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

Optimal. Leaf size=302 \[ \frac{b (A b-a B)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b \left (5 a^4 A b^3+4 a^2 A b^5+10 a^6 A b+4 a^5 b^2 B-4 a^7 B+A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}-\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4}+\frac{A \log (\sin (c+d x))}{a^4 d} \]

[Out]

-(((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4) + (A*Log[Sin[c + d*x]])/(a^4*d) - (
b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]
])/(a^4*(a^2 + b^2)^4*d) + (b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b*(3*a^2*A*b + A*b^3
- 2*a^3*B))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a
^3*b^2*B))/(a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.898587, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3609, 3649, 3651, 3530, 3475} \[ \frac{b (A b-a B)}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right )}{a^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{2 a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b \left (5 a^4 A b^3+4 a^2 A b^5+10 a^6 A b+4 a^5 b^2 B-4 a^7 B+A b^7\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^4}-\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4}+\frac{A \log (\sin (c+d x))}{a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*a^2*b^2*B - b^4*B)*x)/(a^2 + b^2)^4) + (A*Log[Sin[c + d*x]])/(a^4*d) - (
b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]
])/(a^4*(a^2 + b^2)^4*d) + (b*(A*b - a*B))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (b*(3*a^2*A*b + A*b^3
- 2*a^3*B))/(2*a^2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) + (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a
^3*b^2*B))/(a^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3609

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

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 \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=\frac{b (A b-a B)}{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 A \left (a^2+b^2\right )-3 a (A b-a B) \tan (c+d x)+3 b (A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 a \left (a^2+b^2\right )}\\ &=\frac{b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\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 A \left (a^2+b^2\right )^2-6 a^2 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+6 b \left (3 a^2 A b+A b^3-2 a^3 B\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 (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\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 A \left (a^2+b^2\right )^3-6 a^3 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \tan (c+d x)+6 b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\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{\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}+\frac{b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{A \int \cot (c+d x) \, dx}{a^4}-\frac{\left (b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\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{\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) x}{\left (a^2+b^2\right )^4}+\frac{A \log (\sin (c+d x))}{a^4 d}-\frac{b \left (10 a^6 A b+5 a^4 A b^3+4 a^2 A b^5+A b^7-4 a^7 B+4 a^5 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^4 d}+\frac{b (A b-a B)}{3 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right )}{a^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 3.04072, size = 308, normalized size = 1.02 \[ \frac{\frac{2 a b \left (a^2+b^2\right ) (A b-a B)}{(a+b \tan (c+d x))^3}+\frac{6 b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right )}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{3 b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{(a+b \tan (c+d x))^2}+\frac{3 \left (-2 b \left (5 a^4 A b^3+4 a^2 A b^5+10 a^6 A b+4 a^5 b^2 B-4 a^7 B+A b^7\right ) \log (a+b \tan (c+d x))+2 A \left (a^2+b^2\right )^4 \log (\tan (c+d x))-a^4 (a-i b)^4 (A+i B) \log (-\tan (c+d x)+i)-a^4 (a+i b)^4 (A-i B) \log (\tan (c+d x)+i)\right )}{a^2 \left (a^2+b^2\right )^2}}{6 a^2 d \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(-(a^4*(a - I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]]) + 2*A*(a^2 + b^2)^4*Log[Tan[c + d*x]] - a^4*(a + I*b)^
4*(A - I*B)*Log[I + Tan[c + d*x]] - 2*b*(10*a^6*A*b + 5*a^4*A*b^3 + 4*a^2*A*b^5 + A*b^7 - 4*a^7*B + 4*a^5*b^2*
B)*Log[a + b*Tan[c + d*x]]))/(a^2*(a^2 + b^2)^2) + (2*a*b*(a^2 + b^2)*(A*b - a*B))/(a + b*Tan[c + d*x])^3 + (3
*b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a + b*Tan[c + d*x])^2 + (6*b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a
^3*b^2*B))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(6*a^2*(a^2 + b^2)^2*d)

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Maple [B]  time = 0.189, size = 789, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.58947, size = 783, normalized size = 2.59 \begin{align*} \frac{\frac{6 \,{\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\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 (4 \, B a^{7} b - 10 \, A a^{6} b^{2} - 4 \, B a^{5} b^{3} - 5 \, A a^{4} b^{4} - 4 \, A a^{2} b^{6} - A 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 a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A 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{26 \, B a^{7} b - 47 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} - 34 \, A a^{4} b^{4} + 2 \, B a^{3} b^{5} - 11 \, A a^{2} b^{6} + 6 \,{\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (14 \, B a^{6} b^{2} - 27 \, A a^{5} b^{3} - 2 \, B a^{4} b^{4} - 16 \, A a^{3} b^{5} - 5 \, A 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 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 6*(4*B*a^7*b - 10*A*a^6*b^2 - 4*B*a^5*b^3 - 5*A*a^4*b^4 - 4*A*a^2*b^6 - A*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*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A
*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) - (26*B*a^7*b - 47*A*a^6*b^2 + 4
*B*a^5*b^3 - 34*A*a^4*b^4 + 2*B*a^3*b^5 - 11*A*a^2*b^6 + 6*(3*B*a^5*b^3 - 6*A*a^4*b^4 - B*a^3*b^5 - 3*A*a^2*b^
6 - A*b^8)*tan(d*x + c)^2 + 3*(14*B*a^6*b^2 - 27*A*a^5*b^3 - 2*B*a^4*b^4 - 16*A*a^3*b^5 - 5*A*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)*ta
n(d*x + c)) + 6*A*log(tan(d*x + c))/a^4)/d

