3.1.36 \(\int \frac {\cot ^2(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{(a+b \tan (c+d x))^2} \, dx\) [36]

Optimal. Leaf size=137 \[ -\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {B \log (\sin (c+d x))}{a^2 d}-\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b (b B-a C)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

-(2*B*a*b-C*a^2+C*b^2)*x/(a^2+b^2)^2+B*ln(sin(d*x+c))/a^2/d-b*(3*B*a^2*b+B*b^3-2*C*a^3)*ln(a*cos(d*x+c)+b*sin(
d*x+c))/a^2/(a^2+b^2)^2/d+b*(B*b-C*a)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.28, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3713, 3690, 3732, 3611, 3556} \begin {gather*} \frac {b (b B-a C)}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {x \left (a^2 (-C)+2 a b B+b^2 C\right )}{\left (a^2+b^2\right )^2}+\frac {B \log (\sin (c+d x))}{a^2 d}-\frac {b \left (-2 a^3 C+3 a^2 b B+b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 d \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*a*b*B - a^2*C + b^2*C)*x)/(a^2 + b^2)^2) + (B*Log[Sin[c + d*x]])/(a^2*d) - (b*(3*a^2*b*B + b^3*B - 2*a^3
*C)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2*(a^2 + b^2)^2*d) + (b*(b*B - a*C))/(a*(a^2 + b^2)*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 3690

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 3713

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[1/b^2, Int[(a + b*Tan[e + f*x])
^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rule 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 ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {\cot (c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\\ &=\frac {b (b B-a C)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (\left (a^2+b^2\right ) B-a (b B-a C) \tan (c+d x)+b (b B-a C) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {b (b B-a C)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {B \int \cot (c+d x) \, dx}{a^2}-\frac {\left (b \left (3 a^2 b B+b^3 B-2 a^3 C\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {B \log (\sin (c+d x))}{a^2 d}-\frac {b \left (3 a^2 b B+b^3 B-2 a^3 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b (b B-a C)}{a \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.65, size = 159, normalized size = 1.16 \begin {gather*} -\frac {\frac {(B+i C) \log (i-\tan (c+d x))}{(a+i b)^2}-\frac {2 B \log (\tan (c+d x))}{a^2}+\frac {(B-i C) \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 b \left (3 a^2 b B+b^3 B-2 a^3 C\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {2 b (-b B+a C)}{a \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(((B + I*C)*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (2*B*Log[Tan[c + d*x]])/a^2 + ((B - I*C)*Log[I + Tan[c +
d*x]])/(a - I*b)^2 + (2*b*(3*a^2*b*B + b^3*B - 2*a^3*C)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + (2*b*(
-(b*B) + a*C))/(a*(a^2 + b^2)*(a + b*Tan[c + d*x])))/d

