\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [257]

Optimal result

Integrand size = 31, antiderivative size = 263 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\left (\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x\right )+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \log (\cos (c+d x))}{d}-\frac {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{d}-\frac {\left (2 a A b+a^2 B-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}-\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d} \]

[Out]

-(A*a^4-6*A*a^2*b^2+A*b^4-4*B*a^3*b+4*B*a*b^3)*x+(4*A*a^3*b-4*A*a*b^3+B*a^4-6*B*a^2*b^2+B*b^4)*ln(cos(d*x+c))/
d-b*(3*A*a^2*b-A*b^3+B*a^3-3*B*a*b^2)*tan(d*x+c)/d-1/2*(2*A*a*b+B*a^2-B*b^2)*(a+b*tan(d*x+c))^2/d-1/3*(A*b+B*a
)*(a+b*tan(d*x+c))^3/d-1/4*B*(a+b*tan(d*x+c))^4/d+1/30*(6*A*b-B*a)*(a+b*tan(d*x+c))^5/b^2/d+1/6*B*tan(d*x+c)*(
a+b*tan(d*x+c))^5/b/d

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3688, 3711, 3609, 3606, 3556} \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {\left (a^2 B+2 a A b-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {b \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \tan (c+d x)}{d}+\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \log (\cos (c+d x))}{d}-x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}-\frac {(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}-\frac {B (a+b \tan (c+d x))^4}{4 d} \]

[In]

Int[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

-((a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x) + ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b
^4*B)*Log[Cos[c + d*x]])/d - (b*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x])/d - ((2*a*A*b + a^2*B -
b^2*B)*(a + b*Tan[c + d*x])^2)/(2*d) - ((A*b + a*B)*(a + b*Tan[c + d*x])^3)/(3*d) - (B*(a + b*Tan[c + d*x])^4)
/(4*d) + ((6*A*b - a*B)*(a + b*Tan[c + d*x])^5)/(30*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^5)/(6*b*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 3606

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

Rule 3609

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

Rule 3688

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3711

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

Rubi steps \begin{align*} \text {integral}& = \frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac {\int (a+b \tan (c+d x))^4 \left (-a B-6 b B \tan (c+d x)+(6 A b-a B) \tan ^2(c+d x)\right ) \, dx}{6 b} \\ & = \frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac {\int (a+b \tan (c+d x))^4 (-6 A b-6 b B \tan (c+d x)) \, dx}{6 b} \\ & = -\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac {\int (a+b \tan (c+d x))^3 (-6 b (a A-b B)-6 b (A b+a B) \tan (c+d x)) \, dx}{6 b} \\ & = -\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}-\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac {\int (a+b \tan (c+d x))^2 \left (-6 b \left (a^2 A-A b^2-2 a b B\right )-6 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx}{6 b} \\ & = -\frac {\left (2 a A b+a^2 B-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}-\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\frac {\int (a+b \tan (c+d x)) \left (-6 b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-6 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)\right ) \, dx}{6 b} \\ & = -\left (\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x\right )-\frac {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{d}-\frac {\left (2 a A b+a^2 B-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}-\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d}+\left (-4 a^3 A b+4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \int \tan (c+d x) \, dx \\ & = -\left (\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x\right )+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \log (\cos (c+d x))}{d}-\frac {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{d}-\frac {\left (2 a A b+a^2 B-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}-\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}-\frac {B (a+b \tan (c+d x))^4}{4 d}+\frac {(6 A b-a B) (a+b \tan (c+d x))^5}{30 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^5}{6 b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.10 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.10 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\frac {2 (6 A b-a B) (a+b \tan (c+d x))^5}{b}+10 B \tan (c+d x) (a+b \tan (c+d x))^5+10 (A b-a B) \left (3 i (a+i b)^4 \log (i-\tan (c+d x))-3 i (a-i b)^4 \log (i+\tan (c+d x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-2 b^4 \tan ^3(c+d x)\right )+5 B \left (6 i (a+i b)^5 \log (i-\tan (c+d x))-6 (i a+b)^5 \log (i+\tan (c+d x))-60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)+6 b^3 \left (-10 a^2+b^2\right ) \tan ^2(c+d x)-20 a b^4 \tan ^3(c+d x)-3 b^5 \tan ^4(c+d x)\right )}{60 b d} \]