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Fricas [B]  time = 3.56422, size = 2457, normalized size = 8.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(48*B*a^8*b^3 - 75*A*a^7*b^4 + 6*B*a^6*b^5 - 42*A*a^5*b^6 + 2*B*a^4*b^7 - 11*A*a^3*b^8 - (26*B*a^7*b^4 -
47*A*a^6*b^5 - 18*B*a^5*b^6 - 6*A*a^4*b^7 - 3*A*a^2*b^9 + 6*(B*a^8*b^3 - 4*A*a^7*b^4 - 6*B*a^6*b^5 + 4*A*a^5*b
^6 + B*a^4*b^7)*d*x)*tan(d*x + c)^3 - 6*(B*a^11 - 4*A*a^10*b - 6*B*a^9*b^2 + 4*A*a^8*b^3 + B*a^7*b^4)*d*x - 3*
(20*B*a^8*b^3 - 35*A*a^7*b^4 - 22*B*a^6*b^5 + 12*A*a^5*b^6 + 2*B*a^4*b^7 + 5*A*a^3*b^8 + 2*A*a*b^10 + 6*(B*a^9
*b^2 - 4*A*a^8*b^3 - 6*B*a^7*b^4 + 4*A*a^6*b^5 + B*a^5*b^6)*d*x)*tan(d*x + c)^2 - 3*(A*a^11 + 4*A*a^9*b^2 + 6*
A*a^7*b^4 + 4*A*a^5*b^6 + A*a^3*b^8 + (A*a^8*b^3 + 4*A*a^6*b^5 + 6*A*a^4*b^7 + 4*A*a^2*b^9 + A*b^11)*tan(d*x +
c)^3 + 3*(A*a^9*b^2 + 4*A*a^7*b^4 + 6*A*a^5*b^6 + 4*A*a^3*b^8 + A*a*b^10)*tan(d*x + c)^2 + 3*(A*a^10*b + 4*A*
a^8*b^3 + 6*A*a^6*b^5 + 4*A*a^4*b^7 + A*a^2*b^9)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - 3*(4
*B*a^10*b - 10*A*a^9*b^2 - 4*B*a^8*b^3 - 5*A*a^7*b^4 - 4*A*a^5*b^6 - A*a^3*b^8 + (4*B*a^7*b^4 - 10*A*a^6*b^5 -
4*B*a^5*b^6 - 5*A*a^4*b^7 - 4*A*a^2*b^9 - A*b^11)*tan(d*x + c)^3 + 3*(4*B*a^8*b^3 - 10*A*a^7*b^4 - 4*B*a^6*b^
5 - 5*A*a^5*b^6 - 4*A*a^3*b^8 - A*a*b^10)*tan(d*x + c)^2 + 3*(4*B*a^9*b^2 - 10*A*a^8*b^3 - 4*B*a^7*b^4 - 5*A*a
^6*b^5 - 4*A*a^4*b^7 - A*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*(12*B*a^9*b^2 - 20*A*a^8*b^3 - 30*B*a^7*b^4 + 37*A*a^6*b^5 + 2*B*a^5*b^6 + 18*A*a^4*b^7 + 5*A*
a^2*b^9 + 6*(B*a^10*b - 4*A*a^9*b^2 - 6*B*a^8*b^3 + 4*A*a^7*b^4 + B*a^6*b^5)*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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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*tan(d*x+c))/(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.29046, size = 975, normalized size = 3.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) - 3*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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*(4*B*a^7*b^2 - 10*A*a^6*b^3 - 4*B*a^5*b^4 - 5*A*a^4*b^5 - 4*A*a^2*b^7 - A*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) + 6*A*log(abs(tan(d*x +
c)))/a^4 - (44*B*a^7*b^4*tan(d*x + c)^3 - 110*A*a^6*b^5*tan(d*x + c)^3 - 44*B*a^5*b^6*tan(d*x + c)^3 - 55*A*a
^4*b^7*tan(d*x + c)^3 - 44*A*a^2*b^9*tan(d*x + c)^3 - 11*A*b^11*tan(d*x + c)^3 + 150*B*a^8*b^3*tan(d*x + c)^2
- 366*A*a^7*b^4*tan(d*x + c)^2 - 120*B*a^6*b^5*tan(d*x + c)^2 - 219*A*a^5*b^6*tan(d*x + c)^2 - 6*B*a^4*b^7*tan
(d*x + c)^2 - 156*A*a^3*b^8*tan(d*x + c)^2 - 39*A*a*b^10*tan(d*x + c)^2 + 174*B*a^9*b^2*tan(d*x + c) - 411*A*a
^8*b^3*tan(d*x + c) - 96*B*a^7*b^4*tan(d*x + c) - 294*A*a^6*b^5*tan(d*x + c) - 6*B*a^5*b^6*tan(d*x + c) - 195*
A*a^4*b^7*tan(d*x + c) - 48*A*a^2*b^9*tan(d*x + c) + 70*B*a^10*b - 157*A*a^9*b^2 - 14*B*a^8*b^3 - 136*A*a^7*b^
4 + 6*B*a^6*b^5 - 89*A*a^5*b^6 + 2*B*a^4*b^7 - 22*A*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))/d