3.43 \(\int \frac{\cot ^2(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{(a+b \tan (c+d x))^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.679883, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3632, 3609, 3649, 3651, 3530, 3475} \[ \frac{b (b B-a C)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2 b B-2 a^3 C+b^3 B\right )}{a^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b \left (3 a^2 b^3 B+a^3 b^2 C+6 a^4 b B-3 a^5 C+b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3}+\frac{B \log (\sin (c+d x))}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]

[Out]

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

Rule 3632

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

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

Mathematica [C]  time = 2.87756, size = 223, normalized size = 1.04 \[ \frac{\frac{b (b B-a C)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{2 b \left (3 a^2 b B-2 a^3 C+b^3 B\right )}{a^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{2 b \left (3 a^2 b^3 B+a^3 b^2 C+6 a^4 b B-3 a^5 C+b^5 B\right ) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )^3}+\frac{2 B \log (\tan (c+d x))}{a^3}-\frac{(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^3}-\frac{(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2))/(a + b*Tan[c + d*x])^3,x]

[Out]

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

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Maple [B]  time = 0.179, size = 540, normalized size = 2.5 \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)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.83939, size = 502, normalized size = 2.33 \begin{align*} \frac{\frac{2 \,{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B 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 \, C a^{5} b - 6 \, B a^{4} b^{2} - C a^{3} b^{3} - 3 \, B a^{2} b^{4} - B 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 (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C 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 \, C a^{4} b - 7 \, B a^{3} b^{2} + C a^{2} b^{3} - 3 \, B a b^{4} + 2 \,{\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - B 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 \, B \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.61426, size = 1451, normalized size = 6.75 \begin{align*} -\frac{7 \, C a^{5} b^{3} - 9 \, B a^{4} b^{4} + C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 2 \,{\left (C a^{8} - 3 \, B a^{7} b - 3 \, C a^{6} b^{2} + B a^{5} b^{3}\right )} d x -{\left (5 \, C a^{5} b^{3} - 7 \, B a^{4} b^{4} - C a^{3} b^{5} - B a^{2} b^{6} + 2 \,{\left (C a^{6} b^{2} - 3 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + B a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} -{\left (B a^{8} + 3 \, B a^{6} b^{2} + 3 \, B a^{4} b^{4} + B a^{2} b^{6} +{\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + 3 \, B a^{2} b^{6} + B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{7} b + 3 \, B a^{5} b^{3} + 3 \, B a^{3} b^{5} + B 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 \, C a^{7} b - 6 \, B a^{6} b^{2} - C a^{5} b^{3} - 3 \, B a^{4} b^{4} - B a^{2} b^{6} +{\left (3 \, C a^{5} b^{3} - 6 \, B a^{4} b^{4} - C a^{3} b^{5} - 3 \, B a^{2} b^{6} - B b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (3 \, C a^{6} b^{2} - 6 \, B a^{5} b^{3} - C a^{4} b^{4} - 3 \, B a^{3} b^{5} - B 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 \, C a^{6} b^{2} - 4 \, B a^{5} b^{3} - 3 \, C a^{4} b^{4} + 3 \, B a^{3} b^{5} + B a b^{7} + 2 \,{\left (C a^{7} b - 3 \, B a^{6} b^{2} - 3 \, C a^{5} b^{3} + B 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [B]  time = 1.58297, size = 647, normalized size = 3.01 \begin{align*} \frac{\frac{2 \,{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C 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 \, C a^{5} b^{2} - 6 \, B a^{4} b^{3} - C a^{3} b^{4} - 3 \, B a^{2} b^{5} - B 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 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{9 \, C a^{5} b^{3} \tan \left (d x + c\right )^{2} - 18 \, B a^{4} b^{4} \tan \left (d x + c\right )^{2} - 3 \, C a^{3} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a^{2} b^{6} \tan \left (d x + c\right )^{2} - 3 \, B b^{8} \tan \left (d x + c\right )^{2} + 22 \, C a^{6} b^{2} \tan \left (d x + c\right ) - 42 \, B a^{5} b^{3} \tan \left (d x + c\right ) - 2 \, C a^{4} b^{4} \tan \left (d x + c\right ) - 26 \, B a^{3} b^{5} \tan \left (d x + c\right ) - 8 \, B a b^{7} \tan \left (d x + c\right ) + 14 \, C a^{7} b - 25 \, B a^{6} b^{2} + 3 \, C a^{5} b^{3} - 19 \, B a^{4} b^{4} + C a^{3} b^{5} - 6 \, B 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(2*(C*a^3 - 3*B*a^2*b - 3*C*a*b^2 + B*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (B*a^3 + 3*C*a^
2*b - 3*B*a*b^2 - C*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*C*a^5*b^2 - 6*B*a^
4*b^3 - C*a^3*b^4 - 3*B*a^2*b^5 - B*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*B*log(abs(tan(d*x + c)))/a^3 - (9*C*a^5*b^3*tan(d*x + c)^2 - 18*B*a^4*b^4*tan(d*x + c)^2 - 3*C*a^3*b^5*t
an(d*x + c)^2 - 9*B*a^2*b^6*tan(d*x + c)^2 - 3*B*b^8*tan(d*x + c)^2 + 22*C*a^6*b^2*tan(d*x + c) - 42*B*a^5*b^3
*tan(d*x + c) - 2*C*a^4*b^4*tan(d*x + c) - 26*B*a^3*b^5*tan(d*x + c) - 8*B*a*b^7*tan(d*x + c) + 14*C*a^7*b - 2
5*B*a^6*b^2 + 3*C*a^5*b^3 - 19*B*a^4*b^4 + C*a^3*b^5 - 6*B*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