[In]

Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^4*(A + B*Tan[c + d*x]),x]

[Out]

((2*(6*A*b - a*B)*(a + b*Tan[c + d*x])^5)/b + 10*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^5 + 10*(A*b - a*B)*((3*I)
*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6*a^2 + b^2)*Tan[c + d*
x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) + 5*B*((6*I)*(a + I*b)^5*Log[I - Tan[c + d*x]] - 6*(I*a +
b)^5*Log[I + Tan[c + d*x]] - 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] + 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 - 20*
a*b^4*Tan[c + d*x]^3 - 3*b^5*Tan[c + d*x]^4))/(60*b*d)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.04

method	result	size
parts	\(\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \,a^{4} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B \,b^{4} \left (\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(274\)
norman	\(\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) x +\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6 d}+\frac {b \left (6 A \,a^{2} b -A \,b^{3}+4 B \,a^{3}-4 B a \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} \left (4 A a b +6 B \,a^{2}-B \,b^{2}\right ) \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b^{3} \left (A b +4 B a \right ) \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(297\)
derivativedivides	\(\frac {\frac {B \,b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {A \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {4 B a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+A a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )+\frac {3 B \,a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2}-\frac {B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+2 A \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {4 B \,a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {4 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 A \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )-2 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}}{2}-3 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+A \tan \left (d x +c \right ) a^{4}-6 A \,a^{2} b^{2} \tan \left (d x +c \right )+A \,b^{4} \tan \left (d x +c \right )-4 B \,a^{3} b \tan \left (d x +c \right )+4 B a \,b^{3} \tan \left (d x +c \right )+\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(386\)
default	\(\frac {\frac {B \,b^{4} \left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {A \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {4 B a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+A a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )+\frac {3 B \,a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{2}-\frac {B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+2 A \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {4 B \,a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {4 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 A \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )-2 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}}{2}-3 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+\frac {B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+A \tan \left (d x +c \right ) a^{4}-6 A \,a^{2} b^{2} \tan \left (d x +c \right )+A \,b^{4} \tan \left (d x +c \right )-4 B \,a^{3} b \tan \left (d x +c \right )+4 B a \,b^{3} \tan \left (d x +c \right )+\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{4}+6 A \,a^{2} b^{2}-A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(386\)
parallelrisch	\(-\frac {-60 A \,b^{4} \tan \left (d x +c \right )+360 A \,a^{2} b^{2} \tan \left (d x +c \right )+240 B \,a^{3} b \tan \left (d x +c \right )-240 B a \,b^{3} \tan \left (d x +c \right )+20 A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )-30 B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )-60 A \tan \left (d x +c \right ) a^{4}+60 A \,b^{4} d x -360 A \,a^{2} b^{2} d x -240 B \,a^{3} b d x +240 B a \,b^{3} d x +120 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b -120 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}-180 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}-12 A \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )+15 B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )+60 A x \,a^{4} d +30 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}-30 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}-10 B \,b^{4} \left (\tan ^{6}\left (d x +c \right )\right )+30 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}-120 A \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-80 B \,a^{3} b \left (\tan ^{3}\left (d x +c \right )\right )+80 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )-120 A \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )+120 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+180 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-48 B a \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )-60 A a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )-90 B \,a^{2} b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{60 d}\)	\(432\)
risch	\(\text {Expression too large to display}\)	\(1139\)

[In]

