\(\int \cot ^7(c+d x) (a+b \tan (c+d x))^3 (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [24]

Optimal result

Integrand size = 40, antiderivative size = 233 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\left (\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x\right )-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]

[Out]

-(B*a^3-3*B*a*b^2-3*C*a^2*b+C*b^3)*x-(B*a^3-3*B*a*b^2-3*C*a^2*b+C*b^3)*cot(d*x+c)/d+1/2*(3*B*a^2*b-B*b^3+C*a^3
-3*C*a*b^2)*cot(d*x+c)^2/d+1/15*a*(5*B*a^2-12*B*b^2-15*C*a*b)*cot(d*x+c)^3/d-1/20*a^2*(7*B*b+5*C*a)*cot(d*x+c)
^4/d+(3*B*a^2*b-B*b^3+C*a^3-3*C*a*b^2)*ln(sin(d*x+c))/d-1/5*a*B*cot(d*x+c)^5*(a+b*tan(d*x+c))^2/d

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3713, 3686, 3716, 3709, 3610, 3612, 3556} \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {a \left (5 a^2 B-15 a b C-12 b^2 B\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (5 a C+7 b B) \cot ^4(c+d x)}{20 d}+\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right ) \log (\sin (c+d x))}{d}-x \left (a^3 B-3 a^2 b C-3 a b^2 B+b^3 C\right )-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \]

[In]

Int[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

-((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*x) - ((a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Cot[c + d*x])/d + ((3*
a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c + d*x]^2)/(2*d) + (a*(5*a^2*B - 12*b^2*B - 15*a*b*C)*Cot[c + d*x]^3
)/(15*d) - (a^2*(7*b*B + 5*a*C)*Cot[c + d*x]^4)/(20*d) + ((3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Log[Sin[c +
d*x]])/d - (a*B*Cot[c + d*x]^5*(a + b*Tan[c + d*x])^2)/(5*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 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 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 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]

Rubi steps \begin{align*} \text {integral}& = \int \cot ^6(c+d x) (a+b \tan (c+d x))^3 (B+C \tan (c+d x)) \, dx \\ & = -\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^5(c+d x) (a+b \tan (c+d x)) \left (a (7 b B+5 a C)-5 \left (a^2 B-b^2 B-2 a b C\right ) \tan (c+d x)-b (3 a B-5 b C) \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^4(c+d x) \left (-a \left (5 a^2 B-12 b^2 B-15 a b C\right )-5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)-b^2 (3 a B-5 b C) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^3(c+d x) \left (-5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right )+5 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot ^2(c+d x) \left (5 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right )+5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \tan (c+d x)\right ) \, dx \\ & = -\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\frac {1}{5} \int \cot (c+d x) \left (5 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right )-5 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \tan (c+d x)\right ) \, dx \\ & = -\left (\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x\right )-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d}+\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \int \cot (c+d x) \, dx \\ & = -\left (\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) x\right )-\frac {\left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)}{d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)}{2 d}+\frac {a \left (5 a^2 B-12 b^2 B-15 a b C\right ) \cot ^3(c+d x)}{15 d}-\frac {a^2 (7 b B+5 a C) \cot ^4(c+d x)}{20 d}+\frac {\left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\sin (c+d x))}{d}-\frac {a B \cot ^5(c+d x) (a+b \tan (c+d x))^2}{5 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.02 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {-60 \left (a^3 B-3 a b^2 B-3 a^2 b C+b^3 C\right ) \cot (c+d x)+30 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \cot ^2(c+d x)+20 a \left (a^2 B-3 b^2 B-3 a b C\right ) \cot ^3(c+d x)-15 a^2 (3 b B+a C) \cot ^4(c+d x)-12 a^3 B \cot ^5(c+d x)+30 i (a+i b)^3 (B+i C) \log (i-\tan (c+d x))+60 \left (3 a^2 b B-b^3 B+a^3 C-3 a b^2 C\right ) \log (\tan (c+d x))+30 (i a+b)^3 (B-i C) \log (i+\tan (c+d x))}{60 d} \]

[In]

Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^3*(B*Tan[c + d*x] + C*Tan[c + d*x]^2),x]

[Out]

