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

Optimal. Leaf size=287 \[ -\frac {(3 b B-a C) \log (\sin (c+d x))}{a^4 d}-\frac {b \left (2 a^2 B-a b C+3 b^2 B\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3}-\frac {b \left (a^4 B-3 a^3 b C+6 a^2 b^2 B-a b^3 C+3 b^4 B\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \left (-6 a^5 C+10 a^4 b B-3 a^3 b^2 C+9 a^2 b^3 B-a b^4 C+3 b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]

[Out]

-(B*a^3-3*B*a*b^2+3*C*a^2*b-C*b^3)*x/(a^2+b^2)^3-(3*B*b-C*a)*ln(sin(d*x+c))/a^4/d+b^2*(10*B*a^4*b+9*B*a^2*b^3+
3*B*b^5-6*C*a^5-3*C*a^3*b^2-C*a*b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)^3/d-1/2*b*(2*B*a^2+3*B*b^2-C*
a*b)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^2-B*cot(d*x+c)/a/d/(a+b*tan(d*x+c))^2-b*(B*a^4+6*B*a^2*b^2+3*B*b^4-3*C*a
^3*b-C*a*b^3)/a^3/(a^2+b^2)^2/d/(a+b*tan(d*x+c))

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Rubi [A]  time = 0.94, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3632, 3609, 3649, 3651, 3530, 3475} \[ -\frac {b \left (6 a^2 b^2 B-3 a^3 b C+a^4 B-a b^3 C+3 b^4 B\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b \left (2 a^2 B-a b C+3 b^2 B\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b^2 \left (9 a^2 b^3 B-3 a^3 b^2 C+10 a^4 b B-6 a^5 C-a b^4 C+3 b^5 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3}-\frac {(3 b B-a C) \log (\sin (c+d x))}{a^4 d}-\frac {B \cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C]  time = 6.42, size = 288, normalized size = 1.00 \[ -\frac {(3 b B-a C) \log (\tan (c+d x))}{a^4 d}-\frac {B \cot (c+d x)}{a^3 d}-\frac {b^2 (b B-a C)}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {b^2 \left (-3 a^3 C+4 a^2 b B-a b^2 C+2 b^3 B\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \left (-6 a^5 C+10 a^4 b B-3 a^3 b^2 C+9 a^2 b^3 B-a b^4 C+3 b^5 B\right ) \log (a+b \tan (c+d x))}{a^4 d \left (a^2+b^2\right )^3}+\frac {(B+i C) \log (-\tan (c+d x)+i)}{2 d (-b+i a)^3}-\frac {(C+i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.89, size = 917, normalized size = 3.20 \[ -\frac {2 \, B a^{9} + 6 \, B a^{7} b^{2} + 6 \, B a^{5} b^{4} + 2 \, B a^{3} b^{6} + {\left (7 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} b^{7} + 2 \, {\left (B a^{7} b^{2} + 3 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} - C a^{4} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (B a^{7} b^{2} + 4 \, C a^{6} b^{3} - 2 \, B a^{5} b^{4} - 3 \, C a^{4} b^{5} + 6 \, B a^{3} b^{6} - C a^{2} b^{7} + 3 \, B a b^{8} + 2 \, {\left (B a^{8} b + 3 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} - C a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left ({\left (C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 