3.289 \(\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=352 \[ \frac{b \left (11 a^2 A b^3+3 a^4 A b-6 a^3 b^2 B+a^5 (-B)-3 a b^4 B+6 A b^5\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{b \left (5 a^2 A b-2 a^3 B-3 a b^2 B+6 A b^3\right )}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{\left (a^2 A+3 a b B-6 A b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac{b^3 \left (17 a^2 A b^3+15 a^4 A b-9 a^3 b^2 B-10 a^5 B-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2} \]

[Out]

((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3 - ((a^2*A - 6*A*b^2 + 3*a*b*B)*Log[Sin[c + d*x]])/(a
^5*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a*Cos[c + d*x] + b
*Sin[c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b*(5*a^2*A*b + 6*A*b^3 - 2*a^3*B - 3*a*b^2*B))/(2*a^3*(a^2 + b^2)*d*(
a + b*Tan[c + d*x])^2) + ((2*A*b - a*B)*Cot[c + d*x])/(a^2*d*(a + b*Tan[c + d*x])^2) - (A*Cot[c + d*x]^2)/(2*a
*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^4*A*b + 11*a^2*A*b^3 + 6*A*b^5 - a^5*B - 6*a^3*b^2*B - 3*a*b^4*B))/(a^4*(
a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 1.24866, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3609, 3649, 3651, 3530, 3475} \[ \frac{b \left (11 a^2 A b^3+3 a^4 A b-6 a^3 b^2 B+a^5 (-B)-3 a b^4 B+6 A b^5\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{b \left (5 a^2 A b-2 a^3 B-3 a b^2 B+6 A b^3\right )}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{\left (a^2 A+3 a b B-6 A b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac{b^3 \left (17 a^2 A b^3+15 a^4 A b-9 a^3 b^2 B-10 a^5 B-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]

[Out]

((3*a^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)^3 - ((a^2*A - 6*A*b^2 + 3*a*b*B)*Log[Sin[c + d*x]])/(a
^5*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a*Cos[c + d*x] + b
*Sin[c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b*(5*a^2*A*b + 6*A*b^3 - 2*a^3*B - 3*a*b^2*B))/(2*a^3*(a^2 + b^2)*d*(
a + b*Tan[c + d*x])^2) + ((2*A*b - a*B)*Cot[c + d*x])/(a^2*d*(a + b*Tan[c + d*x])^2) - (A*Cot[c + d*x]^2)/(2*a
*d*(a + b*Tan[c + d*x])^2) + (b*(3*a^4*A*b + 11*a^2*A*b^3 + 6*A*b^5 - a^5*B - 6*a^3*b^2*B - 3*a*b^4*B))/(a^4*(
a^2 + b^2)^2*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 ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}-\frac{\int \frac{\cot ^2(c+d x) \left (2 (2 A b-a B)+2 a A \tan (c+d x)+4 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac{\int \frac{\cot (c+d x) \left (-2 \left (a^2 A-6 A b^2+3 a b B\right )-2 a^2 B \tan (c+d x)+6 b (2 A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a^2}\\ &=\frac{b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac{\int \frac{\cot (c+d x) \left (-4 \left (a^2+b^2\right ) \left (a^2 A-6 A b^2+3 a b B\right )+4 a^3 (A b-a B) \tan (c+d x)+4 b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{4 a^3 \left (a^2+b^2\right )}\\ &=\frac{b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{\cot (c+d x) \left (-4 \left (a^2+b^2\right )^2 \left (a^2 A-6 A b^2+3 a b B\right )+4 a^4 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+4 b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{4 a^4 \left (a^2+b^2\right )^2}\\ &=\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac{b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (a^2 A-6 A b^2+3 a b B\right ) \int \cot (c+d x) \, dx}{a^5}-\frac{\left (b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right )\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^3}\\ &=\frac{\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}-\frac{\left (a^2 A-6 A b^2+3 a b B\right ) \log (\sin (c+d x))}{a^5 d}-\frac{b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac{b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac{b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.46483, size = 320, normalized size = 0.91 \[ \frac{b^3 \left (5 a^2 A b-4 a^3 B-2 a b^2 B+3 A b^3\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{b^3 (A b-a B)}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{b^3 \left (17 a^2 A b^3+15 a^4 A b-9 a^3 b^2 B-10 a^5 B-3 a b^4 B+6 A b^5\right ) \log (a+b \tan (c+d x))}{a^5 d \left (a^2+b^2\right )^3}-\frac{\left (a^2 A+3 a b B-6 A b^2\right ) \log (\tan (c+d x))}{a^5 d}+\frac{(3 A b-a B) \cot (c+d x)}{a^4 d}-\frac{A \cot ^2(c+d x)}{2 a^3 d}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{2 d (a+i b)^3}+\frac{(A-i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]

