3.273 \(\int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx\)
Optimal. Leaf size=137 \[ -\frac{\left (a^2 A+a b B-A b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{b^3 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}+\frac{x (A b-a B)}{a^2+b^2}+\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{A \cot ^2(c+d x)}{2 a d} \]
[Out]
((A*b - a*B)*x)/(a^2 + b^2) + ((A*b - a*B)*Cot[c + d*x])/(a^2*d) - (A*Cot[c + d*x]^2)/(2*a*d) - ((a^2*A - A*b^
2 + a*b*B)*Log[Sin[c + d*x]])/(a^3*d) - (b^3*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2
)*d)
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Rubi [A] time = 0.550286, antiderivative size = 137, normalized size of antiderivative = 1.,
number of steps used = 5, 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} \[ -\frac{\left (a^2 A+a b B-A b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{b^3 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}+\frac{x (A b-a B)}{a^2+b^2}+\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{A \cot ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
Int[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]
[Out]
((A*b - a*B)*x)/(a^2 + b^2) + ((A*b - a*B)*Cot[c + d*x])/(a^2*d) - (A*Cot[c + d*x]^2)/(2*a*d) - ((a^2*A - A*b^
2 + a*b*B)*Log[Sin[c + d*x]])/(a^3*d) - (b^3*(A*b - a*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^3*(a^2 + b^2
)*d)
Rule 3609
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n
+ 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))
Rule 3649
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Rule 3651
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[((a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d
))*x)/((a^2 + b^2)*(c^2 + d^2)), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Rule 3530
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=-\frac{A \cot ^2(c+d x)}{2 a d}-\frac{\int \frac{\cot ^2(c+d x) \left (2 (A b-a B)+2 a A \tan (c+d x)+2 A b \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a}\\ &=\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{A \cot ^2(c+d x)}{2 a d}+\frac{\int \frac{\cot (c+d x) \left (-2 \left (a^2 A-A b^2+a b B\right )-2 a^2 B \tan (c+d x)+2 b (A b-a B) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac{(A b-a B) x}{a^2+b^2}+\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{A \cot ^2(c+d x)}{2 a d}-\frac{\left (b^3 (A b-a B)\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac{\left (a^2 A-A b^2+a b B\right ) \int \cot (c+d x) \, dx}{a^3}\\ &=\frac{(A b-a B) x}{a^2+b^2}+\frac{(A b-a B) \cot (c+d x)}{a^2 d}-\frac{A \cot ^2(c+d x)}{2 a d}-\frac{\left (a^2 A-A b^2+a b B\right ) \log (\sin (c+d x))}{a^3 d}-\frac{b^3 (A b-a B) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 1.35034, size = 163, normalized size = 1.19 \[ \frac{\frac{2 b^3 (a B-A b) \log (a+b \tan (c+d x))}{a^3 \left (a^2+b^2\right )}-\frac{2 \left (a^2 A+a b B-A b^2\right ) \log (\tan (c+d x))}{a^3}+\frac{2 (A b-a B) \cot (c+d x)}{a^2}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{a+i b}+\frac{(A-i B) \log (\tan (c+d x)+i)}{a-i b}-\frac{A \cot ^2(c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cot[c + d*x]^3*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x]),x]
[Out]
((2*(A*b - a*B)*Cot[c + d*x])/a^2 - (A*Cot[c + d*x]^2)/a + ((A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b) - (2*(a
^2*A - A*b^2 + a*b*B)*Log[Tan[c + d*x]])/a^3 + ((A - I*B)*Log[I + Tan[c + d*x]])/(a - I*b) + (2*b^3*(-(A*b) +
a*B)*Log[a + b*Tan[c + d*x]])/(a^3*(a^2 + b^2)))/(2*d)
________________________________________________________________________________________
Maple [A] time = 0.114, size = 266, normalized size = 1.9 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Aa}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Bb}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ) b}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{A}{2\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{Ab}{{a}^{2}d\tan \left ( dx+c \right ) }}-{\frac{B}{ad\tan \left ( dx+c \right ) }}-{\frac{A\ln \left ( \tan \left ( dx+c \right ) \right ) }{ad}}+{\frac{A\ln \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{{a}^{3}d}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) Bb}{{a}^{2}d}}-{\frac{{b}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{{a}^{3}d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{{a}^{2}d \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x)
