3.281 \(\int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\)
Optimal. Leaf size=250 \[ \frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{\left (a^2+b^2\right )^2}+\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {b \left (a^3 (-B)+2 a^2 A b-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {b^3 \left (-4 a^3 B+5 a^2 A b-2 a b^2 B+3 A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]
[Out]
(2*A*a*b-B*a^2+B*b^2)*x/(a^2+b^2)^2-(A*a^2-3*A*b^2+2*B*a*b)*ln(sin(d*x+c))/a^4/d-b^3*(5*A*a^2*b+3*A*b^3-4*B*a^
3-2*B*a*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+b^2)^2/d+b*(2*A*a^2*b+3*A*b^3-B*a^3-2*B*a*b^2)/a^3/(a^2+b^
2)/d/(a+b*tan(d*x+c))+1/2*(3*A*b-2*B*a)*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))-1/2*A*cot(d*x+c)^2/a/d/(a+b*tan(d*x+
c))
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Rubi [A] time = 0.86, antiderivative size = 250, normalized size of antiderivative = 1.00,
number of steps used = 6, 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} \[ \frac {b \left (2 a^2 A b+a^3 (-B)-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2 A+2 a b B-3 A b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^3 \left (5 a^2 A b-4 a^3 B-2 a b^2 B+3 A b^3\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}+\frac {x \left (a^2 (-B)+2 a A b+b^2 B\right )}{\left (a^2+b^2\right )^2}+\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
((2*a*A*b - a^2*B + b^2*B)*x)/(a^2 + b^2)^2 - ((a^2*A - 3*A*b^2 + 2*a*b*B)*Log[Sin[c + d*x]])/(a^4*d) - (b^3*(
5*a^2*A*b + 3*A*b^3 - 4*a^3*B - 2*a*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b*(2
*a^2*A*b + 3*A*b^3 - a^3*B - 2*a*b^2*B))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + ((3*A*b - 2*a*B)*Cot[c + d
*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - (A*Cot[c + d*x]^2)/(2*a*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 ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac {\int \frac {\cot ^2(c+d x) \left (3 A b-2 a B+2 a A \tan (c+d x)+3 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a}\\ &=\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2 A-3 A b^2+2 a b B\right )-2 a^2 B \tan (c+d x)+2 b (3 A b-2 a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2}\\ &=\frac {b \left (2 a^2 A b+3 A b^3-a^3 B-2 a b^2 B\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2+b^2\right ) \left (a^2 A-3 A b^2+2 a b B\right )+2 a^3 (A b-a B) \tan (c+d x)+2 b \left (2 a^2 A b+3 A b^3-a^3 B-2 a b^2 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )}\\ &=\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}+\frac {b \left (2 a^2 A b+3 A b^3-a^3 B-2 a b^2 B\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \int \cot (c+d x) \, dx}{a^4}-\frac {\left (b^3 \left (5 a^2 A b+3 A b^3-4 a^3 B-2 a 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 )^2}\\ &=\frac {\left (2 a A b-a^2 B+b^2 B\right ) x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2 A-3 A b^2+2 a b B\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^3 \left (5 a^2 A b+3 A b^3-4 a^3 B-2 a b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^2 d}+\frac {b \left (2 a^2 A b+3 A b^3-a^3 B-2 a b^2 B\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(3 A b-2 a B) \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 4.62, size = 220, normalized size = 0.88 \[ \frac {-\frac {2 (a B-2 A b) \cot (c+d x)}{a^3}-\frac {A \cot ^2(c+d x)}{a^2}-\frac {2 \left (a^2 A+2 a b B-3 A b^2\right ) \log (\tan (c+d x))}{a^4}+\frac {2 b^3 (A b-a B)}{a^3 \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 b^3 \left (4 a^3 B-5 a^2 A b+2 a b^2 B-3 A b^3\right ) \log (a+b \tan (c+d x))}{a^4 \left (a^2+b^2\right )^2}+\frac {(A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^2}+\frac {(A-i B) \log (\tan (c+d x)+i)}{(a-i b)^2}}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^2,x]
[Out]
((-2*(-2*A*b + a*B)*Cot[c + d*x])/a^3 - (A*Cot[c + d*x]^2)/a^2 + ((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2
- (2*(a^2*A - 3*A*b^2 + 2*a*b*B)*Log[Tan[c + d*x]])/a^4 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*
b^3*(-5*a^2*A*b - 3*A*b^3 + 4*a^3*B + 2*a*b^2*B)*Log[a + b*Tan[c + d*x]])/(a^4*(a^2 + b^2)^2) + (2*b^3*(A*b -
a*B))/(a^3*(a^2 + b^2)*(a + b*Tan[c + d*x])))/(2*d)