(-60*(a^3*B - 3*a*b^2*B - 3*a^2*b*C + b^3*C)*Cot[c + d*x] + 30*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Cot[c +
d*x]^2 + 20*a*(a^2*B - 3*b^2*B - 3*a*b*C)*Cot[c + d*x]^3 - 15*a^2*(3*b*B + a*C)*Cot[c + d*x]^4 - 12*a^3*B*Cot
[c + d*x]^5 + (30*I)*(a + I*b)^3*(B + I*C)*Log[I - Tan[c + d*x]] + 60*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*
Log[Tan[c + d*x]] + 30*(I*a + b)^3*(B - I*C)*Log[I + Tan[c + d*x]])/(60*d)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\frac {\left (-90 B \,a^{2} b +30 B \,b^{3}-30 C \,a^{3}+90 C a \,b^{2}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+\left (180 B \,a^{2} b -60 B \,b^{3}+60 C \,a^{3}-180 C a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-12 B \,a^{3} \cot \left (d x +c \right )^{5}+\left (-45 B \,a^{2} b -15 C \,a^{3}\right ) \cot \left (d x +c \right )^{4}+20 a \cot \left (d x +c \right )^{3} \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )+\left (90 B \,a^{2} b -30 B \,b^{3}+30 C \,a^{3}-90 C a \,b^{2}\right ) \cot \left (d x +c \right )^{2}+\left (-60 B \,a^{3}+180 B a \,b^{2}+180 C \,a^{2} b -60 C \,b^{3}\right ) \cot \left (d x +c \right )-60 d x \left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right )}{60 d}\)	\(243\)
derivativedivides	\(\frac {-\frac {-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}}{2 \tan \left (d x +c \right )^{2}}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}}{\tan \left (d x +c \right )}-\frac {B \,a^{3}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{2} \left (3 B b +C a \right )}{4 \tan \left (d x +c \right )^{4}}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {\left (-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(248\)
default	\(\frac {-\frac {-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}}{2 \tan \left (d x +c \right )^{2}}+\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}}{\tan \left (d x +c \right )}-\frac {B \,a^{3}}{5 \tan \left (d x +c \right )^{5}}-\frac {a^{2} \left (3 B b +C a \right )}{4 \tan \left (d x +c \right )^{4}}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right )}{3 \tan \left (d x +c \right )^{3}}+\frac {\left (-3 B \,a^{2} b +B \,b^{3}-C \,a^{3}+3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(248\)
norman	\(\frac {\left (-B \,a^{3}+3 B a \,b^{2}+3 C \,a^{2} b -C \,b^{3}\right ) x \tan \left (d x +c \right )^{6}+\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \tan \left (d x +c \right )^{4}}{2 d}-\frac {\left (B \,a^{3}-3 B a \,b^{2}-3 C \,a^{2} b +C \,b^{3}\right ) \tan \left (d x +c \right )^{5}}{d}-\frac {B \,a^{3} \tan \left (d x +c \right )}{5 d}+\frac {a \left (B \,a^{2}-3 B \,b^{2}-3 C a b \right ) \tan \left (d x +c \right )^{3}}{3 d}-\frac {a^{2} \left (3 B b +C a \right ) \tan \left (d x +c \right )^{2}}{4 d}}{\tan \left (d x +c \right )^{6}}+\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {\left (3 B \,a^{2} b -B \,b^{3}+C \,a^{3}-3 C a \,b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\)	\(275\)
risch	\(-B \,a^{3} x +3 B a \,b^{2} x +3 C \,a^{2} b x -C \,b^{3} x -\frac {6 i B \,a^{2} b c}{d}-\frac {2 i \left (-60 C \,a^{2} b -60 B a \,b^{2}+15 C \,b^{3}+23 B \,a^{3}-70 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-60 C \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-45 i C a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+180 i B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-135 i C a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-180 i B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+135 i C a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+90 i B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-90 i B \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+45 i C a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+90 C \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-90 B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+45 B \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+15 C \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-60 C \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+140 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-90 C \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+270 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+270 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-330 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-330 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+210 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+210 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-60 i C \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-15 i B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+30 i C \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-90 B a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+15 i B \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-30 i C \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-45 i B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+60 i C \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+45 i B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}+i B \,b^{3} x -i C \,a^{3} x +\frac {6 i C a \,b^{2} c}{d}+\frac {2 i B \,b^{3} c}{d}-\frac {2 i C \,a^{3} c}{d}+3 i C a \,b^{2} x -3 i B \,a^{2} b x +\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B b}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) B \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) C a \,b^{2}}{d}\)	\(755\)

