\(\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx\) [455]

Optimal result

Integrand size = 21, antiderivative size = 198 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d} \]

[Out]

-4*a*b*(a^2-b^2)*x-4*a*b*(a^2-b^2)*cot(d*x+c)/d-1/2*(a^4-6*a^2*b^2+b^4)*cot(d*x+c)^2/d+4/3*a*b*(a^2-b^2)*cot(d
*x+c)^3/d+1/12*a^2*(3*a^2-16*b^2)*cot(d*x+c)^4/d-7/15*a^3*b*cot(d*x+c)^5/d-(a^4-6*a^2*b^2+b^4)*ln(sin(d*x+c))/
d-1/6*a^2*cot(d*x+c)^6*(a+b*tan(d*x+c))^2/d

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3646, 3716, 3709, 3610, 3612, 3556} \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-4 a b x \left (a^2-b^2\right )-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d} \]

[In]

Int[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

-4*a*b*(a^2 - b^2)*x - (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d - ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/(2*d) + (
4*a*b*(a^2 - b^2)*Cot[c + d*x]^3)/(3*d) + (a^2*(3*a^2 - 16*b^2)*Cot[c + d*x]^4)/(12*d) - (7*a^3*b*Cot[c + d*x]
^5)/(15*d) - ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2)/(6*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 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^6(c+d x) (a+b \tan (c+d x)) \left (14 a^2 b-6 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx \\ & = -\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^5(c+d x) \left (-2 a^2 \left (3 a^2-16 b^2\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^4(c+d x) \left (-24 a b \left (a^2-b^2\right )+6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^3(c+d x) \left (6 \left (a^4-6 a^2 b^2+b^4\right )+24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = -\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot ^2(c+d x) \left (24 a b \left (a^2-b^2\right )-6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx \\ & = -\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac {1}{6} \int \cot (c+d x) \left (-6 \left (a^4-6 a^2 b^2+b^4\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx \\ & = -4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\left (-a^4+6 a^2 b^2-b^4\right ) \int \cot (c+d x) \, dx \\ & = -4 a b \left (a^2-b^2\right ) x-\frac {4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac {4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac {a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac {7 a^3 b \cot ^5(c+d x)}{15 d}-\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.90 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {4 a (a-b) b (a+b) \cot (c+d x)+\frac {1}{2} \left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)-\frac {4}{3} a (a-b) b (a+b) \cot ^3(c+d x)-\frac {1}{4} a^2 \left (a^2-6 b^2\right ) \cot ^4(c+d x)+\frac {4}{5} a^3 b \cot ^5(c+d x)+\frac {1}{6} a^4 \cot ^6(c+d x)-\frac {1}{2} (a-i b)^4 \log (i-\cot (c+d x))-\frac {1}{2} (a+i b)^4 \log (i+\cot (c+d x))}{d} \]

[In]

Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

-((4*a*(a - b)*b*(a + b)*Cot[c + d*x] + ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/2 - (4*a*(a - b)*b*(a + b)*Co
t[c + d*x]^3)/3 - (a^2*(a^2 - 6*b^2)*Cot[c + d*x]^4)/4 + (4*a^3*b*Cot[c + d*x]^5)/5 + (a^4*Cot[c + d*x]^6)/6 -
((a - I*b)^4*Log[I - Cot[c + d*x]])/2 - ((a + I*b)^4*Log[I + Cot[c + d*x]])/2)/d)

