3.476 \(\int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\)
Optimal. Leaf size=189 \[ \frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}+\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]
[Out]
2*a*b*x/(a^2+b^2)^2-(a^2-3*b^2)*ln(sin(d*x+c))/a^4/d-b^4*(5*a^2+3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/a^4/(a^2+
b^2)^2/d+b^2*(2*a^2+3*b^2)/a^3/(a^2+b^2)/d/(a+b*tan(d*x+c))+3/2*b*cot(d*x+c)/a^2/d/(a+b*tan(d*x+c))-1/2*cot(d*
x+c)^2/a/d/(a+b*tan(d*x+c))
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Rubi [A] time = 0.57, antiderivative size = 189, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used =
{3569, 3649, 3650, 3651, 3530, 3475} \[ \frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^2}+\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]
[Out]
(2*a*b*x)/(a^2 + b^2)^2 - ((a^2 - 3*b^2)*Log[Sin[c + d*x]])/(a^4*d) - (b^4*(5*a^2 + 3*b^2)*Log[a*Cos[c + d*x]
+ b*Sin[c + d*x]])/(a^4*(a^2 + b^2)^2*d) + (b^2*(2*a^2 + 3*b^2))/(a^3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])) + (3
*b*Cot[c + d*x])/(2*a^2*d*(a + b*Tan[c + d*x])) - 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 3569
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && !(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 3650
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*t
an[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 + a^2*C)*(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[A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2)) - a*C*(b*c*(m + 1)
+ a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - b*C)*Tan[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 2)*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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] && ( !IntegerQ[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))^2} \, dx &=-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac {\int \frac {\cot ^2(c+d x) \left (3 b+2 a \tan (c+d x)+3 b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a}\\ &=\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\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^2-3 b^2\right )+6 b^2 \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2}\\ &=\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\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^2-3 b^2\right ) \left (a^2+b^2\right )+2 a^3 b \tan (c+d x)+2 b^2 \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^3 \left (a^2+b^2\right )}\\ &=\frac {2 a b x}{\left (a^2+b^2\right )^2}+\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}-\frac {\left (a^2-3 b^2\right ) \int \cot (c+d x) \, dx}{a^4}-\frac {\left (b^4 \left (5 a^2+3 b^2\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 {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^2 d}+\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 1.26, size = 146, normalized size = 0.77 \[ -\frac {-\frac {4 b \cot (c+d x)}{a^3}+\frac {\cot ^2(c+d x)}{a^2}+\frac {2 b^5}{a^4 \left (a^2+b^2\right ) (a \cot (c+d x)+b)}+\frac {2 b^4 \left (5 a^2+3 b^2\right ) \log (a \cot (c+d x)+b)}{a^4 \left (a^2+b^2\right )^2}-\frac {\log (-\cot (c+d x)+i)}{(a-i b)^2}-\frac {\log (\cot (c+d x)+i)}{(a+i b)^2}}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^3/(a + b*Tan[c + d*x])^2,x]
[Out]
-1/2*((-4*b*Cot[c + d*x])/a^3 + Cot[c + d*x]^2/a^2 + (2*b^5)/(a^4*(a^2 + b^2)*(b + a*Cot[c + d*x])) - Log[I -
Cot[c + d*x]]/(a - I*b)^2 - Log[I + Cot[c + d*x]]/(a + I*b)^2 + (2*b^4*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]]
)/(a^4*(a^2 + b^2)^2))/d
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fricas [B] time = 0.60, size = 386, normalized size = 2.04 \[ -\frac {a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4} - {\left (4 \, a^{5} b^{2} d x - a^{6} b - 2 \, a^{4} b^{3} - 3 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} - {\left (4 \, a^{6} b d x - a^{7} + 2 \, a^{5} b^{2} + 7 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (a^{6} b - a^{4} b^{3} - 5 \, a^{2} b^{5} - 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \tan \left (d x + c\right )^{3} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \tan \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="fricas")
[Out]
-1/2*(a^7 + 2*a^5*b^2 + a^3*b^4 - (4*a^5*b^2*d*x - a^6*b - 2*a^4*b^3 - 3*a^2*b^5)*tan(d*x + c)^3 - (4*a^6*b*d*
x - a^7 + 2*a^5*b^2 + 7*a^3*b^4 + 6*a*b^6)*tan(d*x + c)^2 + ((a^6*b - a^4*b^3 - 5*a^2*b^5 - 3*b^7)*tan(d*x + c
)^3 + (a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)*tan(d*x + c)^2)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1)) + ((5*a^2
*b^5 + 3*b^7)*tan(d*x + c)^3 + (5*a^3*b^4 + 3*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)) - 3*(a^6*b + 2*a^4*b^3 + 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 = 2.18, size = 272, normalized size = 1.44 \[ \frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (5 \, a^{2} b^{5} + 3 \, b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}} + \frac {2 \, {\left (5 \, a^{2} b^{5} \tan \left (d x + c\right ) + 3 \, b^{7} \tan \left (d x + c\right ) + 6 \, a^{3} b^{4} + 4 \, a b^{6}\right )}}{{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}} - \frac {2 \, {\left (a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {3 \, a^{2} \tan \left (d x + c\right )^{2} - 9 \, b^{2} \tan \left (d x + c\right )^{2} + 4 \, a b \tan \left (d x + c\right ) - a^{2}}{a^{4} \tan \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="giac")
[Out]
1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2
*(5*a^2*b^5 + 3*b^7)*log(abs(b*tan(d*x + c) + a))/(a^8*b + 2*a^6*b^3 + a^4*b^5) + 2*(5*a^2*b^5*tan(d*x + c) +
3*b^7*tan(d*x + c) + 6*a^3*b^4 + 4*a*b^6)/((a^8 + 2*a^6*b^2 + a^4*b^4)*(b*tan(d*x + c) + a)) - 2*(a^2 - 3*b^2)
*log(abs(tan(d*x + c)))/a^4 + (3*a^2*tan(d*x + c)^2 - 9*b^2*tan(d*x + c)^2 + 4*a*b*tan(d*x + c) - a^2)/(a^4*ta
n(d*x + c)^2))/d
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maple [A] time = 0.56, size = 240, normalized size = 1.27 \[ \frac {b^{4}}{d \left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {5 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2} a^{2}}-\frac {3 b^{6} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2} a^{4}}-\frac {1}{2 d \,a^{2} \tan \left (d x +c \right )^{2}}-\frac {\ln \left (\tan \left (d x +c \right )\right )}{d \,a^{2}}+\frac {3 \ln \left (\tan \left (d x +c \right )\right ) b^{2}}{d \,a^{4}}+\frac {2 b}{d \,a^{3} \tan \left (d x +c \right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2}}+\frac {2 a b \arctan \left (\tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x)