9 \, B a^{2} b^{7} + C a b^{8} - 3 \, B b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (C a^{8} b - 3 \, B a^{7} b^{2} + 3 \, C a^{6} b^{3} - 9 \, B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 9 \, B a^{3} b^{6} + C a^{2} b^{7} - 3 \, B a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{9} - 3 \, B a^{8} b + 3 \, C a^{7} b^{2} - 9 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} 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 ({\left (6 \, C a^{5} b^{4} - 10 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 9 \, B a^{2} b^{7} + C a b^{8} - 3 \, B b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (6 \, C a^{6} b^{3} - 10 \, B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 9 \, B a^{3} b^{6} + C a^{2} b^{7} - 3 \, B a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (6 \, C a^{7} b^{2} - 10 \, B a^{6} b^{3} + 3 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + C a^{3} b^{6} - 3 \, B a^{2} 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 ) + {\left (4 \, B a^{8} b + 12 \, B a^{6} b^{3} - 9 \, C a^{5} b^{4} + 23 \, B a^{4} b^{5} - 3 \, C a^{3} b^{6} + 9 \, B a^{2} b^{7} + 2 \, {\left (B a^{9} + 3 \, C a^{8} b - 3 \, B a^{7} b^{2} - C a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*B*a^9 + 6*B*a^7*b^2 + 6*B*a^5*b^4 + 2*B*a^3*b^6 + (7*C*a^5*b^4 - 9*B*a^4*b^5 + C*a^3*b^6 - 3*B*a^2*b^7
+ 2*(B*a^7*b^2 + 3*C*a^6*b^3 - 3*B*a^5*b^4 - C*a^4*b^5)*d*x)*tan(d*x + c)^3 + 2*(B*a^7*b^2 + 4*C*a^6*b^3 - 2*
B*a^5*b^4 - 3*C*a^4*b^5 + 6*B*a^3*b^6 - C*a^2*b^7 + 3*B*a*b^8 + 2*(B*a^8*b + 3*C*a^7*b^2 - 3*B*a^6*b^3 - C*a^5
*b^4)*d*x)*tan(d*x + c)^2 - ((C*a^7*b^2 - 3*B*a^6*b^3 + 3*C*a^5*b^4 - 9*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7
+ C*a*b^8 - 3*B*b^9)*tan(d*x + c)^3 + 2*(C*a^8*b - 3*B*a^7*b^2 + 3*C*a^6*b^3 - 9*B*a^5*b^4 + 3*C*a^4*b^5 - 9*B
*a^3*b^6 + C*a^2*b^7 - 3*B*a*b^8)*tan(d*x + c)^2 + (C*a^9 - 3*B*a^8*b + 3*C*a^7*b^2 - 9*B*a^6*b^3 + 3*C*a^5*b^
4 - 9*B*a^4*b^5 + C*a^3*b^6 - 3*B*a^2*b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + ((6*C*a^5*
b^4 - 10*B*a^4*b^5 + 3*C*a^3*b^6 - 9*B*a^2*b^7 + C*a*b^8 - 3*B*b^9)*tan(d*x + c)^3 + 2*(6*C*a^6*b^3 - 10*B*a^5
*b^4 + 3*C*a^4*b^5 - 9*B*a^3*b^6 + C*a^2*b^7 - 3*B*a*b^8)*tan(d*x + c)^2 + (6*C*a^7*b^2 - 10*B*a^6*b^3 + 3*C*a
^5*b^4 - 9*B*a^4*b^5 + C*a^3*b^6 - 3*B*a^2*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)) + (4*B*a^8*b + 12*B*a^6*b^3 - 9*C*a^5*b^4 + 23*B*a^4*b^5 - 3*C*a^3*b^6 + 9*B*a^2*b^7
+ 2*(B*a^9 + 3*C*a^8*b - 3*B*a^7*b^2 - C*a^6*b^3)*d*x)*tan(d*x + c))/((a^10*b^2 + 3*a^8*b^4 + 3*a^6*b^6 + a^4
*b^8)*d*tan(d*x + c)^3 + 2*(a^11*b + 3*a^9*b^3 + 3*a^7*b^5 + a^5*b^7)*d*tan(d*x + c)^2 + (a^12 + 3*a^10*b^2 +
3*a^8*b^4 + a^6*b^6)*d*tan(d*x + c))