[In]

int(cot(d*x+c)^7*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/60*((-90*B*a^2*b+30*B*b^3-30*C*a^3+90*C*a*b^2)*ln(sec(d*x+c)^2)+(180*B*a^2*b-60*B*b^3+60*C*a^3-180*C*a*b^2)*
ln(tan(d*x+c))-12*B*a^3*cot(d*x+c)^5+(-45*B*a^2*b-15*C*a^3)*cot(d*x+c)^4+20*a*cot(d*x+c)^3*(B*a^2-3*B*b^2-3*C*
a*b)+(90*B*a^2*b-30*B*b^3+30*C*a^3-90*C*a*b^2)*cot(d*x+c)^2+(-60*B*a^3+180*B*a*b^2+180*C*a^2*b-60*C*b^3)*cot(d
*x+c)-60*d*x*(B*a^3-3*B*a*b^2-3*C*a^2*b+C*b^3))/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.14 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, C a^{3} + 9 \, B a^{2} b - 6 \, C a b^{2} - 2 \, B b^{3} - 4 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{5} - 60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \tan \left (d x + c\right )^{4} - 12 \, B a^{3} + 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 20 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} - 15 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{60 \, d \tan \left (d x + c\right )^{5}} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/60*(30*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^5 + 15*
(3*C*a^3 + 9*B*a^2*b - 6*C*a*b^2 - 2*B*b^3 - 4*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*d*x)*tan(d*x + c)^5 - 6
0*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*tan(d*x + c)^4 - 12*B*a^3 + 30*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^
3)*tan(d*x + c)^3 + 20*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2)*tan(d*x + c)^2 - 15*(C*a^3 + 3*B*a^2*b)*tan(d*x + c))/(
d*tan(d*x + c)^5)

Sympy [A] (verification not implemented)

Time = 12.36 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.98 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \text {NaN} & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{3} \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \cot ^{7}{\left (c \right )} & \text {for}\: d = 0 \\\text {NaN} & \text {for}\: c = - d x \\- B a^{3} x - \frac {B a^{3}}{d \tan {\left (c + d x \right )}} + \frac {B a^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {B a^{3}}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 B a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 B a^{2} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 B a^{2} b}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {3 B a^{2} b}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 B a b^{2} x + \frac {3 B a b^{2}}{d \tan {\left (c + d x \right )}} - \frac {B a b^{2}}{d \tan ^{3}{\left (c + d x \right )}} + \frac {B b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {B b^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {B b^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {C a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C a^{3} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {C a^{3}}{2 d \tan ^{2}{\left (c + d x \right )}} - \frac {C a^{3}}{4 d \tan ^{4}{\left (c + d x \right )}} + 3 C a^{2} b x + \frac {3 C a^{2} b}{d \tan {\left (c + d x \right )}} - \frac {C a^{2} b}{d \tan ^{3}{\left (c + d x \right )}} + \frac {3 C a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {3 C a b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {3 C a b^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - C b^{3} x - \frac {C b^{3}}{d \tan {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**7*(a+b*tan(d*x+c))**3*(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))**3*(B*tan(c) + C*tan(c)**2)*cot(c)**7, Eq(d, 0)), (nan
, Eq(c, -d*x)), (-B*a**3*x - B*a**3/(d*tan(c + d*x)) + B*a**3/(3*d*tan(c + d*x)**3) - B*a**3/(5*d*tan(c + d*x)
**5) - 3*B*a**2*b*log(tan(c + d*x)**2 + 1)/(2*d) + 3*B*a**2*b*log(tan(c + d*x))/d + 3*B*a**2*b/(2*d*tan(c + d*
x)**2) - 3*B*a**2*b/(4*d*tan(c + d*x)**4) + 3*B*a*b**2*x + 3*B*a*b**2/(d*tan(c + d*x)) - B*a*b**2/(d*tan(c + d
*x)**3) + B*b**3*log(tan(c + d*x)**2 + 1)/(2*d) - B*b**3*log(tan(c + d*x))/d - B*b**3/(2*d*tan(c + d*x)**2) -
C*a**3*log(tan(c + d*x)**2 + 1)/(2*d) + C*a**3*log(tan(c + d*x))/d + C*a**3/(2*d*tan(c + d*x)**2) - C*a**3/(4*
d*tan(c + d*x)**4) + 3*C*a**2*b*x + 3*C*a**2*b/(d*tan(c + d*x)) - C*a**2*b/(d*tan(c + d*x)**3) + 3*C*a*b**2*lo
g(tan(c + d*x)**2 + 1)/(2*d) - 3*C*a*b**2*log(tan(c + d*x))/d - 3*C*a*b**2/(2*d*tan(c + d*x)**2) - C*b**3*x -
C*b**3/(d*tan(c + d*x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.07 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} + 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 60 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {60 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} \tan \left (d x + c\right )^{4} + 12 \, B a^{3} - 30 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 20 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2}\right )} \tan \left (d x + c\right )^{2} + 15 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{5}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/60*(60*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*(d*x + c) + 30*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*log(t
an(d*x + c)^2 + 1) - 60*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*log(tan(d*x + c)) + (60*(B*a^3 - 3*C*a^2*b - 3
*B*a*b^2 + C*b^3)*tan(d*x + c)^4 + 12*B*a^3 - 30*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*tan(d*x + c)^3 - 20*(
B*a^3 - 3*C*a^2*b - 3*B*a*b^2)*tan(d*x + c)^2 + 15*(C*a^3 + 3*B*a^2*b)*tan(d*x + c))/tan(d*x + c)^5)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (225) = 450\).