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fricas [B] time = 0.67, size = 590, normalized size = 2.36 \[ -\frac {A a^{7} + 2 \, A a^{5} b^{2} + A a^{3} b^{4} + {\left (A a^{6} b + 2 \, A a^{4} b^{3} - 2 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} + 2 \, {\left (B a^{6} b - 2 \, A a^{5} b^{2} - B a^{4} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{7} + 2 \, B a^{6} b - 2 \, A a^{5} b^{2} + 4 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4} + 4 \, B a^{2} b^{5} - 6 \, A a b^{6} + 2 \, {\left (B a^{7} - 2 \, A a^{6} b - B a^{5} b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (A a^{6} b + 2 \, B a^{5} b^{2} - A a^{4} b^{3} + 4 \, B a^{3} b^{4} - 5 \, A a^{2} b^{5} + 2 \, B a b^{6} - 3 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{7} + 2 \, B a^{6} b - A a^{5} b^{2} + 4 \, B a^{4} b^{3} - 5 \, A a^{3} b^{4} + 2 \, B a^{2} b^{5} - 3 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (4 \, B a^{3} b^{4} - 5 \, A a^{2} b^{5} + 2 \, B a b^{6} - 3 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, B a^{4} b^{3} - 5 \, A a^{3} b^{4} + 2 \, B a^{2} b^{5} - 3 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (2 \, B a^{7} - 3 \, A a^{6} b + 4 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} + 2 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \tan \left (d x + c\right )^{3} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \tan \left (d x + c\right )^{2}\right )}} \]
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))^2,x, algorithm="fricas")
[Out]
-1/2*(A*a^7 + 2*A*a^5*b^2 + A*a^3*b^4 + (A*a^6*b + 2*A*a^4*b^3 - 2*B*a^3*b^4 + 3*A*a^2*b^5 + 2*(B*a^6*b - 2*A*
a^5*b^2 - B*a^4*b^3)*d*x)*tan(d*x + c)^3 + (A*a^7 + 2*B*a^6*b - 2*A*a^5*b^2 + 4*B*a^4*b^3 - 7*A*a^3*b^4 + 4*B*
a^2*b^5 - 6*A*a*b^6 + 2*(B*a^7 - 2*A*a^6*b - B*a^5*b^2)*d*x)*tan(d*x + c)^2 + ((A*a^6*b + 2*B*a^5*b^2 - A*a^4*
b^3 + 4*B*a^3*b^4 - 5*A*a^2*b^5 + 2*B*a*b^6 - 3*A*b^7)*tan(d*x + c)^3 + (A*a^7 + 2*B*a^6*b - A*a^5*b^2 + 4*B*a
^4*b^3 - 5*A*a^3*b^4 + 2*B*a^2*b^5 - 3*A*a*b^6)*tan(d*x + c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) - ((4
*B*a^3*b^4 - 5*A*a^2*b^5 + 2*B*a*b^6 - 3*A*b^7)*tan(d*x + c)^3 + (4*B*a^4*b^3 - 5*A*a^3*b^4 + 2*B*a^2*b^5 - 3*
A*a*b^6)*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^7
- 3*A*a^6*b + 4*B*a^5*b^2 - 6*A*a^4*b^3 + 2*B*a^3*b^4 - 3*A*a^2*b^5)*tan(d*x + c))/((a^8*b + 2*a^6*b^3 + a^4*b
^5)*d*tan(d*x + c)^3 + (a^9 + 2*a^7*b^2 + a^5*b^4)*d*tan(d*x + c)^2)
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giac [A] time = 1.71, size = 402, normalized size = 1.61 \[ -\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (4 \, B a^{3} b^{4} - 5 \, A a^{2} b^{5} + 2 \, B a b^{6} - 3 \, A b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}} + \frac {2 \, {\left (4 \, B a^{3} b^{4} \tan \left (d x + c\right ) - 5 \, A a^{2} b^{5} \tan \left (d x + c\right ) + 2 \, B a b^{6} \tan \left (d x + c\right ) - 3 \, A b^{7} \tan \left (d x + c\right ) + 5 \, B a^{4} b^{3} - 6 \, A a^{3} b^{4} + 3 \, B a^{2} b^{5} - 4 \, A a b^{6}\right )}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} + \frac {2 \, {\left (A a^{2} + 2 \, B a b - 3 \, A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {3 \, A a^{2} \tan \left (d x + c\right )^{2} + 6 \, B a b \tan \left (d x + c\right )^{2} - 9 \, A b^{2} \tan \left (d x + c\right )^{2} - 2 \, B a^{2} \tan \left (d x + c\right ) + 4 \, A a b \tan \left (d x + c\right ) - A a^{2}}{a^{4} \tan \left (d x + c\right )^{2}}}{2 \, d} \]
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))^2,x, algorithm="giac")
[Out]
-1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (A*a^2 + 2*B*a*b - A*b^2)*log(tan(d*x +
c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2*(4*B*a^3*b^4 - 5*A*a^2*b^5 + 2*B*a*b^6 - 3*A*b^7)*log(abs(b*tan(d*x + c)
+ a))/(a^8*b + 2*a^6*b^3 + a^4*b^5) + 2*(4*B*a^3*b^4*tan(d*x + c) - 5*A*a^2*b^5*tan(d*x + c) + 2*B*a*b^6*tan(
d*x + c) - 3*A*b^7*tan(d*x + c) + 5*B*a^4*b^3 - 6*A*a^3*b^4 + 3*B*a^2*b^5 - 4*A*a*b^6)/((a^8 + 2*a^6*b^2 + a^4
*b^4)*(b*tan(d*x + c) + a)) + 2*(A*a^2 + 2*B*a*b - 3*A*b^2)*log(abs(tan(d*x + c)))/a^4 - (3*A*a^2*tan(d*x + c)
^2 + 6*B*a*b*tan(d*x + c)^2 - 9*A*b^2*tan(d*x + c)^2 - 2*B*a^2*tan(d*x + c) + 4*A*a*b*tan(d*x + c) - A*a^2)/(a
^4*tan(d*x + c)^2))/d