int(tan(d*x+c)^2*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

(A*b^4+4*B*a*b^3)/d*(1/5*tan(d*x+c)^5-1/3*tan(d*x+c)^3+tan(d*x+c)-arctan(tan(d*x+c)))+(4*A*a*b^3+6*B*a^2*b^2)/
d*(1/4*tan(d*x+c)^4-1/2*tan(d*x+c)^2+1/2*ln(1+tan(d*x+c)^2))+(6*A*a^2*b^2+4*B*a^3*b)/d*(1/3*tan(d*x+c)^3-tan(d
*x+c)+arctan(tan(d*x+c)))+(4*A*a^3*b+B*a^4)/d*(1/2*tan(d*x+c)^2-1/2*ln(1+tan(d*x+c)^2))+A*a^4/d*(tan(d*x+c)-ar
ctan(tan(d*x+c)))+B*b^4/d*(1/6*tan(d*x+c)^6-1/4*tan(d*x+c)^4+1/2*tan(d*x+c)^2-1/2*ln(1+tan(d*x+c)^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.10 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {10 \, B b^{4} \tan \left (d x + c\right )^{6} + 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{5} + 15 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{3} - 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x + 30 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{2} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(10*B*b^4*tan(d*x + c)^6 + 12*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^5 + 15*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*t
an(d*x + c)^4 + 20*(4*B*a^3*b + 6*A*a^2*b^2 - 4*B*a*b^3 - A*b^4)*tan(d*x + c)^3 - 60*(A*a^4 - 4*B*a^3*b - 6*A*
a^2*b^2 + 4*B*a*b^3 + A*b^4)*d*x + 30*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c)^2 + 3
0*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*log(1/(tan(d*x + c)^2 + 1)) + 60*(A*a^4 - 4*B*a^3*b -
6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (248) = 496\).

Time = 0.25 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.04 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} - A a^{4} x + \frac {A a^{4} \tan {\left (c + d x \right )}}{d} - \frac {2 A a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 A a^{3} b \tan ^{2}{\left (c + d x \right )}}{d} + 6 A a^{2} b^{2} x + \frac {2 A a^{2} b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac {6 A a^{2} b^{2} \tan {\left (c + d x \right )}}{d} + \frac {2 A a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {A a b^{3} \tan ^{4}{\left (c + d x \right )}}{d} - \frac {2 A a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} - A b^{4} x + \frac {A b^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {A b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {A b^{4} \tan {\left (c + d x \right )}}{d} - \frac {B a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} + 4 B a^{3} b x + \frac {4 B a^{3} b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 B a^{3} b \tan {\left (c + d x \right )}}{d} + \frac {3 B a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 B a^{2} b^{2} \tan ^{4}{\left (c + d x \right )}}{2 d} - \frac {3 B a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} - 4 B a b^{3} x + \frac {4 B a b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 B a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {4 B a b^{3} \tan {\left (c + d x \right )}}{d} - \frac {B b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{4} \tan ^{6}{\left (c + d x \right )}}{6 d} - \frac {B b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} + \frac {B b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \tan ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(tan(d*x+c)**2*(a+b*tan(d*x+c))**4*(A+B*tan(d*x+c)),x)

[Out]

Piecewise((-A*a**4*x + A*a**4*tan(c + d*x)/d - 2*A*a**3*b*log(tan(c + d*x)**2 + 1)/d + 2*A*a**3*b*tan(c + d*x)
**2/d + 6*A*a**2*b**2*x + 2*A*a**2*b**2*tan(c + d*x)**3/d - 6*A*a**2*b**2*tan(c + d*x)/d + 2*A*a*b**3*log(tan(
c + d*x)**2 + 1)/d + A*a*b**3*tan(c + d*x)**4/d - 2*A*a*b**3*tan(c + d*x)**2/d - A*b**4*x + A*b**4*tan(c + d*x
)**5/(5*d) - A*b**4*tan(c + d*x)**3/(3*d) + A*b**4*tan(c + d*x)/d - B*a**4*log(tan(c + d*x)**2 + 1)/(2*d) + B*
a**4*tan(c + d*x)**2/(2*d) + 4*B*a**3*b*x + 4*B*a**3*b*tan(c + d*x)**3/(3*d) - 4*B*a**3*b*tan(c + d*x)/d + 3*B
*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 3*B*a**2*b**2*tan(c + d*x)**4/(2*d) - 3*B*a**2*b**2*tan(c + d*x)**2/d
- 4*B*a*b**3*x + 4*B*a*b**3*tan(c + d*x)**5/(5*d) - 4*B*a*b**3*tan(c + d*x)**3/(3*d) + 4*B*a*b**3*tan(c + d*x)
/d - B*b**4*log(tan(c + d*x)**2 + 1)/(2*d) + B*b**4*tan(c + d*x)**6/(6*d) - B*b**4*tan(c + d*x)**4/(4*d) + B*b
**4*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(A + B*tan(c))*(a + b*tan(c))**4*tan(c)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.10 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {10 \, B b^{4} \tan \left (d x + c\right )^{6} + 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{5} + 15 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{3} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )^{2} - 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )} - 30 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/60*(10*B*b^4*tan(d*x + c)^6 + 12*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^5 + 15*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*t
an(d*x + c)^4 + 20*(4*B*a^3*b + 6*A*a^2*b^2 - 4*B*a*b^3 - A*b^4)*tan(d*x + c)^3 + 30*(B*a^4 + 4*A*a^3*b - 6*B*
a^2*b^2 - 4*A*a*b^3 + B*b^4)*tan(d*x + c)^2 - 60*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*(d*x +
c) - 30*(B*a^4 + 4*A*a^3*b - 6*B*a^2*b^2 - 4*A*a*b^3 + B*b^4)*log(tan(d*x + c)^2 + 1) + 60*(A*a^4 - 4*B*a^3*b
- 6*A*a^2*b^2 + 4*B*a*b^3 + A*b^4)*tan(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6042 vs. \(2 (252) = 504\).