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giac [A]  time = 9.82, size = 560, normalized size = 1.95 \[ -\frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, C a^{5} b^{3} - 10 \, B a^{4} b^{4} + 3 \, C a^{3} b^{5} - 9 \, B a^{2} b^{6} + C a b^{7} - 3 \, B b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} - \frac {18 \, C a^{5} b^{4} \tan \left (d x + c\right )^{2} - 30 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 9 \, C a^{3} b^{6} \tan \left (d x + c\right )^{2} - 27 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 3 \, C a b^{8} \tan \left (d x + c\right )^{2} - 9 \, B b^{9} \tan \left (d x + c\right )^{2} + 42 \, C a^{6} b^{3} \tan \left (d x + c\right ) - 68 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 26 \, C a^{4} b^{5} \tan \left (d x + c\right ) - 66 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 8 \, C a^{2} b^{7} \tan \left (d x + c\right ) - 22 \, B a b^{8} \tan \left (d x + c\right ) + 25 \, C a^{7} b^{2} - 39 \, B a^{6} b^{3} + 19 \, C a^{5} b^{4} - 41 \, B a^{4} b^{5} + 6 \, C a^{3} b^{6} - 14 \, B a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {2 \, {\left (C a - 3 \, B b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {2 \, {\left (C a \tan \left (d x + c\right ) - 3 \, B b \tan \left (d x + c\right ) + B a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(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*(B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (C*a^3 - 3*B*a
^2*b - 3*C*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*(6*C*a^5*b^3 - 10*B*
a^4*b^4 + 3*C*a^3*b^5 - 9*B*a^2*b^6 + C*a*b^7 - 3*B*b^8)*log(abs(b*tan(d*x + c) + a))/(a^10*b + 3*a^8*b^3 + 3*
a^6*b^5 + a^4*b^7) - (18*C*a^5*b^4*tan(d*x + c)^2 - 30*B*a^4*b^5*tan(d*x + c)^2 + 9*C*a^3*b^6*tan(d*x + c)^2 -
27*B*a^2*b^7*tan(d*x + c)^2 + 3*C*a*b^8*tan(d*x + c)^2 - 9*B*b^9*tan(d*x + c)^2 + 42*C*a^6*b^3*tan(d*x + c) -
68*B*a^5*b^4*tan(d*x + c) + 26*C*a^4*b^5*tan(d*x + c) - 66*B*a^3*b^6*tan(d*x + c) + 8*C*a^2*b^7*tan(d*x + c)
- 22*B*a*b^8*tan(d*x + c) + 25*C*a^7*b^2 - 39*B*a^6*b^3 + 19*C*a^5*b^4 - 41*B*a^4*b^5 + 6*C*a^3*b^6 - 14*B*a^2
*b^7)/((a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*(b*tan(d*x + c) + a)^2) - 2*(C*a - 3*B*b)*log(abs(tan(d*x + c)
))/a^4 + 2*(C*a*tan(d*x + c) - 3*B*b*tan(d*x + c) + B*a)/(a^4*tan(d*x + c)))/d

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maple [B]  time = 0.93, size = 651, normalized size = 2.27 \[ -\frac {4 b^{3} B}{d a \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 b^{5} B}{d \,a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{2} C}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{4} C}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {10 \ln \left (a +b \tan \left (d x +c \right )\right ) b^{3} B}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {9 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{2} \left (a^{2}+b^{2}\right )^{3}}+\frac {3 b^{7} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {6 \ln \left (a +b \tan \left (d x +c \right )\right ) C a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) C}{d a \left (a^{2}+b^{2}\right )^{3}}-\frac {b^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) C}{d \,a^{3} \left (a^{2}+b^{2}\right )^{3}}-\frac {b^{3} B}{2 d \,a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} C}{2 d a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {B}{d \,a^{3} \tan \left (d x +c \right )}-\frac {3 \ln \left (\tan \left (d x +c \right )\right ) B b}{d \,a^{4}}+\frac {\ln \left (\tan \left (d x +c \right )\right ) C}{d \,a^{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 {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C \,a^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C a \,b^{2}}{2 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}}-\frac {3 C \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {C \arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^3*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^3,x)

[Out]

-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+3/d/(a^2+b^2)^2/(a+b*tan(
d*x+c))*b^2*C+1/d*b^4/a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))*C+10/d/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*b^3*B+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-6/d/(a^2+b^2)^3*ln(a+b*tan(d*x+
c))*C*a*b^2-3/d*b^4/a/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*C-1/d*b^6/a^3/(a^2+b^2)^3*ln(a+b*tan(d*x+c))*C-1/2/d*b^3/
a^2/(a^2+b^2)/(a+b*tan(d*x+c))^2*B+1/2/d*b^2/a/(a^2+b^2)/(a+b*tan(d*x+c))^2*C-1/d*B/a^3/tan(d*x+c)-3/d/a^4*ln(
tan(d*x+c))*B*b+1/d/a^3*ln(tan(d*x+c))*C+3/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*a^2*b*B-1/2/d/(a^2+b^2)^3*ln(1+t
an(d*x+c)^2)*b^3*B-1/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*C*a^3+3/2/d/(a^2+b^2)^3*ln(1+tan(d*x+c)^2)*C*a*b^2-1/d
/(a^2+b^2)^3*B*arctan(tan(d*x+c))*a^3+3/d/(a^2+b^2)^3*B*arctan(tan(d*x+c))*a*b^2-3/d/(a^2+b^2)^3*C*arctan(tan(
d*x+c))*a^2*b+1/d/(a^2+b^2)^3*C*arctan(tan(d*x+c))*b^3