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maple [A] time = 0.69, size = 457, normalized size = 1.83 \[ -\frac {5 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2}}-\frac {3 b^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{4} \left (a^{2}+b^{2}\right )^{2}}+\frac {4 b^{3} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d a \left (a^{2}+b^{2}\right )^{2}}+\frac {2 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {b^{4} A}{d \,a^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} B}{d \,a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}-\frac {A}{2 a^{2} d \tan \left (d x +c \right )^{2}}+\frac {2 A b}{d \,a^{3} \tan \left (d x +c \right )}-\frac {B}{a^{2} d \tan \left (d x +c \right )}-\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}+\frac {3 \ln \left (\tan \left (d x +c \right )\right ) A \,b^{2}}{d \,a^{4}}-\frac {2 \ln \left (\tan \left (d x +c \right )\right ) B b}{d \,a^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} A}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a b}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {2 A \arctan \left (\tan \left (d x +c \right )\right ) a b}{d \left (a^{2}+b^{2}\right )^{2}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{2}}{d \left (a^{2}+b^{2}\right )^{2}}+\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{2}}{d \left (a^{2}+b^{2}\right )^{2}} \]
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))^2,x)
[Out]
-5/d*b^4/a^2/(a^2+b^2)^2*ln(a+b*tan(d*x+c))*A-3/d*b^6/a^4/(a^2+b^2)^2*ln(a+b*tan(d*x+c))*A+4/d*b^3/a/(a^2+b^2)
^2*ln(a+b*tan(d*x+c))*B+2/d*b^5/a^3/(a^2+b^2)^2*ln(a+b*tan(d*x+c))*B+1/d*b^4/a^3/(a^2+b^2)/(a+b*tan(d*x+c))*A-
1/d*b^3/a^2/(a^2+b^2)/(a+b*tan(d*x+c))*B-1/2/a^2/d*A/tan(d*x+c)^2+2/d/a^3/tan(d*x+c)*A*b-1/a^2/d/tan(d*x+c)*B-
1/a^2/d*A*ln(tan(d*x+c))+3/d/a^4*ln(tan(d*x+c))*A*b^2-2/d/a^3*ln(tan(d*x+c))*B*b+1/2/d/(a^2+b^2)^2*ln(1+tan(d*
x+c)^2)*a^2*A-1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*A*b^2+1/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*B*a*b+2/d/(a^2+b^2
)^2*A*arctan(tan(d*x+c))*a*b-1/d/(a^2+b^2)^2*B*arctan(tan(d*x+c))*a^2+1/d/(a^2+b^2)^2*B*arctan(tan(d*x+c))*b^2
________________________________________________________________________________________
maxima [A] time = 0.82, size = 325, normalized size = 1.30 \[ -\frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (4 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4} + 2 \, B a b^{5} - 3 \, A b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}} - \frac {{\left (A a^{2} + 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {A a^{4} + A a^{2} b^{2} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2} + 2 \, B a b^{3} - 3 \, A b^{4}\right )} \tan \left (d x + c\right )^{2} + {\left (2 \, B a^{4} - 3 \, A a^{3} b + 2 \, B a^{2} b^{2} - 3 \, A a b^{3}\right )} \tan \left (d x + c\right )}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}} + \frac {2 \, {\left (A a^{2} + 2 \, B a b - 3 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
-1/2*(2*(B*a^2 - 2*A*a*b - B*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - 2*(4*B*a^3*b^3 - 5*A*a^2*b^4 + 2*B*a*b^5
- 3*A*b^6)*log(b*tan(d*x + c) + a)/(a^8 + 2*a^6*b^2 + a^4*b^4) - (A*a^2 + 2*B*a*b - A*b^2)*log(tan(d*x + c)^2
+ 1)/(a^4 + 2*a^2*b^2 + b^4) + (A*a^4 + A*a^2*b^2 + 2*(B*a^3*b - 2*A*a^2*b^2 + 2*B*a*b^3 - 3*A*b^4)*tan(d*x +
c)^2 + (2*B*a^4 - 3*A*a^3*b + 2*B*a^2*b^2 - 3*A*a*b^3)*tan(d*x + c))/((a^5*b + a^3*b^3)*tan(d*x + c)^3 + (a^6
+ a^4*b^2)*tan(d*x + c)^2) + 2*(A*a^2 + 2*B*a*b - 3*A*b^2)*log(tan(d*x + c))/a^4)/d
________________________________________________________________________________________
mupad [B] time = 10.66, size = 284, normalized size = 1.14 \[ \frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-B\,a^3\,b+2\,A\,a^2\,b^2-2\,B\,a\,b^3+3\,A\,b^4\right )}{a^3\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3+a\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^2+2\,B\,a\,b-3\,A\,b^2\right )}{a^4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-4\,B\,a^3\,b^3+5\,A\,a^2\,b^4-2\,B\,a\,b^5+3\,A\,b^6\right )}{d\,\left (a^8+2\,a^6\,b^2+a^4\,b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cot(c + d*x)^3*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^2,x)
[Out]
((tan(c + d*x)*(3*A*b - 2*B*a))/(2*a^2) - A/(2*a) + (tan(c + d*x)^2*(3*A*b^4 + 2*A*a^2*b^2 - 2*B*a*b^3 - B*a^3
*b))/(a^3*(a^2 + b^2)))/(d*(a*tan(c + d*x)^2 + b*tan(c + d*x)^3)) - (log(tan(c + d*x))*(A*a^2 - 3*A*b^2 + 2*B*
a*b))/(a^4*d) + (log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(a*b*2i + a^2 - b^2)) - (log(a + b*tan(c + d*x))*(3*A
*b^6 + 5*A*a^2*b^4 - 4*B*a^3*b^3 - 2*B*a*b^5))/(d*(a^8 + a^4*b^4 + 2*a^6*b^2)) + (log(tan(c + d*x) + 1i)*(A*1i