Time = 7.74 (sec) , antiderivative size = 6042, normalized size of antiderivative = 22.97 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^4*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/60*(60*A*a^4*d*x*tan(d*x)^6*tan(c)^6 - 240*B*a^3*b*d*x*tan(d*x)^6*tan(c)^6 - 360*A*a^2*b^2*d*x*tan(d*x)^6*t
an(c)^6 + 240*B*a*b^3*d*x*tan(d*x)^6*tan(c)^6 + 60*A*b^4*d*x*tan(d*x)^6*tan(c)^6 - 30*B*a^4*log(4*(tan(d*x)^2*
tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 - 120
*A*a^3*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)
)*tan(d*x)^6*tan(c)^6 + 180*B*a^2*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 + 120*A*a*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c
) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 - 30*B*b^4*log(4*(tan(d*x)^2*tan
(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^6*tan(c)^6 - 360*A*
a^4*d*x*tan(d*x)^5*tan(c)^5 + 1440*B*a^3*b*d*x*tan(d*x)^5*tan(c)^5 + 2160*A*a^2*b^2*d*x*tan(d*x)^5*tan(c)^5 -
1440*B*a*b^3*d*x*tan(d*x)^5*tan(c)^5 - 360*A*b^4*d*x*tan(d*x)^5*tan(c)^5 - 30*B*a^4*tan(d*x)^6*tan(c)^6 - 120*
A*a^3*b*tan(d*x)^6*tan(c)^6 + 270*B*a^2*b^2*tan(d*x)^6*tan(c)^6 + 180*A*a*b^3*tan(d*x)^6*tan(c)^6 - 55*B*b^4*t
an(d*x)^6*tan(c)^6 + 180*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(
d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 + 720*A*a^3*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/
(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 1080*B*a^2*b^2*log(4*(tan(d*x)^2*tan(
c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 - 720*A*a
*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*ta
n(d*x)^5*tan(c)^5 + 180*B*b^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d
*x)^2 + tan(c)^2 + 1))*tan(d*x)^5*tan(c)^5 + 60*A*a^4*tan(d*x)^6*tan(c)^5 - 240*B*a^3*b*tan(d*x)^6*tan(c)^5 -
360*A*a^2*b^2*tan(d*x)^6*tan(c)^5 + 240*B*a*b^3*tan(d*x)^6*tan(c)^5 + 60*A*b^4*tan(d*x)^6*tan(c)^5 + 60*A*a^4*
tan(d*x)^5*tan(c)^6 - 240*B*a^3*b*tan(d*x)^5*tan(c)^6 - 360*A*a^2*b^2*tan(d*x)^5*tan(c)^6 + 240*B*a*b^3*tan(d*
x)^5*tan(c)^6 + 60*A*b^4*tan(d*x)^5*tan(c)^6 + 900*A*a^4*d*x*tan(d*x)^4*tan(c)^4 - 3600*B*a^3*b*d*x*tan(d*x)^4
*tan(c)^4 - 5400*A*a^2*b^2*d*x*tan(d*x)^4*tan(c)^4 + 3600*B*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 900*A*b^4*d*x*tan(
d*x)^4*tan(c)^4 - 30*B*a^4*tan(d*x)^6*tan(c)^4 - 120*A*a^3*b*tan(d*x)^6*tan(c)^4 + 180*B*a^2*b^2*tan(d*x)^6*ta
n(c)^4 + 120*A*a*b^3*tan(d*x)^6*tan(c)^4 - 30*B*b^4*tan(d*x)^6*tan(c)^4 + 120*B*a^4*tan(d*x)^5*tan(c)^5 + 480*
A*a^3*b*tan(d*x)^5*tan(c)^5 - 1260*B*a^2*b^2*tan(d*x)^5*tan(c)^5 - 840*A*a*b^3*tan(d*x)^5*tan(c)^5 + 270*B*b^4
*tan(d*x)^5*tan(c)^5 - 30*B*a^4*tan(d*x)^4*tan(c)^6 - 120*A*a^3*b*tan(d*x)^4*tan(c)^6 + 180*B*a^2*b^2*tan(d*x)
^4*tan(c)^6 + 120*A*a*b^3*tan(d*x)^4*tan(c)^6 - 30*B*b^4*tan(d*x)^4*tan(c)^6 + 80*B*a^3*b*tan(d*x)^6*tan(c)^3
+ 120*A*a^2*b^2*tan(d*x)^6*tan(c)^3 - 80*B*a*b^3*tan(d*x)^6*tan(c)^3 - 20*A*b^4*tan(d*x)^6*tan(c)^3 - 450*B*a^
