)^4*Log[Tan[c + d*x]] - a^4*(a + I*b)^4*Log[I + Ta
n[c + d*x]] - 2*b^2*(10*a^6 + 5*a^4*b^2 + 4*a^2*b^4 + b^6)*Log[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2) + (2*
a*b^2*(a^2 + b^2))/(a + b*Tan[c + d*x])^3 + (3*(3*a^2*b^2 + b^4))/(a + b*Tan[c + d*x])^2 + (6*(6*a^4*b^2 + 3*a
^2*b^4 + b^6))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(6*a^2*(a^2 + b^2)^2*d)

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fricas [B]  time = 0.70, size = 793, normalized size = 3.51 \[ \frac {75 \, a^{7} b^{4} + 42 \, a^{5} b^{6} + 11 \, a^{3} b^{8} - {\left (47 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + 24 \, {\left (a^{7} b^{4} - a^{5} b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 24 \, {\left (a^{10} b - a^{8} b^{3}\right )} d x - 3 \, {\left (35 \, a^{7} b^{4} - 12 \, a^{5} b^{6} - 5 \, a^{3} b^{8} - 2 \, a b^{10} + 24 \, {\left (a^{8} b^{3} - a^{6} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (10 \, a^{9} b^{2} + 5 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} + {\left (10 \, a^{6} b^{5} + 5 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (10 \, a^{7} b^{4} + 5 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (10 \, a^{8} b^{3} + 5 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )\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 ) - 3 \, {\left (20 \, a^{8} b^{3} - 37 \, a^{6} b^{5} - 18 \, a^{4} b^{7} - 5 \, a^{2} b^{9} + 24 \, {\left (a^{9} b^{2} - a^{7} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{12} b^{3} + 4 \, a^{10} b^{5} + 6 \, a^{8} b^{7} + 4 \, a^{6} b^{9} + a^{4} b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{13} b^{2} + 4 \, a^{11} b^{4} + 6 \, a^{9} b^{6} + 4 \, a^{7} b^{8} + a^{5} b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{14} b + 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} + 4 \, a^{8} b^{7} + a^{6} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{15} + 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} + 4 \, a^{9} b^{6} + a^{7} b^{8}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(75*a^7*b^4 + 42*a^5*b^6 + 11*a^3*b^8 - (47*a^6*b^5 + 6*a^4*b^7 + 3*a^2*b^9 + 24*(a^7*b^4 - a^5*b^6)*d*x)*
tan(d*x + c)^3 - 24*(a^10*b - a^8*b^3)*d*x - 3*(35*a^7*b^4 - 12*a^5*b^6 - 5*a^3*b^8 - 2*a*b^10 + 24*(a^8*b^3 -
a^6*b^5)*d*x)*tan(d*x + c)^2 + 3*(a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8 + (a^8*b^3 + 4*a^6*b^5 +
6*a^4*b^7 + 4*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d
*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x
+ c)^2 + 1)) - 3*(10*a^9*b^2 + 5*a^7*b^4 + 4*a^5*b^6 + a^3*b^8 + (10*a^6*b^5 + 5*a^4*b^7 + 4*a^2*b^9 + b^11)*
tan(d*x + c)^3 + 3*(10*a^7*b^4 + 5*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d*x + c)^2 + 3*(10*a^8*b^3 + 5*a^6*b^5 +
4*a^4*b^7 + a^2*b^9)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) -
3*(20*a^8*b^3 - 37*a^6*b^5 - 18*a^4*b^7 - 5*a^2*b^9 + 24*(a^9*b^2 - a^7*b^4)*d*x)*tan(d*x + c))/((a^12*b^3 +
4*a^10*b^5 + 6*a^8*b^7 + 4*a^6*b^9 + a^4*b^11)*d*tan(d*x + c)^3 + 3*(a^13*b^2 + 4*a^11*b^4 + 6*a^9*b^6 + 4*a^7
*b^8 + a^5*b^10)*d*tan(d*x + c)^2 + 3*(a^14*b + 4*a^12*b^3 + 6*a^10*b^5 + 4*a^8*b^7 + a^6*b^9)*d*tan(d*x + c)
+ (a^15 + 4*a^13*b^2 + 6*a^11*b^4 + 4*a^9*b^6 + a^7*b^8)*d)