+ B))/(2*d*(2*a*b + a^2*1i - b^2*1i))
________________________________________________________________________________________
sympy [A] time = 10.76, size = 9840, normalized size = 39.36 \[ \text {result too large to display} \]
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))**2,x)
[Out]
Piecewise((zoo*A*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((-A*log(tan(c + d*x)**2 + 1)/(2*d) + A*log(ta
n(c + d*x))/d + A/(2*d*tan(c + d*x)**2) - A/(4*d*tan(c + d*x)**4) + B*x + B/(d*tan(c + d*x)) - B/(3*d*tan(c +
d*x)**3))/b**2, Eq(a, 0)), (15*I*A*d*x*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3
+ 4*b**2*d*tan(c + d*x)**2) + 30*A*d*x*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**
3 + 4*b**2*d*tan(c + d*x)**2) - 15*I*A*d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x
)**3 + 4*b**2*d*tan(c + d*x)**2) + 8*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8
*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 16*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-4*b*
*2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 8*A*log(tan(c + d*x)**2 + 1)*t
an(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 16*A*log(
tan(c + d*x))*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)*
*2) + 32*I*A*log(tan(c + d*x))*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**
2*d*tan(c + d*x)**2) + 16*A*log(tan(c + d*x))*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c +
d*x)**3 + 4*b**2*d*tan(c + d*x)**2) + 15*I*A*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d
*x)**3 + 4*b**2*d*tan(c + d*x)**2) + 22*A*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)
**3 + 4*b**2*d*tan(c + d*x)**2) - 4*I*A*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 +
4*b**2*d*tan(c + d*x)**2) + 2*A/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*
x)**2) - 9*B*d*x*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*
x)**2) + 18*I*B*d*x*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c +
d*x)**2) + 9*B*d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c +
d*x)**2) + 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x
)**3 + 4*b**2*d*tan(c + d*x)**2) + 8*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8
*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(-4*b**
2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 8*I*B*log(tan(c + d*x))*tan(c +
d*x)**4/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 16*B*log(tan(c
+ d*x))*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) +
8*I*B*log(tan(c + d*x))*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan
(c + d*x)**2) - 9*B*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c +
d*x)**2) + 14*I*B*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c +
d*x)**2) + 4*B*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**4 + 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2
), Eq(a, -I*b)), (-15*I*A*d*x*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2
*d*tan(c + d*x)**2) + 30*A*d*x*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**
2*d*tan(c + d*x)**2) + 15*I*A*d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*
b**2*d*tan(c + d*x)**2) + 8*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d
*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) + 16*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-4*b**2*d*tan(
c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 8*A*log(tan(c + d*x)**2 + 1)*tan(c + d*
x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 16*A*log(tan(c + d
*x))*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 32*
I*A*log(tan(c + d*x))*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c
+ d*x)**2) + 16*A*log(tan(c + d*x))*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 +
4*b**2*d*tan(c + d*x)**2) - 15*I*A*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 +
4*b**2*d*tan(c + d*x)**2) + 22*A*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b
**2*d*tan(c + d*x)**2) + 4*I*A*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d
*tan(c + d*x)**2) + 2*A/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) -
9*B*d*x*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) -
18*I*B*d*x*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2)
+ 9*B*d*x*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2)
- 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**4/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*