4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d
*x)^4*tan(c)^4 - 1800*A*a^3*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d
*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 2700*B*a^2*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1
)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 1800*A*a*b^3*log(4*(tan(d*x)^2*tan(
c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 - 450*B*b
^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(
d*x)^4*tan(c)^4 - 300*A*a^4*tan(d*x)^5*tan(c)^4 + 1440*B*a^3*b*tan(d*x)^5*tan(c)^4 + 2160*A*a^2*b^2*tan(d*x)^5
*tan(c)^4 - 1440*B*a*b^3*tan(d*x)^5*tan(c)^4 - 360*A*b^4*tan(d*x)^5*tan(c)^4 - 300*A*a^4*tan(d*x)^4*tan(c)^5 +
1440*B*a^3*b*tan(d*x)^4*tan(c)^5 + 2160*A*a^2*b^2*tan(d*x)^4*tan(c)^5 - 1440*B*a*b^3*tan(d*x)^4*tan(c)^5 - 36
0*A*b^4*tan(d*x)^4*tan(c)^5 + 80*B*a^3*b*tan(d*x)^3*tan(c)^6 + 120*A*a^2*b^2*tan(d*x)^3*tan(c)^6 - 80*B*a*b^3*
tan(d*x)^3*tan(c)^6 - 20*A*b^4*tan(d*x)^3*tan(c)^6 - 90*B*a^2*b^2*tan(d*x)^6*tan(c)^2 - 60*A*a*b^3*tan(d*x)^6*
tan(c)^2 + 15*B*b^4*tan(d*x)^6*tan(c)^2 - 1200*A*a^4*d*x*tan(d*x)^3*tan(c)^3 + 4800*B*a^3*b*d*x*tan(d*x)^3*tan
(c)^3 + 7200*A*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 - 4800*B*a*b^3*d*x*tan(d*x)^3*tan(c)^3 - 1200*A*b^4*d*x*tan(d*x
)^3*tan(c)^3 + 120*B*a^4*tan(d*x)^5*tan(c)^3 + 480*A*a^3*b*tan(d*x)^5*tan(c)^3 - 1080*B*a^2*b^2*tan(d*x)^5*tan
(c)^3 - 720*A*a*b^3*tan(d*x)^5*tan(c)^3 + 180*B*b^4*tan(d*x)^5*tan(c)^3 - 210*B*a^4*tan(d*x)^4*tan(c)^4 - 840*
A*a^3*b*tan(d*x)^4*tan(c)^4 + 2070*B*a^2*b^2*tan(d*x)^4*tan(c)^4 + 1380*A*a*b^3*tan(d*x)^4*tan(c)^4 - 495*B*b^
4*tan(d*x)^4*tan(c)^4 + 120*B*a^4*tan(d*x)^3*tan(c)^5 + 480*A*a^3*b*tan(d*x)^3*tan(c)^5 - 1080*B*a^2*b^2*tan(d
*x)^3*tan(c)^5 - 720*A*a*b^3*tan(d*x)^3*tan(c)^5 + 180*B*b^4*tan(d*x)^3*tan(c)^5 - 90*B*a^2*b^2*tan(d*x)^2*tan
(c)^6 - 60*A*a*b^3*tan(d*x)^2*tan(c)^6 + 15*B*b^4*tan(d*x)^2*tan(c)^6 + 48*B*a*b^3*tan(d*x)^6*tan(c) + 12*A*b^
4*tan(d*x)^6*tan(c) - 240*B*a^3*b*tan(d*x)^5*tan(c)^2 - 360*A*a^2*b^2*tan(d*x)^5*tan(c)^2 + 480*B*a*b^3*tan(d*
x)^5*tan(c)^2 + 120*A*b^4*tan(d*x)^5*tan(c)^2 + 600*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/
(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 2400*A*a^3*b*log(4*(tan(d*x)^2*tan(c)
^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 - 3600*B*a^
2*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*t
an(d*x)^3*tan(c)^3 - 2400*A*a*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + t
an(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 600*B*b^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)
/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^3*tan(c)^3 + 600*A*a^4*tan(d*x)^4*tan(c)^3 - 3120
*B*a^3*b*tan(d*x)^4*tan(c)^3 - 4680*A*a^2*b^2*tan(d*x)^4*tan(c)^3 + 3600*B*a*b^3*tan(d*x)^4*tan(c)^3 + 900*A*b