[Out]

((3*A*b - a*B)*Cot[c + d*x])/(a^4*d) - (A*Cot[c + d*x]^2)/(2*a^3*d) + ((A + I*B)*Log[I - Tan[c + d*x]])/(2*(a
+ I*b)^3*d) - ((a^2*A - 6*A*b^2 + 3*a*b*B)*Log[Tan[c + d*x]])/(a^5*d) + ((A - I*B)*Log[I + Tan[c + d*x]])/(2*(
a - I*b)^3*d) - (b^3*(15*a^4*A*b + 17*a^2*A*b^3 + 6*A*b^5 - 10*a^5*B - 9*a^3*b^2*B - 3*a*b^4*B)*Log[a + b*Tan[
c + d*x]])/(a^5*(a^2 + b^2)^3*d) + (b^3*(A*b - a*B))/(2*a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + (b^3*(5*a^
2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B))/(a^4*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

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Maple [B]  time = 0.177, size = 713, normalized size = 2. \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)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)

[Out]

3/d/a^4/tan(d*x+c)*A*b+6/d/a^5*ln(tan(d*x+c))*A*b^2-3/d/a^4*ln(tan(d*x+c))*B*b-1/2/d/a^3*A/tan(d*x+c)^2-1/d/a^
3/tan(d*x+c)*B-1/d/a^3*A*ln(tan(d*x+c))+5/d*b^4/a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))*A-1/2/d/(a^2+b^2)^3*ln(1+tan(
d*x+c)^2)*B*b^3-1/d/(a^2+b^2)^3*A*arctan(tan(d*x+c))*b^3-1/d/(a^2+b^2)^3*B*arctan(tan(d*x+c))*a^3+1/2/d/(a^2+b
^2)^3*ln(1+tan(d*x+c)^2)*A*a^3+10/d/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*B*b^3-3/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*
A*a*b^2+3/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*B*a^2*b+3/d/(a^2+b^2)^3*A*arctan(tan(d*x+c))*a^2*b+3/d/(a^2+b^2)^
3*B*arctan(tan(d*x+c))*a*b^2-6/d*b^8/a^5/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*A+9/d*b^5/a^2/(a^2+b^2)^3*ln(a+b*tan(d
*x+c))*B+3/d*b^7/a^4/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*B+1/2/d*b^4/a^3/(a^2+b^2)/(a+b*tan(d*x+c))^2*A-1/2/d*b^3/a
^2/(a^2+b^2)/(a+b*tan(d*x+c))^2*B+3/d*b^6/a^4/(a^2+b^2)^2/(a+b*tan(d*x+c))*A-4/d*b^3/a/(a^2+b^2)^2/(a+b*tan(d*
x+c))*B-2/d*b^5/a^3/(a^2+b^2)^2/(a+b*tan(d*x+c))*B-15/d*b^4/a/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*A-17/d*b^6/a^3/(a
^2+b^2)^3*ln(a+b*tan(d*x+c))*A