b**2*d*tan(c + d*x)**2) + 8*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d
*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) + 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(-4*b**2*d*tan(c
+ d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) + 8*I*B*log(tan(c + d*x))*tan(c + d*x)**4/
(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 16*B*log(tan(c + d*x))*t
an(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2) - 8*I*B*log
(tan(c + d*x))*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)
**2) - 9*B*tan(c + d*x)**3/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2)
- 14*I*B*tan(c + d*x)**2/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2)
+ 4*B*tan(c + d*x)/(-4*b**2*d*tan(c + d*x)**4 - 8*I*b**2*d*tan(c + d*x)**3 + 4*b**2*d*tan(c + d*x)**2), Eq(a,
I*b)), (zoo*A*x/a**2, Eq(c, -d*x)), (x*(A + B*tan(c))*cot(c)**3/(a + b*tan(c))**2, Eq(d, 0)), ((A*log(tan(c +
d*x)**2 + 1)/(2*d) - A*log(tan(c + d*x))/d - A/(2*d*tan(c + d*x)**2) - B*x - B/(d*tan(c + d*x)))/a**2, Eq(b, 0
)), (A*a**7*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 +
4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*
tan(c + d*x)**3) - 2*A*a**7*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d
*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**
4*b**5*d*tan(c + d*x)**3) - A*a**7/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(
c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) +
4*A*a**6*b*d*x*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c +
d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + A*
a**6*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**
7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c
+ d*x)**3) - 2*A*a**6*b*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)
**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b
**5*d*tan(c + d*x)**3) + 3*A*a**6*b*tan(c + d*x)/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a*
*7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(
c + d*x)**3) + 4*A*a**5*b**2*d*x*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a*
*7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(
c + d*x)**3) - A*a**5*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan
(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 +
2*a**4*b**5*d*tan(c + d*x)**3) + 2*A*a**5*b**2*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 +
2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan
(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 4*A*a**5*b**2*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a*
*8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c +
d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 2*A*a**5*b**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)*
*3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b*
*5*d*tan(c + d*x)**3) - A*a**4*b**3*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**
8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c +
d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 2*A*a**4*b**3*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d
*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b
**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 6*A*a**4*b**3*tan(c + d*x)/(2*a**9*d*tan(c + d*x)**2
+ 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*t
an(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 10*A*a**3*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a*
*9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)
**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 10*A*a**3*b**4*log(tan(c + d*x))*tan(c
+ d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3
*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 10*A*a**3*b**4*tan(c + d
*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*
tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - A*a**3*b**4/(2*a**9*d*tan(c
+ d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a*
*5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 10*A*a**2*b**5*log(a/b + tan(c + d*x))*tan(c + d*
x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*t
an(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 10*A*a**2*b**5*log(tan(c + d
*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 +
4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 3*A*a**2*b**5
*tan(c + d*x)/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*