^4*tan(d*x)^4*tan(c)^3 + 600*A*a^4*tan(d*x)^3*tan(c)^4 - 3120*B*a^3*b*tan(d*x)^3*tan(c)^4 - 4680*A*a^2*b^2*tan
(d*x)^3*tan(c)^4 + 3600*B*a*b^3*tan(d*x)^3*tan(c)^4 + 900*A*b^4*tan(d*x)^3*tan(c)^4 - 240*B*a^3*b*tan(d*x)^2*t
an(c)^5 - 360*A*a^2*b^2*tan(d*x)^2*tan(c)^5 + 480*B*a*b^3*tan(d*x)^2*tan(c)^5 + 120*A*b^4*tan(d*x)^2*tan(c)^5
+ 48*B*a*b^3*tan(d*x)*tan(c)^6 + 12*A*b^4*tan(d*x)*tan(c)^6 - 10*B*b^4*tan(d*x)^6 + 180*B*a^2*b^2*tan(d*x)^5*t
an(c) + 120*A*a*b^3*tan(d*x)^5*tan(c) - 90*B*b^4*tan(d*x)^5*tan(c) + 900*A*a^4*d*x*tan(d*x)^2*tan(c)^2 - 3600*
B*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 5400*A*a^2*b^2*d*x*tan(d*x)^2*tan(c)^2 + 3600*B*a*b^3*d*x*tan(d*x)^2*tan(c)^
2 + 900*A*b^4*d*x*tan(d*x)^2*tan(c)^2 - 180*B*a^4*tan(d*x)^4*tan(c)^2 - 720*A*a^3*b*tan(d*x)^4*tan(c)^2 + 1800
*B*a^2*b^2*tan(d*x)^4*tan(c)^2 + 1200*A*a*b^3*tan(d*x)^4*tan(c)^2 - 450*B*b^4*tan(d*x)^4*tan(c)^2 + 240*B*a^4*
tan(d*x)^3*tan(c)^3 + 960*A*a^3*b*tan(d*x)^3*tan(c)^3 - 2160*B*a^2*b^2*tan(d*x)^3*tan(c)^3 - 1440*A*a*b^3*tan(
d*x)^3*tan(c)^3 + 360*B*b^4*tan(d*x)^3*tan(c)^3 - 180*B*a^4*tan(d*x)^2*tan(c)^4 - 720*A*a^3*b*tan(d*x)^2*tan(c
)^4 + 1800*B*a^2*b^2*tan(d*x)^2*tan(c)^4 + 1200*A*a*b^3*tan(d*x)^2*tan(c)^4 - 450*B*b^4*tan(d*x)^2*tan(c)^4 +
180*B*a^2*b^2*tan(d*x)*tan(c)^5 + 120*A*a*b^3*tan(d*x)*tan(c)^5 - 90*B*b^4*tan(d*x)*tan(c)^5 - 10*B*b^4*tan(c)
^6 - 48*B*a*b^3*tan(d*x)^5 - 12*A*b^4*tan(d*x)^5 + 240*B*a^3*b*tan(d*x)^4*tan(c) + 360*A*a^2*b^2*tan(d*x)^4*ta
n(c) - 480*B*a*b^3*tan(d*x)^4*tan(c) - 120*A*b^4*tan(d*x)^4*tan(c) - 450*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*
tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 - 1800*A*a^3*b*log
(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2
*tan(c)^2 + 2700*B*a^2*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)
^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 + 1800*A*a*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(ta
n(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 - 450*B*b^4*log(4*(tan(d*x)^2*tan(c)^2 - 2
*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 - 600*A*a^4*tan(d
*x)^3*tan(c)^2 + 3120*B*a^3*b*tan(d*x)^3*tan(c)^2 + 4680*A*a^2*b^2*tan(d*x)^3*tan(c)^2 - 3600*B*a*b^3*tan(d*x)
^3*tan(c)^2 - 900*A*b^4*tan(d*x)^3*tan(c)^2 - 600*A*a^4*tan(d*x)^2*tan(c)^3 + 3120*B*a^3*b*tan(d*x)^2*tan(c)^3
+ 4680*A*a^2*b^2*tan(d*x)^2*tan(c)^3 - 3600*B*a*b^3*tan(d*x)^2*tan(c)^3 - 900*A*b^4*tan(d*x)^2*tan(c)^3 + 240
*B*a^3*b*tan(d*x)*tan(c)^4 + 360*A*a^2*b^2*tan(d*x)*tan(c)^4 - 480*B*a*b^3*tan(d*x)*tan(c)^4 - 120*A*b^4*tan(d
*x)*tan(c)^4 - 48*B*a*b^3*tan(c)^5 - 12*A*b^4*tan(c)^5 - 90*B*a^2*b^2*tan(d*x)^4 - 60*A*a*b^3*tan(d*x)^4 + 15*
B*b^4*tan(d*x)^4 - 360*A*a^4*d*x*tan(d*x)*tan(c) + 1440*B*a^3*b*d*x*tan(d*x)*tan(c) + 2160*A*a^2*b^2*d*x*tan(d
*x)*tan(c) - 1440*B*a*b^3*d*x*tan(d*x)*tan(c) - 360*A*b^4*d*x*tan(d*x)*tan(c) + 120*B*a^4*tan(d*x)^3*tan(c) +