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98

method	result	size
parallelrisch	\(\frac {30 \left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (\sec ^{2}\left (d x +c \right )\right )+60 \left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )-10 \left (\cot ^{6}\left (d x +c \right )\right ) a^{4}-48 \left (\cot ^{5}\left (d x +c \right )\right ) a^{3} b +15 \left (a^{4}-6 a^{2} b^{2}\right ) \left (\cot ^{4}\left (d x +c \right )\right )+80 \left (a^{3} b -a \,b^{3}\right ) \left (\cot ^{3}\left (d x +c \right )\right )+30 \left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \left (\cot ^{2}\left (d x +c \right )\right )+240 \left (-a^{3} b +a \,b^{3}\right ) \cot \left (d x +c \right )-240 a b d x \left (a -b \right ) \left (a +b \right )}{60 d}\)	\(195\)
derivativedivides	\(\frac {-\frac {a^{4}}{6 \tan \left (d x +c \right )^{6}}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {a^{4}-6 a^{2} b^{2}+b^{4}}{2 \tan \left (d x +c \right )^{2}}+\frac {a^{2} \left (a^{2}-6 b^{2}\right )}{4 \tan \left (d x +c \right )^{4}}-\frac {4 a^{3} b}{5 \tan \left (d x +c \right )^{5}}+\frac {4 a b \left (a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}-\frac {4 a b \left (a^{2}-b^{2}\right )}{\tan \left (d x +c \right )}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(198\)
default	\(\frac {-\frac {a^{4}}{6 \tan \left (d x +c \right )^{6}}+\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )-\frac {a^{4}-6 a^{2} b^{2}+b^{4}}{2 \tan \left (d x +c \right )^{2}}+\frac {a^{2} \left (a^{2}-6 b^{2}\right )}{4 \tan \left (d x +c \right )^{4}}-\frac {4 a^{3} b}{5 \tan \left (d x +c \right )^{5}}+\frac {4 a b \left (a^{2}-b^{2}\right )}{3 \tan \left (d x +c \right )^{3}}-\frac {4 a b \left (a^{2}-b^{2}\right )}{\tan \left (d x +c \right )}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(198\)
norman	\(\frac {-\frac {a^{4}}{6 d}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \left (a^{2}-6 b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{4 d}-\frac {4 a^{3} b \tan \left (d x +c \right )}{5 d}+\frac {4 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {4 a b \left (a^{2}-b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{d}-4 a b \left (a^{2}-b^{2}\right ) x \left (\tan ^{6}\left (d x +c \right )\right )}{\tan \left (d x +c \right )^{6}}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(216\)
risch	\(-4 a^{3} b x +4 a \,b^{3} x +i a^{4} x -6 i a^{2} b^{2} x +i b^{4} x +\frac {2 i a^{4} c}{d}-\frac {12 i a^{2} b^{2} c}{d}+\frac {2 i b^{4} c}{d}+\frac {72 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+72 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-24 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-\frac {248 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}}{5}-96 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+48 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+112 i a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {320 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{3}-96 i a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-64 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-\frac {368 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}}{3}+72 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}-12 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+12 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-8 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+\frac {68 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}}{3}-12 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}-8 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{10 i \left (d x +c \right )}+2 b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-\frac {32 i a \,b^{3}}{3}-24 a^{2} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+\frac {184 i a^{3} b}{15}+16 i a \,b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-24 i a^{3} b \,{\mathrm e}^{10 i \left (d x +c \right )}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {6 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{4}}{d}\)	\(554\)

[In]

int(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/60*(30*(a^4-6*a^2*b^2+b^4)*ln(sec(d*x+c)^2)+60*(-a^4+6*a^2*b^2-b^4)*ln(tan(d*x+c))-10*cot(d*x+c)^6*a^4-48*co
t(d*x+c)^5*a^3*b+15*(a^4-6*a^2*b^2)*cot(d*x+c)^4+80*(a^3*b-a*b^3)*cot(d*x+c)^3+30*(-a^4+6*a^2*b^2-b^4)*cot(d*x
+c)^2+240*(-a^3*b+a*b^3)*cot(d*x+c)-240*a*b*d*x*(a-b)*(a+b))/d

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.08 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{6} + 5 \, {\left (11 \, a^{4} - 54 \, a^{2} b^{2} + 6 \, b^{4} + 48 \, {\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{6} + 240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, d \tan \left (d x + c\right )^{6}} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/60*(30*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^6 + 5*(11*a^4 - 54*a^2
*b^2 + 6*b^4 + 48*(a^3*b - a*b^3)*d*x)*tan(d*x + c)^6 + 240*(a^3*b - a*b^3)*tan(d*x + c)^5 + 48*a^3*b*tan(d*x
+ c) + 30*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^4 + 10*a^4 - 80*(a^3*b - a*b^3)*tan(d*x + c)^3 - 15*(a^4 - 6*a^
2*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^6)

Sympy [A] (verification not implemented)

Time = 5.39 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.71 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\begin {cases} \tilde {\infty } a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{7}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{4} x & \text {for}\: c = - d x \\\frac {a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {a^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {a^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} + \frac {a^{4}}{4 d \tan ^{4}{\left (c + d x \right )}} - \frac {a^{4}}{6 d \tan ^{6}{\left (c + d x \right )}} - 4 a^{3} b x - \frac {4 a^{3} b}{d \tan {\left (c + d x \right )}} + \frac {4 a^{3} b}{3 d \tan ^{3}{\left (c + d x \right )}} - \frac {4 a^{3} b}{5 d \tan ^{5}{\left (c + d x \right )}} - \frac {3 a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {6 a^{2} b^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + \frac {3 a^{2} b^{2}}{d \tan ^{2}{\left (c + d x \right )}} - \frac {3 a^{2} b^{2}}{2 d \tan ^{4}{\left (c + d x \right )}} + 4 a b^{3} x + \frac {4 a b^{3}}{d \tan {\left (c + d x \right )}} - \frac {4 a b^{3}}{3 d \tan ^{3}{\left (c + d x \right )}} + \frac {b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac {b^{4} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} - \frac {b^{4}}{2 d \tan ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(cot(d*x+c)**7*(a+b*tan(d*x+c))**4,x)

[Out]