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giac [B]  time = 3.35, size = 476, normalized size = 2.11 \[ -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, a^{6} b^{3} + 5 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 4 \, a^{10} b^{3} + 6 \, a^{8} b^{5} + 4 \, a^{6} b^{7} + a^{4} b^{9}} - \frac {110 \, a^{6} b^{5} \tan \left (d x + c\right )^{3} + 55 \, a^{4} b^{7} \tan \left (d x + c\right )^{3} + 44 \, a^{2} b^{9} \tan \left (d x + c\right )^{3} + 11 \, b^{11} \tan \left (d x + c\right )^{3} + 366 \, a^{7} b^{4} \tan \left (d x + c\right )^{2} + 219 \, a^{5} b^{6} \tan \left (d x + c\right )^{2} + 156 \, a^{3} b^{8} \tan \left (d x + c\right )^{2} + 39 \, a b^{10} \tan \left (d x + c\right )^{2} + 411 \, a^{8} b^{3} \tan \left (d x + c\right ) + 294 \, a^{6} b^{5} \tan \left (d x + c\right ) + 195 \, a^{4} b^{7} \tan \left (d x + c\right ) + 48 \, a^{2} b^{9} \tan \left (d x + c\right ) + 157 \, a^{9} b^{2} + 136 \, a^{7} b^{4} + 89 \, a^{5} b^{6} + 22 \, a^{3} b^{8}}{{\left (a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}} - \frac {6 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)
*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(10*a^6*b^3 + 5*a^4*b^5 + 4*a^2*b
^7 + b^9)*log(abs(b*tan(d*x + c) + a))/(a^12*b + 4*a^10*b^3 + 6*a^8*b^5 + 4*a^6*b^7 + a^4*b^9) - (110*a^6*b^5*
tan(d*x + c)^3 + 55*a^4*b^7*tan(d*x + c)^3 + 44*a^2*b^9*tan(d*x + c)^3 + 11*b^11*tan(d*x + c)^3 + 366*a^7*b^4*
tan(d*x + c)^2 + 219*a^5*b^6*tan(d*x + c)^2 + 156*a^3*b^8*tan(d*x + c)^2 + 39*a*b^10*tan(d*x + c)^2 + 411*a^8*
b^3*tan(d*x + c) + 294*a^6*b^5*tan(d*x + c) + 195*a^4*b^7*tan(d*x + c) + 48*a^2*b^9*tan(d*x + c) + 157*a^9*b^2
+ 136*a^7*b^4 + 89*a^5*b^6 + 22*a^3*b^8)/((a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8)*(b*tan(d*x +
c) + a)^3) - 6*log(abs(tan(d*x + c)))/a^4)/d

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maple [B]  time = 0.62, size = 460, normalized size = 2.04 \[ \frac {b^{2}}{3 a \left (a^{2}+b^{2}\right ) d \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {3 b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{4}}{2 d \left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {6 a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{4}}{d \left (a^{2}+b^{2}\right )^{3} a \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{6}}{d \left (a^{2}+b^{2}\right )^{3} a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {10 a^{2} b^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {5 \ln \left (a +b \tan \left (d x +c \right )\right ) b^{4}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {4 b^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4} a^{2}}-\frac {b^{8} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4} a^{4}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{d \,a^{4}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}}{2 d \left (a^{2}+b^{2}\right )^{4}}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}}{2 d \left (a^{2}+b^{2}\right )^{4}}-\frac {4 \arctan \left (\tan \left (d x +c \right )\right ) a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {4 \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^3+3/2/d/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*b^2+1/2/d*b^4/(a^2+b^2)^2/a^2/(a
+b*tan(d*x+c))^2+6/d*a/(a^2+b^2)^3/(a+b*tan(d*x+c))*b^2+3/d*b^4/(a^2+b^2)^3/a/(a+b*tan(d*x+c))+1/d*b^6/(a^2+b^
2)^3/a^3/(a+b*tan(d*x+c))-10/d*a^2/(a^2+b^2)^4*b^2*ln(a+b*tan(d*x+c))-5/d/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*b^4-4
/d*b^6/(a^2+b^2)^4/a^2*ln(a+b*tan(d*x+c))-1/d*b^8/(a^2+b^2)^4/a^4*ln(a+b*tan(d*x+c))+1/d/a^4*ln(tan(d*x+c))-1/
2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^4+3/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^2*b^2-1/2/d/(a^2+b^2)^4*ln(1+tan(d
*x+c)^2)*b^4-4/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^3*b+4/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a*b^3