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Maxima [A]  time = 1.75714, size = 730, normalized size = 2.07 \begin{align*} -\frac{\frac{2 \,{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (10 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} + 9 \, B a^{3} b^{5} - 17 \, A a^{2} b^{6} + 3 \, B a b^{7} - 6 \, A b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}} - \frac{{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{A a^{7} + 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \,{\left (B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} + 3 \, B a b^{6} - 6 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} +{\left (4 \, B a^{6} b - 11 \, A a^{5} b^{2} + 17 \, B a^{4} b^{3} - 33 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 18 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{7} - 2 \, A a^{6} b + 2 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3} + B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{2}} + \frac{2 \,{\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2*(2*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(10*B*a^5*b^
3 - 15*A*a^4*b^4 + 9*B*a^3*b^5 - 17*A*a^2*b^6 + 3*B*a*b^7 - 6*A*b^8)*log(b*tan(d*x + c) + a)/(a^11 + 3*a^9*b^2
+ 3*a^7*b^4 + a^5*b^6) - (A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3
*a^2*b^4 + b^6) + (A*a^7 + 2*A*a^5*b^2 + A*a^3*b^4 + 2*(B*a^5*b^2 - 3*A*a^4*b^3 + 6*B*a^3*b^4 - 11*A*a^2*b^5 +
3*B*a*b^6 - 6*A*b^7)*tan(d*x + c)^3 + (4*B*a^6*b - 11*A*a^5*b^2 + 17*B*a^4*b^3 - 33*A*a^3*b^4 + 9*B*a^2*b^5 -
18*A*a*b^6)*tan(d*x + c)^2 + 2*(B*a^7 - 2*A*a^6*b + 2*B*a^5*b^2 - 4*A*a^4*b^3 + B*a^3*b^4 - 2*A*a^2*b^5)*tan(
d*x + c))/((a^8*b^2 + 2*a^6*b^4 + a^4*b^6)*tan(d*x + c)^4 + 2*(a^9*b + 2*a^7*b^3 + a^5*b^5)*tan(d*x + c)^3 + (
a^10 + 2*a^8*b^2 + a^6*b^4)*tan(d*x + c)^2) + 2*(A*a^2 + 3*B*a*b - 6*A*b^2)*log(tan(d*x + c))/a^5)/d

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Fricas [B]  time = 3.23821, size = 2338, normalized size = 6.64 \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)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(A*a^10 + 3*A*a^8*b^2 + 3*A*a^6*b^4 + A*a^4*b^6 + (A*a^8*b^2 + 3*A*a^6*b^4 - 9*B*a^5*b^5 + 14*A*a^4*b^6 -
3*B*a^3*b^7 + 6*A*a^2*b^8 + 2*(B*a^8*b^2 - 3*A*a^7*b^3 - 3*B*a^6*b^4 + A*a^5*b^5)*d*x)*tan(d*x + c)^4 + 2*(A*
a^9*b + B*a^8*b^2 - 2*B*a^6*b^4 + 6*B*a^4*b^6 - 11*A*a^3*b^7 + 3*B*a^2*b^8 - 6*A*a*b^9 + 2*(B*a^9*b - 3*A*a^8*
b^2 - 3*B*a^7*b^3 + A*a^6*b^4)*d*x)*tan(d*x + c)^3 + (A*a^10 + 4*B*a^9*b - 8*A*a^8*b^2 + 12*B*a^7*b^3 - 30*A*a
^6*b^4 + 23*B*a^5*b^5 - 45*A*a^4*b^6 + 9*B*a^3*b^7 - 18*A*a^2*b^8 + 2*(B*a^10 - 3*A*a^9*b - 3*B*a^8*b^2 + A*a^
7*b^3)*d*x)*tan(d*x + c)^2 + ((A*a^8*b^2 + 3*B*a^7*b^3 - 3*A*a^6*b^4 + 9*B*a^5*b^5 - 15*A*a^4*b^6 + 9*B*a^3*b^
7 - 17*A*a^2*b^8 + 3*B*a*b^9 - 6*A*b^10)*tan(d*x + c)^4 + 2*(A*a^9*b + 3*B*a^8*b^2 - 3*A*a^7*b^3 + 9*B*a^6*b^4
- 15*A*a^5*b^5 + 9*B*a^4*b^6 - 17*A*a^3*b^7 + 3*B*a^2*b^8 - 6*A*a*b^9)*tan(d*x + c)^3 + (A*a^10 + 3*B*a^9*b -
3*A*a^8*b^2 + 9*B*a^7*b^3 - 15*A*a^6*b^4 + 9*B*a^5*b^5 - 17*A*a^4*b^6 + 3*B*a^3*b^7 - 6*A*a^2*b^8)*tan(d*x +
c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - ((10*B*a^5*b^5 - 15*A*a^4*b^6 + 9*B*a^3*b^7 - 17*A*a^2*b^8 +
3*B*a*b^9 - 6*A*b^10)*tan(d*x + c)^4 + 2*(10*B*a^6*b^4 - 15*A*a^5*b^5 + 9*B*a^4*b^6 - 17*A*a^3*b^7 + 3*B*a^2*b
^8 - 6*A*a*b^9)*tan(d*x + c)^3 + (10*B*a^7*b^3 - 15*A*a^6*b^4 + 9*B*a^5*b^5 - 17*A*a^4*b^6 + 3*B*a^3*b^7 - 6*A
*a^2*b^8)*tan(d*x + c)^2)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + 2*(B*a^1
0 - 2*A*a^9*b + 3*B*a^8*b^2 - 6*A*a^7*b^3 + 3*B*a^6*b^4 - 6*A*a^5*b^5 + B*a^4*b^6 - 2*A*a^3*b^7)*tan(d*x + c))
/((a^11*b^2 + 3*a^9*b^4 + 3*a^7*b^6 + a^5*b^8)*d*tan(d*x + c)^4 + 2*(a^12*b + 3*a^10*b^3 + 3*a^8*b^5 + a^6*b^7
)*d*tan(d*x + c)^3 + (a^13 + 3*a^11*b^2 + 3*a^9*b^4 + a^7*b^6)*d*tan(d*x + c)^2)