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maxima [A]  time = 0.72, size = 454, normalized size = 1.58 \[ -\frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, C a^{5} b^{2} - 10 \, B a^{4} b^{3} + 3 \, C a^{3} b^{4} - 9 \, B a^{2} b^{5} + C a b^{6} - 3 \, B b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, B a^{6} + 4 \, B a^{4} b^{2} + 2 \, B a^{2} b^{4} + 2 \, {\left (B a^{4} b^{2} - 3 \, C a^{3} b^{3} + 6 \, B a^{2} b^{4} - C a b^{5} + 3 \, B b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, B a^{5} b - 7 \, C a^{4} b^{2} + 17 \, B a^{3} b^{3} - 3 \, C a^{2} b^{4} + 9 \, B a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (C a - 3 \, B b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^3*(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*(B*a^3 + 3*C*a^2*b - 3*B*a*b^2 - C*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(6*C*a^5*b^2
- 10*B*a^4*b^3 + 3*C*a^3*b^4 - 9*B*a^2*b^5 + C*a*b^6 - 3*B*b^7)*log(b*tan(d*x + c) + a)/(a^10 + 3*a^8*b^2 + 3
*a^6*b^4 + a^4*b^6) + (C*a^3 - 3*B*a^2*b - 3*C*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*B*a^6 + 4*B*a^4*b^2 + 2*B*a^2*b^4 + 2*(B*a^4*b^2 - 3*C*a^3*b^3 + 6*B*a^2*b^4 - C*a*b^5 + 3*B*
b^6)*tan(d*x + c)^2 + (4*B*a^5*b - 7*C*a^4*b^2 + 17*B*a^3*b^3 - 3*C*a^2*b^4 + 9*B*a*b^5)*tan(d*x + c))/((a^7*b
^2 + 2*a^5*b^4 + a^3*b^6)*tan(d*x + c)^3 + 2*(a^8*b + 2*a^6*b^3 + a^4*b^5)*tan(d*x + c)^2 + (a^9 + 2*a^7*b^2 +
a^5*b^4)*tan(d*x + c)) - 2*(C*a - 3*B*b)*log(tan(d*x + c))/a^4)/d

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mupad [B]  time = 13.99, size = 380, normalized size = 1.32 \[ \frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-6\,C\,a^5+10\,B\,a^4\,b-3\,C\,a^3\,b^2+9\,B\,a^2\,b^3-C\,a\,b^4+3\,B\,b^5\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,B\,b-C\,a\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {B}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (B\,a^4\,b^2-3\,C\,a^3\,b^3+6\,B\,a^2\,b^4-C\,a\,b^5+3\,B\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,B\,a^4\,b-7\,C\,a^3\,b^2+17\,B\,a^2\,b^3-3\,C\,a\,b^4+9\,B\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cot(c + d*x)^3*(B*tan(c + d*x) + C*tan(c + d*x)^2))/(a + b*tan(c + d*x))^3,x)

[Out]

(b^2*log(a + b*tan(c + d*x))*(3*B*b^5 - 6*C*a^5 + 9*B*a^2*b^3 - 3*C*a^3*b^2 + 10*B*a^4*b - C*a*b^4))/(a^4*d*(a
^2 + b^2)^3) - (log(tan(c + d*x) - 1i)*(B*1i - C))/(2*d*(3*a*b^2 - a^2*b*3i - a^3 + b^3*1i)) - (log(tan(c + d*
x))*(3*B*b - C*a))/(a^4*d) - (log(tan(c + d*x) + 1i)*(B - C*1i))/(2*d*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) - (
B/a + (tan(c + d*x)^2*(3*B*b^6 + 6*B*a^2*b^4 + B*a^4*b^2 - 3*C*a^3*b^3 - C*a*b^5))/(a^3*(a^4 + b^4 + 2*a^2*b^2
)) + (tan(c + d*x)*(9*B*b^5 + 17*B*a^2*b^3 - 7*C*a^3*b^2 + 4*B*a^4*b - 3*C*a*b^4))/(2*a^2*(a^4 + b^4 + 2*a^2*b
^2)))/(d*(a^2*tan(c + d*x) + b^2*tan(c + d*x)^3 + 2*a*b*tan(c + d*x)^2))

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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)**3*(B*tan(d*x+c)+C*tan(d*x+c)**2)/(a+b*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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