3.5.76 \(\int \frac {\cot ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [476]

Optimal. Leaf size=189 \[ \frac {2 a b x}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}-\frac {b^4 \left (5 a^2+3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^2 d}+\frac {b^2 \left (2 a^2+3 b^2\right )}{a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {3 b \cot (c+d x)}{2 a^2 d (a+b \tan (c+d x))}-\frac {\cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))} \]

[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 = 0.39, 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 = {3650, 3730, 3731, 3732, 3611, 3556} \begin {gather*} \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))} \end {gather*}

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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], 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 3650

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 3730

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*Ta
n[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 3731

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 3732

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] Result contains complex when optimal does not.
time = 1.28, size = 146, normalized size = 0.77 \begin {gather*} -\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 ) (b+a \cot (c+d x))}-\frac {\log (i-\cot (c+d x))}{(a-i b)^2}-\frac {\log (i+\cot (c+d x))}{(a+i b)^2}+\frac {2 b^4 \left (5 a^2+3 b^2\right ) \log (b+a \cot (c+d x))}{a^4 \left (a^2+b^2\right )^2}}{2 d} \end {gather*}

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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Maple [A]
time = 0.33, size = 166, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {-\frac {1}{2 a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \tan \left (d x +c \right )}+\frac {b^{4}}{\left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(166\)
default	\(\frac {-\frac {1}{2 a^{2} \tan \left (d x +c \right )^{2}}+\frac {\left (-a^{2}+3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4}}+\frac {2 b}{a^{3} \tan \left (d x +c \right )}+\frac {b^{4}}{\left (a^{2}+b^{2}\right ) a^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2} a^{4}}+\frac {\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+2 a b \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(166\)
norman	\(\frac {\frac {\left (-2 a^{2} b^{2}-3 b^{4}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{d \,a^{4} \left (a^{2}+b^{2}\right )}-\frac {1}{2 d a}+\frac {3 b \tan \left (d x +c \right )}{2 a^{2} d}+\frac {2 a^{2} b x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 b^{2} a x \left (\tan ^{3}\left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{\tan \left (d x +c \right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{4} d}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \left (5 a^{2}+3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{4} d}\)	\(266\)
risch	\(\frac {i x}{2 i a b -a^{2}+b^{2}}+\frac {10 i b^{4} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {10 i b^{4} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}+\frac {6 i b^{6} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{4}}+\frac {6 i b^{6} c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{4} d}+\frac {2 i x}{a^{2}}+\frac {2 i c}{a^{2} d}-\frac {6 i b^{2} x}{a^{4}}-\frac {6 i b^{2} c}{a^{4} d}+\frac {2 i \left (2 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+3 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+3 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-4 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+4 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{2} b^{3}+3 b^{5}+2 a^{4} b \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-i a +b \right ) \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) a^{3} d}-\frac {5 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}-\frac {3 b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{4} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}\)	\(585\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/a^2/tan(d*x+c)^2+(-a^2+3*b^2)/a^4*ln(tan(d*x+c))+2/a^3*b/tan(d*x+c)+b^4/(a^2+b^2)/a^3/(a+b*tan(d*x+c
))-b^4*(5*a^2+3*b^2)/(a^2+b^2)^2/a^4*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^2*(1/2*(a^2-b^2)*ln(1+tan(d*x+c)^2)+2*a*b*
arctan(tan(d*x+c))))

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Maxima [A]
time = 0.50, size = 240, normalized size = 1.27 \begin {gather*} \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} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (185) = 370\).
time = 1.18, size = 386, normalized size = 2.04 \begin {gather*} -\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 )}} \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 2.90, size = 5222, normalized size = 27.63 \begin {gather*} \text {Too large to display} \end {gather*}

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))/a**2, Eq(b, 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(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*lo
g(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*l
og(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)), (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*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) + 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*ta
n(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 + ...

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Giac [A]
time = 0.94, size = 272, normalized size = 1.44 \begin {gather*} \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} \end {gather*}

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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Mupad [B]
time = 4.46, size = 222, normalized size = 1.17 \begin {gather*} \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 )} \end {gather*}

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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