480*A*a^3*b*tan(d*x)^3*tan(c) - 1080*B*a^2*b^2*tan(d*x)^3*tan(c) - 720*A*a*b^3*tan(d*x)^3*tan(c) + 180*B*b^4*t
an(d*x)^3*tan(c) - 210*B*a^4*tan(d*x)^2*tan(c)^2 - 840*A*a^3*b*tan(d*x)^2*tan(c)^2 + 2070*B*a^2*b^2*tan(d*x)^2
*tan(c)^2 + 1380*A*a*b^3*tan(d*x)^2*tan(c)^2 - 495*B*b^4*tan(d*x)^2*tan(c)^2 + 120*B*a^4*tan(d*x)*tan(c)^3 + 4
80*A*a^3*b*tan(d*x)*tan(c)^3 - 1080*B*a^2*b^2*tan(d*x)*tan(c)^3 - 720*A*a*b^3*tan(d*x)*tan(c)^3 + 180*B*b^4*ta
n(d*x)*tan(c)^3 - 90*B*a^2*b^2*tan(c)^4 - 60*A*a*b^3*tan(c)^4 + 15*B*b^4*tan(c)^4 - 80*B*a^3*b*tan(d*x)^3 - 12
0*A*a^2*b^2*tan(d*x)^3 + 80*B*a*b^3*tan(d*x)^3 + 20*A*b^4*tan(d*x)^3 + 180*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 -
2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c) + 720*A*a^3*b*log(4*
(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(
c) - 1080*B*a^2*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + ta
n(c)^2 + 1))*tan(d*x)*tan(c) - 720*A*a*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan
(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c) + 180*B*b^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c)
+ 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c) + 300*A*a^4*tan(d*x)^2*tan(c) - 1440*
B*a^3*b*tan(d*x)^2*tan(c) - 2160*A*a^2*b^2*tan(d*x)^2*tan(c) + 1440*B*a*b^3*tan(d*x)^2*tan(c) + 360*A*b^4*tan(
d*x)^2*tan(c) + 300*A*a^4*tan(d*x)*tan(c)^2 - 1440*B*a^3*b*tan(d*x)*tan(c)^2 - 2160*A*a^2*b^2*tan(d*x)*tan(c)^
2 + 1440*B*a*b^3*tan(d*x)*tan(c)^2 + 360*A*b^4*tan(d*x)*tan(c)^2 - 80*B*a^3*b*tan(c)^3 - 120*A*a^2*b^2*tan(c)^
3 + 80*B*a*b^3*tan(c)^3 + 20*A*b^4*tan(c)^3 + 60*A*a^4*d*x - 240*B*a^3*b*d*x - 360*A*a^2*b^2*d*x + 240*B*a*b^3
*d*x + 60*A*b^4*d*x - 30*B*a^4*tan(d*x)^2 - 120*A*a^3*b*tan(d*x)^2 + 180*B*a^2*b^2*tan(d*x)^2 + 120*A*a*b^3*ta
n(d*x)^2 - 30*B*b^4*tan(d*x)^2 + 120*B*a^4*tan(d*x)*tan(c) + 480*A*a^3*b*tan(d*x)*tan(c) - 1260*B*a^2*b^2*tan(
d*x)*tan(c) - 840*A*a*b^3*tan(d*x)*tan(c) + 270*B*b^4*tan(d*x)*tan(c) - 30*B*a^4*tan(c)^2 - 120*A*a^3*b*tan(c)
^2 + 180*B*a^2*b^2*tan(c)^2 + 120*A*a*b^3*tan(c)^2 - 30*B*b^4*tan(c)^2 - 30*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 -
2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) - 120*A*a^3*b*log(4*(tan(d*x)^2*tan
(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) + 180*B*a^2*b^2*log(4*(tan(d
*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) + 120*A*a*b^3*log(4
*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) - 30*B*b^4*l
og(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) - 60*A*a
^4*tan(d*x) + 240*B*a^3*b*tan(d*x) + 360*A*a^2*b^2*tan(d*x) - 240*B*a*b^3*tan(d*x) - 60*A*b^4*tan(d*x) - 60*A*
a^4*tan(c) + 240*B*a^3*b*tan(c) + 360*A*a^2*b^2*tan(c) - 240*B*a*b^3*tan(c) - 60*A*b^4*tan(c) - 30*B*a^4 - 120
*A*a^3*b + 270*B*a^2*b^2 + 180*A*a*b^3 - 55*B*b^4)/(d*tan(d*x)^6*tan(c)^6 - 6*d*tan(d*x)^5*tan(c)^5 + 15*d*tan
(d*x)^4*tan(c)^4 - 20*d*tan(d*x)^3*tan(c)^3 + 15*d*tan(d*x)^2*tan(c)^2 - 6*d*tan(d*x)*tan(c) + d)

