\(\int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [264]

Optimal result

Integrand size = 31, antiderivative size = 225 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac {a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac {a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\sin (c+d x))}{d}-\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d} \]

[Out]

(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*x+1/6*a*(24*A*a^2*b-19*A*b^3+6*B*a^3-34*B*a*b^2)*cot(d*x+c)/d+1/
12*a^2*(6*A*a^2-13*A*b^2-16*B*a*b)*cot(d*x+c)^2/d+(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^3)*ln(sin(d*x+c))
/d-1/12*a*(7*A*b+4*B*a)*cot(d*x+c)^3*(a+b*tan(d*x+c))^2/d-1/4*a*A*cot(d*x+c)^4*(a+b*tan(d*x+c))^3/d

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3686, 3726, 3716, 3709, 3612, 3556} \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {a^2 \left (6 a^2 A-16 a b B-13 A b^2\right ) \cot ^2(c+d x)}{12 d}+\frac {a \left (6 a^3 B+24 a^2 A b-34 a b^2 B-19 A b^3\right ) \cot (c+d x)}{6 d}+\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \log (\sin (c+d x))}{d}+x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )-\frac {a (4 a B+7 A b) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d} \]

[In]

Int[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*B)*x + (a*(24*a^2*A*b - 19*A*b^3 + 6*a^3*B - 34*a*b^2*B)*Co
t[c + d*x])/(6*d) + (a^2*(6*a^2*A - 13*A*b^2 - 16*a*b*B)*Cot[c + d*x]^2)/(12*d) + ((a^4*A - 6*a^2*A*b^2 + A*b^
4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Sin[c + d*x]])/d - (a*(7*A*b + 4*a*B)*Cot[c + d*x]^3*(a + b*Tan[c + d*x])^2)/(1
2*d) - (a*A*Cot[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(4*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 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 3686

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*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e
+ f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*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] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

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 3716

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

Rule 3726

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*d^2 + c*(c*C - B*d))*(a + b*Ta
n[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), I
nt[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1)*Simp[A*d*(b*d*m - a*c*(n + 1)) + (c*C - B*d)*(b*c
*m + a*d*(n + 1)) - d*(n + 1)*((A - C)*(b*c - a*d) + B*(a*c + b*d))*Tan[e + f*x] - b*(d*(B*c - A*d)*(m + n + 1
) - C*(c^2*m - d^2*(n + 1)))*Tan[e + f*x]^2, 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] && GtQ[m, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) (a+b \tan (c+d x))^2 \left (a (7 A b+4 a B)-4 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-b (a A-4 b B) \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac {1}{12} \int \cot ^3(c+d x) (a+b \tan (c+d x)) \left (-2 a \left (6 a^2 A-13 A b^2-16 a b B\right )-12 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-2 b \left (5 a A b+2 a^2 B-6 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac {1}{12} \int \cot ^2(c+d x) \left (-2 a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right )+12 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \tan (c+d x)-2 b^2 \left (5 a A b+2 a^2 B-6 b^2 B\right ) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac {a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\frac {1}{12} \int \cot (c+d x) \left (12 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right )+12 \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \tan (c+d x)\right ) \, dx \\ & = \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac {a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac {a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}-\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d}+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \int \cot (c+d x) \, dx \\ & = \left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x+\frac {a \left (24 a^2 A b-19 A b^3+6 a^3 B-34 a b^2 B\right ) \cot (c+d x)}{6 d}+\frac {a^2 \left (6 a^2 A-13 A b^2-16 a b B\right ) \cot ^2(c+d x)}{12 d}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\sin (c+d x))}{d}-\frac {a (7 A b+4 a B) \cot ^3(c+d x) (a+b \tan (c+d x))^2}{12 d}-\frac {a A \cot ^4(c+d x) (a+b \tan (c+d x))^3}{4 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.94 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 a \left (4 a^2 A b-4 A b^3+a^3 B-6 a b^2 B\right ) \cot (c+d x)+6 a^2 \left (a^2 A-6 A b^2-4 a b B\right ) \cot ^2(c+d x)-4 a^3 (4 A b+a B) \cot ^3(c+d x)-3 a^4 A \cot ^4(c+d x)-6 (a+i b)^4 (A+i B) \log (i-\tan (c+d x))+12 \left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\tan (c+d x))-6 (a-i b)^4 (A-i B) \log (i+\tan (c+d x))}{12 d} \]

[In]

Integrate[Cot[c + d*x]^5*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

(12*a*(4*a^2*A*b - 4*A*b^3 + a^3*B - 6*a*b^2*B)*Cot[c + d*x] + 6*a^2*(a^2*A - 6*A*b^2 - 4*a*b*B)*Cot[c + d*x]^
2 - 4*a^3*(4*A*b + a*B)*Cot[c + d*x]^3 - 3*a^4*A*Cot[c + d*x]^4 - 6*(a + I*b)^4*(A + I*B)*Log[I - Tan[c + d*x]
] + 12*(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*Log[Tan[c + d*x]] - 6*(a - I*b)^4*(A - I*B)*Log[I
+ Tan[c + d*x]])/(12*d)