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maxima [A]  time = 0.81, size = 444, normalized size = 1.96 \[ -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (10 \, a^{6} b^{2} + 5 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 4 \, a^{10} b^{2} + 6 \, a^{8} b^{4} + 4 \, a^{6} b^{6} + a^{4} b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {47 \, a^{6} b^{2} + 34 \, a^{4} b^{4} + 11 \, a^{2} b^{6} + 6 \, {\left (6 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (27 \, a^{5} b^{3} + 16 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \tan \left (d x + c\right )}{a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6} + {\left (a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} \tan \left (d x + c\right )} - \frac {6 \, \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 6*(10*a^6*b^2 + 5*a^4*b^4
+ 4*a^2*b^6 + b^8)*log(b*tan(d*x + c) + a)/(a^12 + 4*a^10*b^2 + 6*a^8*b^4 + 4*a^6*b^6 + a^4*b^8) + 3*(a^4 - 6
*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - (47*a^6*b^2 + 34*a^4
*b^4 + 11*a^2*b^6 + 6*(6*a^4*b^4 + 3*a^2*b^6 + b^8)*tan(d*x + c)^2 + 3*(27*a^5*b^3 + 16*a^3*b^5 + 5*a*b^7)*tan
(d*x + c))/(a^12 + 3*a^10*b^2 + 3*a^8*b^4 + a^6*b^6 + (a^9*b^3 + 3*a^7*b^5 + 3*a^5*b^7 + a^3*b^9)*tan(d*x + c)
^3 + 3*(a^10*b^2 + 3*a^8*b^4 + 3*a^6*b^6 + a^4*b^8)*tan(d*x + c)^2 + 3*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b
^7)*tan(d*x + c)) - 6*log(tan(d*x + c))/a^4)/d

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mupad [B]  time = 4.87, size = 385, normalized size = 1.70 \[ \frac {\frac {47\,a^4\,b^2+34\,a^2\,b^4+11\,b^6}{6\,a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (27\,a^4\,b^3+16\,a^2\,b^5+5\,b^7\right )}{2\,a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (6\,a^4\,b^4+3\,a^2\,b^6+b^8\right )}{a^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^4\,d}-\frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (10\,a^6+5\,a^4\,b^2+4\,a^2\,b^4+b^6\right )}{a^4\,d\,{\left (a^2+b^2\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((11*b^6 + 34*a^2*b^4 + 47*a^4*b^2)/(6*a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)*(5*b^7 + 16*a^2*
b^5 + 27*a^4*b^3))/(2*a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^2*(b^8 + 3*a^2*b^6 + 6*a^4*b^4)
)/(a^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^3*tan(c + d*x)^3 + 3*a*b^2*tan(c + d*x)^2 + 3*a^2*b*t
an(c + d*x))) - (log(tan(c + d*x) + 1i)*1i)/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) - log(tan
(c + d*x) - 1i)/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) + log(tan(c + d*x))/(a^4*d) - (b^2*log(a +
b*tan(c + d*x))*(10*a^6 + b^6 + 4*a^2*b^4 + 5*a^4*b^2))/(a^4*d*(a^2 + b^2)^4)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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