\(\int \cot ^5(c+d x) (a+b \tan (c+d x))^2 (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [15]

Optimal result

Integrand size = 40, antiderivative size = 118 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\left (a^2 B-b^2 B-2 a b C\right ) x+\frac {\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac {a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 C-a (2 b B+a C)\right ) \log (\sin (c+d x))}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3713, 3685, 3709, 3610, 3612, 3556} \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\left (a^2 B-2 a b C-b^2 B\right ) \cot (c+d x)}{d}+x \left (a^2 B-2 a b C-b^2 B\right )-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 C-a (a C+2 b B)\right ) \log (\sin (c+d x))}{d}-\frac {a (a C+2 b B) \cot ^2(c+d x)}{2 d} \]

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

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

Rule 3556

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

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[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] && NeQ[a*c + b*d, 0]

Rule 3685

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

Rule 3709

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

Rubi steps \begin{align*} \text {integral}& = \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 (B+C \tan (c+d x)) \, dx \\ & = -\frac {a^2 B \cot ^3(c+d x)}{3 d}+\int \cot ^3(c+d x) \left (a (2 b B+a C)-\left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)+b^2 C \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\int \cot ^2(c+d x) \left (-a^2 B+b^2 B+2 a b C+\left (b^2 C-a (2 b B+a C)\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac {a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\int \cot (c+d x) \left (b^2 C-a (2 b B+a C)+\left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)\right ) \, dx \\ & = \left (a^2 B-b^2 B-2 a b C\right ) x+\frac {\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac {a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\left (b^2 C-a (2 b B+a C)\right ) \int \cot (c+d x) \, dx \\ & = \left (a^2 B-b^2 B-2 a b C\right ) x+\frac {\left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)}{d}-\frac {a (2 b B+a C) \cot ^2(c+d x)}{2 d}-\frac {a^2 B \cot ^3(c+d x)}{3 d}+\frac {\left (b^2 C-a (2 b B+a C)\right ) \log (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.29 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {6 \left (a^2 B-b^2 B-2 a b C\right ) \cot (c+d x)-3 a (2 b B+a C) \cot ^2(c+d x)-2 a^2 B \cot ^3(c+d x)+3 (a+i b)^2 (-i B+C) \log (i-\tan (c+d x))-6 \left (2 a b B+a^2 C-b^2 C\right ) \log (\tan (c+d x))+3 (a-i b)^2 (i B+C) \log (i+\tan (c+d x))}{6 d} \]

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(6*(a^2*B - b^2*B - 2*a*b*C)*Cot[c + d*x] - 3*a*(2*b*B + a*C)*Cot[c + d*x]^2 - 2*a^2*B*Cot[c + d*x]^3 + 3*(a +
I*b)^2*((-I)*B + C)*Log[I - Tan[c + d*x]] - 6*(2*a*b*B + a^2*C - b^2*C)*Log[Tan[c + d*x]] + 3*(a - I*b)^2*(I*
B + C)*Log[I + Tan[c + d*x]])/(6*d)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {B \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+C \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+2 B a b \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 C a b \left (-\cot \left (d x +c \right )-d x -c \right )+B \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+C \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(136\)
default	\(\frac {B \,b^{2} \left (-\cot \left (d x +c \right )-d x -c \right )+C \,b^{2} \ln \left (\sin \left (d x +c \right )\right )+2 B a b \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )+2 C a b \left (-\cot \left (d x +c \right )-d x -c \right )+B \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+C \,a^{2} \left (-\frac {\cot \left (d x +c \right )^{2}}{2}-\ln \left (\sin \left (d x +c \right )\right )\right )}{d}\)	\(136\)
parallelrisch	\(\frac {3 \left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+6 \left (-2 B a b -C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-2 B \,a^{2} \cot \left (d x +c \right )^{3}+3 \left (-2 B a b -C \,a^{2}\right ) \cot \left (d x +c \right )^{2}+6 \cot \left (d x +c \right ) \left (B \,a^{2}-B \,b^{2}-2 C a b \right )+6 d x \left (B \,a^{2}-B \,b^{2}-2 C a b \right )}{6 d}\)	\(143\)
norman	\(\frac {\frac {\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) \tan \left (d x +c \right )^{3}}{d}+\left (B \,a^{2}-B \,b^{2}-2 C a b \right ) x \tan \left (d x +c \right )^{4}-\frac {B \,a^{2} \tan \left (d x +c \right )}{3 d}-\frac {a \left (2 B b +C a \right ) \tan \left (d x +c \right )^{2}}{2 d}}{\tan \left (d x +c \right )^{4}}-\frac {\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (2 B a b +C \,a^{2}-C \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\)	\(167\)
risch	\(B \,a^{2} x -B \,b^{2} x -2 C a b x +\frac {4 i B a b c}{d}-i C \,b^{2} x +i C \,a^{2} x -\frac {2 i \left (6 i B a b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 B \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 B \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+6 C a b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 i B a b \,{\mathrm e}^{2 i \left (d x +c \right )}-3 i C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 B \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 B \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 C a b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 B \,a^{2}+3 B \,b^{2}+6 C a b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}+\frac {2 i C \,a^{2} c}{d}+2 i B a b x -\frac {2 i C \,b^{2} c}{d}-\frac {2 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C \,b^{2}}{d}\)	\(324\)

