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

Optimal. Leaf size=215 \[ \frac {A \log (\sin (c+d x))}{a^3 d}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.62, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b-2 a^3 B+A b^3\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b \left (3 a^2 A b^3+6 a^4 A b+a^3 b^2 B-3 a^5 B+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3}-\frac {x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac {A \log (\sin (c+d x))}{a^3 d} \]

Antiderivative was successfully verified.

[In]

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

Rubi steps

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

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Mathematica [C]  time = 3.58, size = 254, normalized size = 1.18 \[ \frac {\frac {4 a b (A b-a B)}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 A b^2}{a^2+a b \tan (c+d x)}+\frac {2 A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}-\frac {2 b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{(a+b \tan (c+d x))^2}-\frac {a (a-i b) (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}-\frac {a (a+i b) (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 a d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.94, size = 683, normalized size = 3.18 \[ -\frac {7 \, B a^{5} b^{3} - 9 \, A a^{4} b^{4} + B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 2 \, {\left (B a^{8} - 3 \, A a^{7} b - 3 \, B a^{6} b^{2} + A a^{5} b^{3}\right )} d x - {\left (5 \, B a^{5} b^{3} - 7 \, A a^{4} b^{4} - B a^{3} b^{5} - A a^{2} b^{6} + 2 \, {\left (B a^{6} b^{2} - 3 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + A a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (A a^{8} + 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} + A a^{2} b^{6} + {\left (A a^{6} b^{2} + 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} + A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (A a^{7} b + 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} + A a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, B a^{7} b - 6 \, A a^{6} b^{2} - B a^{5} b^{3} - 3 \, A a^{4} b^{4} - A a^{2} b^{6} + {\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{6} b^{2} - 6 \, A a^{5} b^{3} - B a^{4} b^{4} - 3 \, A a^{3} b^{5} - A a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, B a^{6} b^{2} - 4 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + 3 \, A a^{3} b^{5} + A a b^{7} + 2 \, {\left (B a^{7} b - 3 \, A a^{6} b^{2} - 3 \, B a^{5} b^{3} + A a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.56, size = 479, normalized size = 2.23 \[ \frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} - B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}} + \frac {2 \, A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {9 \, B a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, A a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, B a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, A a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, A b^{8} \tan \left (d x + c\right )^{2} + 22 \, B a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, A a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, B a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, A a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, A a b^{7} \tan \left (d x + c\right ) + 14 \, B a^{7} b - 25 \, A a^{6} b^{2} + 3 \, B a^{5} b^{3} - 19 \, A a^{4} b^{4} + B a^{3} b^{5} - 6 \, A a^{2} b^{6}}{{\left (a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]

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))^3,x, algorithm="giac")

[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) - (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) + 2*(3*B*a^5*b^2 - 6*A*a^
4*b^3 - B*a^3*b^4 - 3*A*a^2*b^5 - A*b^7)*log(abs(b*tan(d*x + c) + a))/(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7
) + 2*A*log(abs(tan(d*x + c)))/a^3 - (9*B*a^5*b^3*tan(d*x + c)^2 - 18*A*a^4*b^4*tan(d*x + c)^2 - 3*B*a^3*b^5*t
an(d*x + c)^2 - 9*A*a^2*b^6*tan(d*x + c)^2 - 3*A*b^8*tan(d*x + c)^2 + 22*B*a^6*b^2*tan(d*x + c) - 42*A*a^5*b^3
*tan(d*x + c) - 2*B*a^4*b^4*tan(d*x + c) - 26*A*a^3*b^5*tan(d*x + c) - 8*A*a*b^7*tan(d*x + c) + 14*B*a^7*b - 2
5*A*a^6*b^2 + 3*B*a^5*b^3 - 19*A*a^4*b^4 + B*a^3*b^5 - 6*A*a^2*b^6)/((a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6)*(
b*tan(d*x + c) + a)^2))/d

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maple [B]  time = 0.78, size = 540, normalized size = 2.51 \[ \frac {3 A \,b^{2}}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{4} A}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 B a b}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {6 a \,b^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d a \left (a^{2}+b^{2}\right )^{3}}-\frac {b^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {3 a^{2} b \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {\ln \left (a +b \tan \left (d x +c \right )\right ) b^{3} B}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {b^{2} A}{2 d a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {b B}{2 d \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A a \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b B}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{3} B}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 A \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {A \arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 B \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}} \]

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))^3,x)

[Out]

3/d/(a^2+b^2)^2/(a+b*tan(d*x+c))*A*b^2+1/d*b^4/a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))*A-2/d/(a^2+b^2)^2/(a+b*tan(d*x
+c))*B*a*b-6/d*a/(a^2+b^2)^3*b^2*ln(a+b*tan(d*x+c))*A-3/d*b^4/a/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*A-1/d*b^6/a^3/(
a^2+b^2)^3*ln(a+b*tan(d*x+c))*A+3/d*a^2/(a^2+b^2)^3*b*ln(a+b*tan(d*x+c))*B-1/d/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*
b^3*B+1/2/d*b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))^2*A-1/2/d*b/(a^2+b^2)/(a+b*tan(d*x+c))^2*B+1/a^3/d*A*ln(tan(d*x+c
))-1/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*A*a^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*l
n(1+tan(d*x+c)^2)*a^2*b*B+1/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*b^3*B-3/d/(a^2+b^2)^3*A*arctan(tan(d*x+c))*a^2*
b+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^3-3/d/(a^2+b^2)^3*B*arctan(t
an(d*x+c))*a*b^2

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maxima [A]  time = 0.64, size = 372, normalized size = 1.73 \[ \frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, B a^{5} b - 6 \, A a^{4} b^{2} - B a^{3} b^{3} - 3 \, A a^{2} b^{4} - A b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, B a^{4} b - 7 \, A a^{3} b^{2} + B a^{2} b^{3} - 3 \, A a b^{4} + 2 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \]

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))^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*(3*B*a^5*b -
6*A*a^4*b^2 - B*a^3*b^3 - 3*A*a^2*b^4 - A*b^6)*log(b*tan(d*x + c) + a)/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*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) - (5*B*
a^4*b - 7*A*a^3*b^2 + B*a^2*b^3 - 3*A*a*b^4 + 2*(2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*tan(d*x + c))/(a^8 + 2*a^6
*b^2 + a^4*b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2*a^5*b^3 + a^3*b^5)*tan(d*x + c)
) + 2*A*log(tan(d*x + c))/a^3)/d

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mupad [B]  time = 8.38, size = 315, normalized size = 1.47 \[ \frac {\frac {-5\,B\,a^3\,b+7\,A\,a^2\,b^2-B\,a\,b^3+3\,A\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^5+6\,A\,a^4\,b+B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3*A*b^4 + 7*A*a^2*b^2 - B*a*b^3 - 5*B*a^3*b)/(2*a*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c + d*x)*(A*b^5 + 3*A*a^2*
b^3 - 2*B*a^3*b^2))/(a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) + (A*lo
g(tan(c + d*x)))/(a^3*d) + (log(tan(c + d*x) - 1i)*(A*1i - B))/(2*d*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3)) + (lo
g(tan(c + d*x) + 1i)*(A - B*1i))/(2*d*(3*a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (b*log(a + b*tan(c + d*x))*(A*b^5
- 3*B*a^5 + 3*A*a^2*b^3 + B*a^3*b^2 + 6*A*a^4*b))/(a^3*d*(a^2 + b^2)^3)

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

[Out]

Exception raised: AttributeError

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