Mupad [B] (verification not implemented)

Time = 7.98 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.14 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,a^4+A\,b^4+4\,B\,a\,b^3-2\,a^2\,b\,\left (3\,A\,b+2\,B\,a\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {A\,b^4}{3}+\frac {4\,B\,a\,b^3}{3}-\frac {2\,a^2\,b\,\left (3\,A\,b+2\,B\,a\right )}{3}\right )}{d}-x\,\left (A\,a^4-4\,B\,a^3\,b-6\,A\,a^2\,b^2+4\,B\,a\,b^3+A\,b^4\right )+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (\frac {A\,b^4}{5}+\frac {4\,B\,a\,b^3}{5}\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {B\,a^4}{2}+2\,A\,a^3\,b-3\,B\,a^2\,b^2-2\,A\,a\,b^3+\frac {B\,b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {B\,b^4}{4}-\frac {a\,b^2\,\left (2\,A\,b+3\,B\,a\right )}{2}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {B\,a^4}{2}+\frac {B\,b^4}{2}+2\,A\,a^3\,b-a\,b^2\,\left (2\,A\,b+3\,B\,a\right )\right )}{d}+\frac {B\,b^4\,{\mathrm {tan}\left (c+d\,x\right )}^6}{6\,d} \]

[In]

int(tan(c + d*x)^2*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^4,x)

[Out]

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