[Out]
1/d*b^4/(a^2+b^2)/a^3/(a+b*tan(d*x+c))-5/d*b^4/(a^2+b^2)^2/a^2*ln(a+b*tan(d*x+c))-3/d*b^6/(a^2+b^2)^2/a^4*ln(a
+b*tan(d*x+c))-1/2/d/a^2/tan(d*x+c)^2-1/d/a^2*ln(tan(d*x+c))+3/d/a^4*ln(tan(d*x+c))*b^2+2/d/a^3*b/tan(d*x+c)+1
/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*a^2-1/2/d/(a^2+b^2)^2*ln(1+tan(d*x+c)^2)*b^2+2/d/(a^2+b^2)^2*a*b*arctan(ta
n(d*x+c))
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maxima [A] time = 0.69, size = 240, normalized size = 1.27 \[ \frac {\frac {4 \, {\left (d x + c\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (5 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {a^{4} + a^{2} b^{2} - 2 \, {\left (2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )}{{\left (a^{5} b + a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{6} + a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}} - \frac {2 \, {\left (a^{2} - 3 \, b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^3/(a+b*tan(d*x+c))^2,x, algorithm="maxima")
[Out]
1/2*(4*(d*x + c)*a*b/(a^4 + 2*a^2*b^2 + b^4) - 2*(5*a^2*b^4 + 3*b^6)*log(b*tan(d*x + c) + a)/(a^8 + 2*a^6*b^2
+ a^4*b^4) + (a^2 - b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - (a^4 + a^2*b^2 - 2*(2*a^2*b^2 + 3*b
^4)*tan(d*x + c)^2 - 3*(a^3*b + 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^2 - 3*b^2)*log(tan(d*x + c))/a^4)/d
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mupad [B] time = 4.46, size = 222, normalized size = 1.17 \[ \frac {\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )}{2\,a^2}-\frac {1}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2\,b^2+3\,b^4\right )}{a^3\,\left (a^2+b^2\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3+a\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-3\,b^2\right )}{a^4\,d}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (5\,a^2\,b^4+3\,b^6\right )}{d\,\left (a^8+2\,a^6\,b^2+a^4\,b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(c + d*x)^3/(a + b*tan(c + d*x))^2,x)
[Out]
((3*b*tan(c + d*x))/(2*a^2) - 1/(2*a) + (tan(c + d*x)^2*(3*b^4 + 2*a^2*b^2))/(a^3*(a^2 + b^2)))/(d*(a*tan(c +
d*x)^2 + b*tan(c + d*x)^3)) + log(tan(c + d*x) - 1i)/(2*d*(a*b*2i + a^2 - b^2)) + (log(tan(c + d*x) + 1i)*1i)/
(2*d*(2*a*b + a^2*1i - b^2*1i)) - (log(tan(c + d*x))*(a^2 - 3*b^2))/(a^4*d) - (log(a + b*tan(c + d*x))*(3*b^6
+ 5*a^2*b^4))/(d*(a^8 + a^4*b^4 + 2*a^6*b^2))
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sympy [A] time = 7.30, size = 5222, normalized size = 27.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**3/(a+b*tan(d*x+c))**2,x)
[Out]
Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(c, 0) & Eq(d, 0)), ((-log(tan(c + d*x)**2 + 1)/(2*d) + log(tan(c +
d*x))/d + 1/(2*d*tan(c + d*x)**2) - 1/(4*d*tan(c + d*x)**4))/b**2, Eq(a, 0)), (15*I*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*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*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*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*log(ta
n(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*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*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*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*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*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*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*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/(-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*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*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*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*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*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*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*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*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*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*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*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*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/(-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*x/a**2, Eq(c, -d*x)), (x*cot(c)**3/(a + b*tan(c))**2, Eq(d, 0)), ((log(tan(c + d*x)**2 +
1)/(2*d) - log(tan(c + d*x))/d - 1/(2*d*tan(c + d*x)**2))/a**2, Eq(b, 0)), (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**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**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**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**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**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**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**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**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**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**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**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**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**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**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*tan(c + d*x)**2 + 2*a**4*b**5*d*tan(c + d*x)**3) - 10*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**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**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**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**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*ta
n(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**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**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*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*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*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*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*ta
n(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**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), True))
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