[In]

int(cot(d*x+c)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.33 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {3 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{3} + 3 \, {\left (C a^{2} + 2 \, B a b - 2 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, B a^{2} - 6 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{6 \, d \tan \left (d x + c\right )^{3}} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (107) = 214\).

Time = 2.31 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.14 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \text {NaN} & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{2} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x \\B a^{2} x + \frac {B a^{2}}{d \tan {\left (c + d x \right )}} - \frac {B a^{2}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {B a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {2 B a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B a b}{d \tan ^{2}{\left (c + d x \right )}} - B b^{2} x - \frac {B b^{2}}{d \tan {\left (c + d x \right )}} + \frac {C a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {C a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {C a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 C a b x - \frac {2 C a b}{d \tan {\left (c + d x \right )}} - \frac {C b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**5*(a+b*tan(d*x+c))**2*(B*tan(d*x+c)+C*tan(d*x+c)**2),x)

[Out]

Piecewise((nan, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**2*(B*tan(c) + C*tan(c)**2)*cot(c)**5, Eq(d, 0)), (nan
, Eq(c, -d*x)), (B*a**2*x + B*a**2/(d*tan(c + d*x)) - B*a**2/(3*d*tan(c + d*x)**3) + B*a*b*log(tan(c + d*x)**2
+ 1)/d - 2*B*a*b*log(tan(c + d*x))/d - B*a*b/(d*tan(c + d*x)**2) - B*b**2*x - B*b**2/(d*tan(c + d*x)) + C*a**
2*log(tan(c + d*x)**2 + 1)/(2*d) - C*a**2*log(tan(c + d*x))/d - C*a**2/(2*d*tan(c + d*x)**2) - 2*C*a*b*x - 2*C
*a*b/(d*tan(c + d*x)) - C*b**2*log(tan(c + d*x)**2 + 1)/(2*d) + C*b**2*log(tan(c + d*x))/d, True))

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.26 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {2 \, B a^{2} - 6 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (C a^{2} + 2 \, B a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{3}}}{6 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (114) = 228\).

Time = 0.91 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.83 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, {\left (B a^{2} - 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )} + 24 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 24 \, {\left (C a^{2} + 2 \, B a b - C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {44 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 88 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 44 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^2*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

1/24*(B*a^2*tan(1/2*d*x + 1/2*c)^3 - 3*C*a^2*tan(1/2*d*x + 1/2*c)^2 - 6*B*a*b*tan(1/2*d*x + 1/2*c)^2 - 15*B*a^
2*tan(1/2*d*x + 1/2*c) + 24*C*a*b*tan(1/2*d*x + 1/2*c) + 12*B*b^2*tan(1/2*d*x + 1/2*c) + 24*(B*a^2 - 2*C*a*b -
B*b^2)*(d*x + c) + 24*(C*a^2 + 2*B*a*b - C*b^2)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 24*(C*a^2 + 2*B*a*b - C*b^2
)*log(abs(tan(1/2*d*x + 1/2*c))) + (44*C*a^2*tan(1/2*d*x + 1/2*c)^3 + 88*B*a*b*tan(1/2*d*x + 1/2*c)^3 - 44*C*b
^2*tan(1/2*d*x + 1/2*c)^3 + 15*B*a^2*tan(1/2*d*x + 1/2*c)^2 - 24*C*a*b*tan(1/2*d*x + 1/2*c)^2 - 12*B*b^2*tan(1
/2*d*x + 1/2*c)^2 - 3*C*a^2*tan(1/2*d*x + 1/2*c) - 6*B*a*b*tan(1/2*d*x + 1/2*c) - B*a^2)/tan(1/2*d*x + 1/2*c)^
3)/d

Mupad [B] (verification not implemented)

Time = 8.74 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.32 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^2 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left (\frac {B\,a^2}{3}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-B\,a^2+2\,C\,a\,b+B\,b^2\right )+\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^2}{2}+B\,b\,a\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (C\,a^2+2\,B\,a\,b-C\,b^2\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d} \]

[In]

int(cot(c + d*x)^5*(B*tan(c + d*x) + C*tan(c + d*x)^2)*(a + b*tan(c + d*x))^2,x)

[Out]

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