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {\frac {\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{4 \tan \left (d x +c \right )^{4}}-\frac {a^{3} \left (4 A b +B a \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{\tan \left (d x +c \right )}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{2 \tan \left (d x +c \right )^{2}}}{d}\)	\(240\)
default	\(\frac {\frac {\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {A \,a^{4}}{4 \tan \left (d x +c \right )^{4}}-\frac {a^{3} \left (4 A b +B a \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right )}{\tan \left (d x +c \right )}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )}{2 \tan \left (d x +c \right )^{2}}}{d}\)	\(240\)
parallelrisch	\(\frac {6 \left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+12 \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )-3 A \left (\cot ^{4}\left (d x +c \right )\right ) a^{4}+4 \left (-4 A \,a^{3} b -B \,a^{4}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+6 a^{2} \left (\cot ^{2}\left (d x +c \right )\right ) \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right )+12 \left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}\right ) \cot \left (d x +c \right )+48 d \left (A \,a^{3} b -A a \,b^{3}+\frac {1}{4} B \,a^{4}-\frac {3}{2} B \,a^{2} b^{2}+\frac {1}{4} B \,b^{4}\right ) x}{12 d}\)	\(241\)
norman	\(\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) x \left (\tan ^{4}\left (d x +c \right )\right )+\frac {a \left (4 A \,a^{2} b -4 A \,b^{3}+B \,a^{3}-6 B a \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{d}-\frac {A \,a^{4}}{4 d}+\frac {a^{2} \left (A \,a^{2}-6 A \,b^{2}-4 B a b \right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \left (4 A b +B a \right ) \tan \left (d x +c \right )}{3 d}}{\tan \left (d x +c \right )^{4}}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(254\)
risch	\(\frac {8 i B \,a^{3} b c}{d}-i A \,a^{4} x -4 i B a \,b^{3} x -\frac {4 i a \left (8 A \,a^{2} b -9 B a \,b^{2}+2 B \,a^{3}-3 i A \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i A \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i A \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,b^{3}+9 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-12 A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+24 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-20 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-27 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+27 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-5 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+18 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+6 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-3 B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+6 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-18 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-12 i B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+9 i A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+6 i B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-18 i A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 i A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+6 i B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {A \,a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{4}}{d}-\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) A \,b^{2}}{d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}+4 A \,a^{3} b x -4 A a \,b^{3} x -6 B \,a^{2} b^{2} x +B \,a^{4} x +B \,b^{4} x +\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B a \,b^{3}}{d}-\frac {2 i a^{4} A c}{d}-\frac {8 i B a \,b^{3} c}{d}-\frac {2 i A \,b^{4} c}{d}+\frac {12 i A \,a^{2} b^{2} c}{d}-i A \,b^{4} x +6 i A \,a^{2} b^{2} x +4 i B \,a^{3} b x\)	\(636\)

[In]

int(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2*(-A*a^4+6*A*a^2*b^2-A*b^4+4*B*a^3*b-4*B*a*b^3)*ln(1+tan(d*x+c)^2)+(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*
b^2+B*b^4)*arctan(tan(d*x+c))+(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^3)*ln(tan(d*x+c))-1/4*A*a^4/tan(d*x+c
)^4-1/3*a^3*(4*A*b+B*a)/tan(d*x+c)^3+a*(4*A*a^2*b-4*A*b^3+B*a^3-6*B*a*b^2)/tan(d*x+c)+1/2*a^2*(A*a^2-6*A*b^2-4
*B*a*b)/tan(d*x+c)^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{4} - 3 \, A a^{4} + 3 \, {\left (3 \, A a^{4} - 8 \, B a^{3} b - 12 \, A a^{2} b^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} - 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{12 \, d \tan \left (d x + c\right )^{4}} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(6*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x
+ c)^4 - 3*A*a^4 + 3*(3*A*a^4 - 8*B*a^3*b - 12*A*a^2*b^2 + 4*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B
*b^4)*d*x)*tan(d*x + c)^4 + 12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*tan(d*x + c)^3 + 6*(A*a^4 - 4*B*a
^3*b - 6*A*a^2*b^2)*tan(d*x + c)^2 - 4*(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/(d*tan(d*x + c)^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (226) = 452\).