Piecewise((zoo*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**4*cot(c)**7, Eq(d, 0)), (zoo*a**4*x, Eq(c, -d*
x)), (a**4*log(tan(c + d*x)**2 + 1)/(2*d) - a**4*log(tan(c + d*x))/d - a**4/(2*d*tan(c + d*x)**2) + a**4/(4*d*
tan(c + d*x)**4) - a**4/(6*d*tan(c + d*x)**6) - 4*a**3*b*x - 4*a**3*b/(d*tan(c + d*x)) + 4*a**3*b/(3*d*tan(c +
d*x)**3) - 4*a**3*b/(5*d*tan(c + d*x)**5) - 3*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 6*a**2*b**2*log(tan(c +
d*x))/d + 3*a**2*b**2/(d*tan(c + d*x)**2) - 3*a**2*b**2/(2*d*tan(c + d*x)**4) + 4*a*b**3*x + 4*a*b**3/(d*tan(c
+ d*x)) - 4*a*b**3/(3*d*tan(c + d*x)**3) + b**4*log(tan(c + d*x)**2 + 1)/(2*d) - b**4*log(tan(c + d*x))/d - b
**4/(2*d*tan(c + d*x)**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {240 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac {240 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \, {\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \, {\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/60*(240*(a^3*b - a*b^3)*(d*x + c) - 30*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1) + 60*(a^4 - 6*a^2*b^
2 + b^4)*log(tan(d*x + c)) + (240*(a^3*b - a*b^3)*tan(d*x + c)^5 + 48*a^3*b*tan(d*x + c) + 30*(a^4 - 6*a^2*b^2
+ b^4)*tan(d*x + c)^4 + 10*a^4 - 80*(a^3*b - a*b^3)*tan(d*x + c)^3 - 15*(a^4 - 6*a^2*b^2)*tan(d*x + c)^2)/tan
(d*x + c)^6)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 506 vs. \(2 (188) = 376\).

Time = 1.34 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.56 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {5 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 435 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2160 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7680 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )} - 1920 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) + 1920 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {4704 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 28224 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4704 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 5280 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4800 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 435 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2160 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 320 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/1920*(5*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 60*a^4*tan(1/2*d*x + 1/2*c)^4 + 180*
a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 435*a^4
*tan(1/2*d*x + 1/2*c)^2 - 2160*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*b^4*tan(1/2*d*x + 1/2*c)^2 - 5280*a^3*b*ta
n(1/2*d*x + 1/2*c) + 4800*a*b^3*tan(1/2*d*x + 1/2*c) + 7680*(a^3*b - a*b^3)*(d*x + c) - 1920*(a^4 - 6*a^2*b^2
+ b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 1920*(a^4 - 6*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c))) - (4704*a
^4*tan(1/2*d*x + 1/2*c)^6 - 28224*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 4704*b^4*tan(1/2*d*x + 1/2*c)^6 - 5280*a^3*
b*tan(1/2*d*x + 1/2*c)^5 + 4800*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 435*a^4*tan(1/2*d*x + 1/2*c)^4 + 2160*a^2*b^2*t
an(1/2*d*x + 1/2*c)^4 - 240*b^4*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*
d*x + 1/2*c)^3 + 60*a^4*tan(1/2*d*x + 1/2*c)^2 - 180*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 48*a^3*b*tan(1/2*d*x + 1
/2*c) - 5*a^4)/tan(1/2*d*x + 1/2*c)^6)/d

Mupad [B] (verification not implemented)

Time = 5.64 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.02 \[ \int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^4}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^4}{2\,d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^6\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {4\,a\,b^3}{3}-\frac {4\,a^3\,b}{3}\right )-{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (4\,a\,b^3-4\,a^3\,b\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4}{4}-\frac {3\,a^2\,b^2}{2}\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {a^4}{2}-3\,a^2\,b^2+\frac {b^4}{2}\right )+\frac {a^4}{6}+\frac {4\,a^3\,b\,\mathrm {tan}\left (c+d\,x\right )}{5}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4-6\,a^2\,b^2+b^4\right )}{d} \]

[In]

int(cot(c + d*x)^7*(a + b*tan(c + d*x))^4,x)

[Out]

(log(tan(c + d*x) - 1i)*(a + b*1i)^4)/(2*d) + (log(tan(c + d*x) + 1i)*(a*1i + b)^4)/(2*d) - (cot(c + d*x)^6*(t
an(c + d*x)^3*((4*a*b^3)/3 - (4*a^3*b)/3) - tan(c + d*x)^5*(4*a*b^3 - 4*a^3*b) - tan(c + d*x)^2*(a^4/4 - (3*a^
2*b^2)/2) + tan(c + d*x)^4*(a^4/2 + b^4/2 - 3*a^2*b^2) + a^4/6 + (4*a^3*b*tan(c + d*x))/5))/d - (log(tan(c + d
*x))*(a^4 + b^4 - 6*a^2*b^2))/d