[Out]
1/2/d/(a^2+b^2)*ln(1+tan(d*x+c)^2)*A*a+1/2/d/(a^2+b^2)*ln(1+tan(d*x+c)^2)*B*b+1/d/(a^2+b^2)*A*arctan(tan(d*x+c
))*b-1/d/(a^2+b^2)*B*arctan(tan(d*x+c))*a-1/2/d/a*A/tan(d*x+c)^2+1/d/a^2/tan(d*x+c)*A*b-1/d/a/tan(d*x+c)*B-1/a
/d*A*ln(tan(d*x+c))+1/d/a^3*ln(tan(d*x+c))*A*b^2-1/d/a^2*ln(tan(d*x+c))*B*b-1/d*b^4/a^3/(a^2+b^2)*ln(a+b*tan(d
*x+c))*A+1/d*b^3/a^2/(a^2+b^2)*ln(a+b*tan(d*x+c))*B
________________________________________________________________________________________
Maxima [A] time = 1.47878, size = 213, normalized size = 1.55 \begin{align*} -\frac{\frac{2 \,{\left (B a - A b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} - \frac{2 \,{\left (B a b^{3} - A b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5} + a^{3} b^{2}} - \frac{{\left (A a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (A a^{2} + B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} + \frac{A a + 2 \,{\left (B a - A b\right )} \tan \left (d x + c\right )}{a^{2} \tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="maxima")
[Out]
-1/2*(2*(B*a - A*b)*(d*x + c)/(a^2 + b^2) - 2*(B*a*b^3 - A*b^4)*log(b*tan(d*x + c) + a)/(a^5 + a^3*b^2) - (A*a
+ B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) + 2*(A*a^2 + B*a*b - A*b^2)*log(tan(d*x + c))/a^3 + (A*a + 2*(B*a
- A*b)*tan(d*x + c))/(a^2*tan(d*x + c)^2))/d
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Fricas [A] time = 2.03542, size = 518, normalized size = 3.78 \begin{align*} -\frac{A a^{4} + A a^{2} b^{2} +{\left (A a^{4} + B a^{3} b + B a b^{3} - A b^{4}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} -{\left (B a b^{3} - A b^{4}\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 ) \tan \left (d x + c\right )^{2} +{\left (A a^{4} + A a^{2} b^{2} + 2 \,{\left (B a^{4} - A a^{3} b\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{4} - A a^{3} b + B a^{2} b^{2} - A a b^{3}\right )} \tan \left (d x + c\right )}{2 \,{\left (a^{5} + a^{3} b^{2}\right )} d \tan \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="fricas")
[Out]
-1/2*(A*a^4 + A*a^2*b^2 + (A*a^4 + B*a^3*b + B*a*b^3 - A*b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^2 - (B*a*b^3 - A*b^4)*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^2 + (A*a^4 + A*a^2*b^2 + 2*(B*a^4 - A*a^3*b)*d*x)*tan(d*x + c)^2 + 2*(B*a^4 - A*a^3*b + B*a^2*b^2 - A*a*b
^3)*tan(d*x + c))/((a^5 + a^3*b^2)*d*tan(d*x + c)^2)
________________________________________________________________________________________
Sympy [A] time = 159.793, size = 2594, normalized size = 18.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x)
[Out]
Piecewise((zoo*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(tan(c
+ d*x))/d - A/(2*d*tan(c + d*x)**2) - B*x - B/(d*tan(c + d*x)))/a, Eq(b, 0)), ((A*x + A/(d*tan(c + d*x)) - A/
(3*d*tan(c + d*x)**3) + B*log(tan(c + d*x)**2 + 1)/(2*d) - B*log(tan(c + d*x))/d - B/(2*d*tan(c + d*x)**2))/b,
Eq(a, 0)), (3*A*d*x*tan(c + d*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*I*A*d*x*tan(c + d*
x)**2/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 2*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-2*
b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 2*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(-2*b*d*tan(c +
d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 4*I*A*log(tan(c + d*x))*tan(c + d*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d
*tan(c + d*x)**2) + 4*A*log(tan(c + d*x))*tan(c + d*x)**2/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) -
3*I*A*tan(c + d*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - I*A*tan(c + d*x)/(-2*b*d*tan(c + d
*x)**3 + 2*I*b*d*tan(c + d*x)**2) + A/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*I*B*d*x*tan(c + d
*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*B*d*x*tan(c + d*x)**2/(-2*b*d*tan(c + d*x)**3 +
2*I*b*d*tan(c + d*x)**2) + B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c
+ d*x)**2) - I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) -
2*B*log(tan(c + d*x))*tan(c + d*x)**3/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*I*B*log(tan(c +
d*x))*tan(c + d*x)**2/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*B*tan(c + d*x)**3/(-2*b*d*tan(c +