Time = 1.57 (sec) , antiderivative size = 670, normalized size of antiderivative = 2.88 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {6 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 180 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 540 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 360 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 960 \, {\left (B a^{3} - 3 \, C a^{2} b - 3 \, B a b^{2} + C b^{3}\right )} {\left (d x + c\right )} - 960 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 960 \, {\left (C a^{3} + 3 \, B a^{2} b - 3 \, C a b^{2} - B b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2192 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6576 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2192 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 660 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1800 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, C b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 180 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 540 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 70 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 45 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^3*(B*tan(d*x+c)+C*tan(d*x+c)^2),x, algorithm="giac")

[Out]

1/960*(6*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^3*tan(1/2*d*x + 1/2*c)^4 - 45*B*a^2*b*tan(1/2*d*x + 1/2*c)^4 -
70*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 120*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 120*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 18
0*C*a^3*tan(1/2*d*x + 1/2*c)^2 + 540*B*a^2*b*tan(1/2*d*x + 1/2*c)^2 - 360*C*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 120
*B*b^3*tan(1/2*d*x + 1/2*c)^2 + 660*B*a^3*tan(1/2*d*x + 1/2*c) - 1800*C*a^2*b*tan(1/2*d*x + 1/2*c) - 1800*B*a*
b^2*tan(1/2*d*x + 1/2*c) + 480*C*b^3*tan(1/2*d*x + 1/2*c) - 960*(B*a^3 - 3*C*a^2*b - 3*B*a*b^2 + C*b^3)*(d*x +
c) - 960*(C*a^3 + 3*B*a^2*b - 3*C*a*b^2 - B*b^3)*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 960*(C*a^3 + 3*B*a^2*b - 3
*C*a*b^2 - B*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (2192*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 6576*B*a^2*b*tan(1/2*d
*x + 1/2*c)^5 - 6576*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 2192*B*b^3*tan(1/2*d*x + 1/2*c)^5 + 660*B*a^3*tan(1/2*d*
x + 1/2*c)^4 - 1800*C*a^2*b*tan(1/2*d*x + 1/2*c)^4 - 1800*B*a*b^2*tan(1/2*d*x + 1/2*c)^4 + 480*C*b^3*tan(1/2*d
*x + 1/2*c)^4 - 180*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 540*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 360*C*a*b^2*tan(1/2*d*
x + 1/2*c)^3 + 120*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 70*B*a^3*tan(1/2*d*x + 1/2*c)^2 + 120*C*a^2*b*tan(1/2*d*x +
1/2*c)^2 + 120*B*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 15*C*a^3*tan(1/2*d*x + 1/2*c) + 45*B*a^2*b*tan(1/2*d*x + 1/2*c
) + 6*B*a^3)/tan(1/2*d*x + 1/2*c)^5)/d

Mupad [B] (verification not implemented)

Time = 8.43 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^3 \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {{\mathrm {cot}\left (c+d\,x\right )}^5\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {C\,a^3}{4}+\frac {3\,B\,b\,a^2}{4}\right )+\frac {B\,a^3}{5}+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {B\,a^3}{3}+C\,a^2\,b+B\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (B\,a^3-3\,C\,a^2\,b-3\,B\,a\,b^2+C\,b^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-\frac {C\,a^3}{2}-\frac {3\,B\,a^2\,b}{2}+\frac {3\,C\,a\,b^2}{2}+\frac {B\,b^3}{2}\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-C\,a^3-3\,B\,a^2\,b+3\,C\,a\,b^2+B\,b^3\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{2\,d} \]

[In]

int(cot(c + d*x)^7*(B*tan(c + d*x) + C*tan(c + d*x)^2)*(a + b*tan(c + d*x))^3,x)

[Out]

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