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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)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.37698, size = 1096, normalized size = 3.11 \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)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

-1/4*(4*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(A*a^3 + 3*B
*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 4*(10*B*a^5*b^4 - 15
*A*a^4*b^5 + 9*B*a^3*b^6 - 17*A*a^2*b^7 + 3*B*a*b^8 - 6*A*b^9)*log(abs(b*tan(d*x + c) + a))/(a^11*b + 3*a^9*b^
3 + 3*a^7*b^5 + a^5*b^7) - (3*A*a^7*b^2*tan(d*x + c)^4 + 9*B*a^6*b^3*tan(d*x + c)^4 - 9*A*a^5*b^4*tan(d*x + c)
^4 - 3*B*a^4*b^5*tan(d*x + c)^4 + 6*A*a^8*b*tan(d*x + c)^3 + 14*B*a^7*b^2*tan(d*x + c)^3 - 6*A*a^6*b^3*tan(d*x
+ c)^3 - 34*B*a^5*b^4*tan(d*x + c)^3 + 56*A*a^4*b^5*tan(d*x + c)^3 - 36*B*a^3*b^6*tan(d*x + c)^3 + 68*A*a^2*b
^7*tan(d*x + c)^3 - 12*B*a*b^8*tan(d*x + c)^3 + 24*A*b^9*tan(d*x + c)^3 + 3*A*a^9*tan(d*x + c)^2 + B*a^8*b*tan
(d*x + c)^2 + 13*A*a^7*b^2*tan(d*x + c)^2 - 45*B*a^6*b^3*tan(d*x + c)^2 + 88*A*a^5*b^4*tan(d*x + c)^2 - 52*B*a
^4*b^5*tan(d*x + c)^2 + 102*A*a^3*b^6*tan(d*x + c)^2 - 18*B*a^2*b^7*tan(d*x + c)^2 + 36*A*a*b^8*tan(d*x + c)^2
- 4*B*a^9*tan(d*x + c) + 8*A*a^8*b*tan(d*x + c) - 12*B*a^7*b^2*tan(d*x + c) + 24*A*a^6*b^3*tan(d*x + c) - 12*
B*a^5*b^4*tan(d*x + c) + 24*A*a^4*b^5*tan(d*x + c) - 4*B*a^3*b^6*tan(d*x + c) + 8*A*a^2*b^7*tan(d*x + c) - 2*A
*a^9 - 6*A*a^7*b^2 - 6*A*a^5*b^4 - 2*A*a^3*b^6)/((a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*(b*tan(d*x + c)^2 +
a*tan(d*x + c))^2) + 4*(A*a^2 + 3*B*a*b - 6*A*b^2)*log(abs(tan(d*x + c)))/a^5)/d