d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*B*tan(c + d*x)/(-2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2), Eq(
a, -I*b)), (-3*A*d*x*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*I*A*d*x*tan(c + d*x
)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 2*I*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d
*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*A*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)
**3 + 2*I*b*d*tan(c + d*x)**2) + 4*I*A*log(tan(c + d*x))*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(
c + d*x)**2) - 4*A*log(tan(c + d*x))*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*I*A
*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - I*A*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 +
2*I*b*d*tan(c + d*x)**2) - A/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 3*I*B*d*x*tan(c + d*x)**3/(2
*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*B*d*x*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan
(c + d*x)**2) - B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) -
I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*B*log(tan(
c + d*x))*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) + 2*I*B*log(tan(c + d*x))*tan(c +
d*x)**2/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2) - 3*B*tan(c + d*x)**3/(2*b*d*tan(c + d*x)**3 + 2*I*b
*d*tan(c + d*x)**2) - 2*B*tan(c + d*x)/(2*b*d*tan(c + d*x)**3 + 2*I*b*d*tan(c + d*x)**2), Eq(a, I*b)), (zoo*A*
x/a, Eq(c, -d*x)), (x*(A + B*tan(c))*cot(c)**3/(a + b*tan(c)), Eq(d, 0)), (A*a**4*log(tan(c + d*x)**2 + 1)*tan
(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*A*a**4*log(tan(c + d*x))*tan(c + d
*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - A*a**4/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*
b**2*d*tan(c + d*x)**2) + 2*A*a**3*b*d*x*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x
)**2) + 2*A*a**3*b*tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - A*a**2*b**2/(2*a*
*5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + 2*A*a*b**3*tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*
a**3*b**2*d*tan(c + d*x)**2) - 2*A*b**4*log(a/b + tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*
a**3*b**2*d*tan(c + d*x)**2) + 2*A*b**4*log(tan(c + d*x))*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b
**2*d*tan(c + d*x)**2) - 2*B*a**4*d*x*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**
2) - 2*B*a**4*tan(c + d*x)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + B*a**3*b*log(tan(c + d
*x)**2 + 1)*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*B*a**3*b*log(tan(c
+ d*x))*tan(c + d*x)**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*B*a**2*b**2*tan(c + d*x
)/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) + 2*B*a*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)
**2/(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2) - 2*B*a*b**3*log(tan(c + d*x))*tan(c + d*x)**2/
(2*a**5*d*tan(c + d*x)**2 + 2*a**3*b**2*d*tan(c + d*x)**2), True))
________________________________________________________________________________________
Giac [A] time = 1.29315, size = 289, normalized size = 2.11 \begin{align*} -\frac{\frac{2 \,{\left (B a - A b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} - \frac{{\left (A a + B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac{2 \,{\left (B a b^{4} - A b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5} b + a^{3} b^{3}} + \frac{2 \,{\left (A a^{2} + B a b - A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{3 \, A a^{2} \tan \left (d x + c\right )^{2} + 3 \, B a b \tan \left (d x + c\right )^{2} - 3 \, A b^{2} \tan \left (d x + c\right )^{2} - 2 \, B a^{2} \tan \left (d x + c\right ) + 2 \, A a b \tan \left (d x + c\right ) - A a^{2}}{a^{3} \tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c)),x, algorithm="giac")
[Out]
-1/2*(2*(B*a - A*b)*(d*x + c)/(a^2 + b^2) - (A*a + B*b)*log(tan(d*x + c)^2 + 1)/(a^2 + b^2) - 2*(B*a*b^4 - A*b
^5)*log(abs(b*tan(d*x + c) + a))/(a^5*b + a^3*b^3) + 2*(A*a^2 + B*a*b - A*b^2)*log(abs(tan(d*x + c)))/a^3 - (3
*A*a^2*tan(d*x + c)^2 + 3*B*a*b*tan(d*x + c)^2 - 3*A*b^2*tan(d*x + c)^2 - 2*B*a^2*tan(d*x + c) + 2*A*a*b*tan(d
*x + c) - A*a^2)/(a^3*tan(d*x + c)^2))/d