b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 6*A*a*b**6*log(a/b +
tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c +
d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 6*A
*a*b**6*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2
*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x
)**3) + 6*A*a*b**6*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(
c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) -
6*A*b**7*log(a/b + tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a
**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan
(c + d*x)**3) + 6*A*b**7*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)
**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b
**5*d*tan(c + d*x)**3) - 2*B*a**7*d*x*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 +
4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d
*tan(c + d*x)**3) - 2*B*a**7*tan(c + d*x)/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2
*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x
)**3) - 2*B*a**6*b*d*x*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*
tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**
3) + 2*B*a**6*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**
3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**
5*d*tan(c + d*x)**3) - 4*B*a**6*b*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan
(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 +
2*a**4*b**5*d*tan(c + d*x)**3) - 2*B*a**6*b*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*
x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4
*b**5*d*tan(c + d*x)**3) + 2*B*a**5*b**2*d*x*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*
x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4
*b**5*d*tan(c + d*x)**3) + 2*B*a**5*b**2*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 +
2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan
(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 4*B*a**5*b**2*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(
c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a
**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 4*B*a**5*b**2*tan(c + d*x)/(2*a**9*d*tan(c + d*x
)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**
4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 2*B*a**4*b**3*d*x*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x
)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**
4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 8*B*a**4*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(
2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c +
d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 8*B*a**4*b**3*log(tan(c + d*x))*tan
(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b
**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 6*B*a**4*b**3*tan(c +
d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*
d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) + 8*B*a**3*b**4*log(a/b + t
an(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*
x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 8*B*a
**3*b**4*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**
2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*
x)**3) - 2*B*a**3*b**4*tan(c + d*x)/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan
(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3)
+ 4*B*a**2*b**5*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3
+ 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5
*d*tan(c + d*x)**3) - 4*B*a**2*b**5*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*t
an(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2
+ 2*a**4*b**5*d*tan(c + d*x)**3) - 4*B*a**2*b**5*tan(c + d*x)**2/(2*a**9*d*tan(c + d*x)**2 + 2*a**8*b*d*tan(c
+ d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*tan(c + d*x)**2 + 2
*a**4*b**5*d*tan(c + d*x)**3) + 4*B*a*b**6*log(a/b + tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c + d*x)**2 +
2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**5*b**4*d*ta
n(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 4*B*a*b**6*log(tan(c + d*x))*tan(c + d*x)**3/(2*a**9*d*tan(c
+ d*x)**2 + 2*a**8*b*d*tan(c + d*x)**3 + 4*a**7*b**2*d*tan(c + d*x)**2 + 4*a**6*b**3*d*tan(c + d*x)**3 + 2*a**
5*b**4*d*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3), True))
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