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Maple [A]
time = 0.50, size = 163, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {\frac {\left (-a^{2} B +b^{2} B -2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\left (B b -C a \right ) b}{a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(163\)
default	\(\frac {\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{2}}+\frac {\frac {\left (-a^{2} B +b^{2} B -2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\left (B b -C a \right ) b}{a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(163\)
norman	\(\frac {-\frac {a \left (2 B a b -C \,a^{2}+b^{2} C \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (2 B a b -C \,a^{2}+b^{2} C \right ) x \left (\tan ^{2}\left (d x +c \right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\left (b^{2} B -C a b \right ) b \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{2} \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right )}+\frac {B \ln \left (\tan \left (d x +c \right )\right )}{a^{2} d}-\frac {\left (a^{2} B -b^{2} B +2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b \left (3 B \,a^{2} b +B \,b^{3}-2 C \,a^{3}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}\)	\(272\)
risch	\(\frac {6 i B \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {x C}{2 i b a -a^{2}+b^{2}}+\frac {6 i B \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 i C a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i B c}{a^{2} d}+\frac {2 i b^{4} B x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}-\frac {2 i x B}{a^{2}}-\frac {i x B}{2 i b a -a^{2}+b^{2}}-\frac {4 i C a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2} C}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 i b^{4} B c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}+\frac {2 i b^{3} B}{\left (-i a +b \right ) d a \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2} B}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2} d}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C a b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} d}\)	\(535\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 208, normalized size = 1.52 \begin {gather*} \frac {\frac {2 \, {\left (C a^{2} - 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, C a^{3} b - 3 \, B a^{2} b^{2} - B b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}} - \frac {{\left (B a^{2} + 2 \, C a b - B 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 (C a b - B b^{2}\right )}}{a^{4} + a^{2} b^{2} + {\left (a^{3} b + a b^{3}\right )} \tan \left (d x + c\right )} + \frac {2 \, B \log \left (\tan \left (d x + c\right )\right )}{a^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(C*a^2 - 2*B*a*b - C*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*C*a^3*b - 3*B*a^2*b^2 - B*b^4)*log(b
*tan(d*x + c) + a)/(a^6 + 2*a^4*b^2 + a^2*b^4) - (B*a^2 + 2*C*a*b - B*b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^
2*b^2 + b^4) - 2*(C*a*b - B*b^2)/(a^4 + a^2*b^2 + (a^3*b + a*b^3)*tan(d*x + c)) + 2*B*log(tan(d*x + c))/a^2)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (137) = 274\).
time = 8.04, size = 323, normalized size = 2.36 \begin {gather*} -\frac {2 \, C a^{2} b^{3} - 2 \, B a b^{4} - 2 \, {\left (C a^{5} - 2 \, B a^{4} b - C a^{3} b^{2}\right )} d x - {\left (B a^{5} + 2 \, B a^{3} b^{2} + B a b^{4} + {\left (B a^{4} b + 2 \, B a^{2} b^{3} + B b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} - B a b^{4} + {\left (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - B b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} + {\left (C a^{4} b - 2 \, B a^{3} b^{2} - C a^{2} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 4.02, size = 4461, normalized size = 32.56 \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)**2*(B*tan(d*x+c)+C*tan(d*x+c)**2)/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x*(B*tan(c) + C*tan(c)**2)*cot(c)**2/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-B*log(tan(c
+ d*x)**2 + 1)/(2*d) + B*log(tan(c + d*x))/d + C*x)/a**2, Eq(b, 0)), ((B*log(tan(c + d*x)**2 + 1)/(2*d) - B*l
og(tan(c + d*x))/d - B/(2*d*tan(c + d*x)**2) - C*x - C/(d*tan(c + d*x)))/b**2, Eq(a, 0)), (3*I*B*d*x*tan(c + d
*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*B*d*x*tan(c + d*x)/(4*b**2*d*tan(c
+ d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*I*B*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x
) - 4*b**2*d) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*
x) - 4*b**2*d) - 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*
x) - 4*b**2*d) - 2*B*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d)
- 4*B*log(tan(c + d*x))*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*I*
B*log(tan(c + d*x))*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*B*log(tan
(c + d*x))/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 3*I*B*tan(c + d*x)/(4*b**2*d*tan(
c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*B/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) -
4*b**2*d) - C*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*I*C*d*x*
tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + C*d*x/(4*b**2*d*tan(c + d*x)**2
- 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - C*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) -
4*b**2*d) + 2*I*C/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b)), (-3*I*B*d*x*t
an(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*B*d*x*tan(c + d*x)/(4*b**2*
d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 3*I*B*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan
(c + d*x) - 4*b**2*d) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*ta
n(c + d*x) - 4*b**2*d) + 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*ta
n(c + d*x) - 4*b**2*d) - 2*B*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*
b**2*d) - 4*B*log(tan(c + d*x))*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d
) - 8*I*B*log(tan(c + d*x))*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*B
*log(tan(c + d*x))/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*I*B*tan(c + d*x)/(4*b**
2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*B/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c +
d*x) - 4*b**2*d) - C*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 2*
I*C*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + C*d*x/(4*b**2*d*tan(c +
d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - C*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c +
d*x) - 4*b**2*d) - 2*I*C/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, I*b)), (x*(B*
tan(c) + C*tan(c)**2)*cot(c)**2/(a + b*tan(c))**2, Eq(d, 0)), (-B*a**5*log(tan(c + d*x)**2 + 1)/(2*a**7*d + 2*
a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x
)) + 2*B*a**5*log(tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*
x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 4*B*a**4*b*d*x/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5
*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - B*a**4*b*log(tan(c + d*x)
**2 + 1)*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**
3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*B*a**4*b*log(tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(
c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 4*B*a**3
*b**2*d*x*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a*
*3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) - 6*B*a**3*b**2*log(a/b + tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c +
d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + B*a**3*b**2
*log(tan(c + d*x)**2 + 1)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2
*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 4*B*a**3*b**2*log(tan(c + d*x))/(2*a**7*d + 2*a**6*b*d*tan(c + d*
x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5*d*tan(c + d*x)) + 2*B*a**3*b**2/
(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a**3*b**4*d + 2*a**2*b**5
*d*tan(c + d*x)) - 6*B*a**2*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**7*d + 2*a**6*b*d*tan(c + d*x) + 4*
a**5*b**2*d + 4*a**4*b**3*d*tan(c + d*x) + 2*a*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (137) = 274\).
time = 1.30, size = 279, normalized size = 2.04 \begin {gather*} \frac {\frac {2 \, {\left (C a^{2} - 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (B a^{2} + 2 \, C a b - B 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 (2 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - B b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}} + \frac {2 \, B \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (2 \, C a^{3} b^{2} \tan \left (d x + c\right ) - 3 \, B a^{2} b^{3} \tan \left (d x + c\right ) - B b^{5} \tan \left (d x + c\right ) + 3 \, C a^{4} b - 4 \, B a^{3} b^{2} + C a^{2} b^{3} - 2 \, B a b^{4}\right )}}{{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(C*a^2 - 2*B*a*b - C*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (B*a^2 + 2*C*a*b - B*b^2)*log(tan(d*x + c
)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*C*a^3*b^2 - 3*B*a^2*b^3 - B*b^5)*log(abs(b*tan(d*x + c) + a))/(a^6*b +
2*a^4*b^3 + a^2*b^5) + 2*B*log(abs(tan(d*x + c)))/a^2 - 2*(2*C*a^3*b^2*tan(d*x + c) - 3*B*a^2*b^3*tan(d*x + c
) - B*b^5*tan(d*x + c) + 3*C*a^4*b - 4*B*a^3*b^2 + C*a^2*b^3 - 2*B*a*b^4)/((a^6 + 2*a^4*b^2 + a^2*b^4)*(b*tan(
d*x + c) + a)))/d

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Mupad [B]
time = 10.69, size = 180, normalized size = 1.31 \begin {gather*} \frac {B\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\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 )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}+\frac {B\,b^2-C\,a\,b}{a\,d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-2\,C\,a^3+3\,B\,a^2\,b+B\,b^3\right )}{a^2\,d\,{\left (a^2+b^2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*log(tan(c + d*x)))/(a^2*d) - (log(tan(c + d*x) - 1i)*(B + C*1i))/(2*d*(a*b*2i + a^2 - b^2)) - (log(tan(c +
d*x) + 1i)*(B*1i + C))/(2*d*(2*a*b + a^2*1i - b^2*1i)) + (B*b^2 - C*a*b)/(a*d*(a^2 + b^2)*(a + b*tan(c + d*x))
) - (b*log(a + b*tan(c + d*x))*(B*b^3 - 2*C*a^3 + 3*B*a^2*b))/(a^2*d*(a^2 + b^2)^2)

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