Time = 4.17 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.04 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \tilde {\infty } A a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{5}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } A a^{4} x & \text {for}\: c = - d x \\- \frac {A a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {A a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {A a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} + 4 A a^{3} b x + \frac {4 A a^{3} b}{d \tan {\left (c + d x \right )}} - \frac {4 A a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {3 A a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {6 A a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 A a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - 4 A a b^{3} x - \frac {4 A a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {A b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + B a^{4} x + \frac {B a^{4}}{d \tan {\left (c + d x \right )}} - \frac {B a^{4}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {2 B a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - \frac {4 B a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {2 B a^{3} b}{d \tan ^{2}{\left (c + d x \right )}} - 6 B a^{2} b^{2} x - \frac {6 B a^{2} b^{2}}{d \tan {\left (c + d x \right )}} - \frac {2 B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 B a b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + B b^{4} x & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**5*(a+b*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

Piecewise((zoo*A*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(A + B*tan(c))*(a + b*tan(c))**4*cot(c)**5, Eq(d, 0)), (zoo*
A*a**4*x, Eq(c, -d*x)), (-A*a**4*log(tan(c + d*x)**2 + 1)/(2*d) + A*a**4*log(tan(c + d*x))/d + A*a**4/(2*d*tan
(c + d*x)**2) - A*a**4/(4*d*tan(c + d*x)**4) + 4*A*a**3*b*x + 4*A*a**3*b/(d*tan(c + d*x)) - 4*A*a**3*b/(3*d*ta
n(c + d*x)**3) + 3*A*a**2*b**2*log(tan(c + d*x)**2 + 1)/d - 6*A*a**2*b**2*log(tan(c + d*x))/d - 3*A*a**2*b**2/
(d*tan(c + d*x)**2) - 4*A*a*b**3*x - 4*A*a*b**3/(d*tan(c + d*x)) - A*b**4*log(tan(c + d*x)**2 + 1)/(2*d) + A*b
**4*log(tan(c + d*x))/d + B*a**4*x + B*a**4/(d*tan(c + d*x)) - B*a**4/(3*d*tan(c + d*x)**3) + 2*B*a**3*b*log(t
an(c + d*x)**2 + 1)/d - 4*B*a**3*b*log(tan(c + d*x))/d - 2*B*a**3*b/(d*tan(c + d*x)**2) - 6*B*a**2*b**2*x - 6*
B*a**2*b**2/(d*tan(c + d*x)) - 2*B*a*b**3*log(tan(c + d*x)**2 + 1)/d + 4*B*a*b**3*log(tan(c + d*x))/d + B*b**4
*x, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) - \frac {3 \, A a^{4} - 12 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} - 6 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4}}}{12 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) - 6*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2
+ 4*B*a*b^3 + A*b^4)*log(tan(d*x + c)^2 + 1) + 12*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(ta
n(d*x + c)) - (3*A*a^4 - 12*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3)*tan(d*x + c)^3 - 6*(A*a^4 - 4*B*a^3*
b - 6*A*a^2*b^2)*tan(d*x + c)^2 + 4*(B*a^4 + 4*A*a^3*b)*tan(d*x + c))/tan(d*x + c)^4)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (217) = 434\).

Time = 1.68 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.60 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 576 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 192 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} + 192 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 192 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {400 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1600 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2400 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1600 \, B a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, A b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 576 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 32 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]

[In]

integrate(cot(d*x+c)^5*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/192*(3*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 8*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 -
36*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96*B*a^3*b*tan(1/2*d*x + 1/2*c)^2 + 144*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 1
20*B*a^4*tan(1/2*d*x + 1/2*c) + 480*A*a^3*b*tan(1/2*d*x + 1/2*c) - 576*B*a^2*b^2*tan(1/2*d*x + 1/2*c) - 384*A*
a*b^3*tan(1/2*d*x + 1/2*c) - 192*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*(d*x + c) + 192*(A*a^4
- 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) - 192*(A*a^4 - 4*B*a^3*b - 6*A*
a^2*b^2 + 4*B*a*b^3 + A*b^4)*log(abs(tan(1/2*d*x + 1/2*c))) + (400*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 1600*B*a^3*b
*tan(1/2*d*x + 1/2*c)^4 - 2400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1600*B*a*b^3*tan(1/2*d*x + 1/2*c)^4 + 400*A*
b^4*tan(1/2*d*x + 1/2*c)^4 - 120*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 480*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 576*B*a^2
*b^2*tan(1/2*d*x + 1/2*c)^3 + 384*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 36*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96*B*a^3*
b*tan(1/2*d*x + 1/2*c)^2 + 144*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*B*a^4*tan(1/2*d*x + 1/2*c) + 32*A*a^3*b*ta
n(1/2*d*x + 1/2*c) + 3*A*a^4)/tan(1/2*d*x + 1/2*c)^4)/d

Mupad [B] (verification not implemented)

Time = 8.51 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.97 \[ \int \cot ^5(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^4-4\,B\,a^3\,b-6\,A\,a^2\,b^2+4\,B\,a\,b^3+A\,b^4\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^4\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+\frac {A\,a^4}{4}-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {A\,a^4}{2}+2\,B\,a^3\,b+3\,A\,a^2\,b^2\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4}{2\,d} \]

[In]

int(cot(c + d*x)^5*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^4,x)

[Out]

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