\(\int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\) [248]

Optimal result

Integrand size = 31, antiderivative size = 201 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\cos (c+d x))}{d}-\frac {b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{d}-\frac {(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac {B (a+b \tan (c+d x))^3}{3 d}+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d} \]

[Out]

-(A*a^3-3*A*a*b^2-3*B*a^2*b+B*b^3)*x+(3*A*a^2*b-A*b^3+B*a^3-3*B*a*b^2)*ln(cos(d*x+c))/d-b*(2*A*a*b+B*a^2-B*b^2
)*tan(d*x+c)/d-1/2*(A*b+B*a)*(a+b*tan(d*x+c))^2/d-1/3*B*(a+b*tan(d*x+c))^3/d+1/20*(5*A*b-B*a)*(a+b*tan(d*x+c))
^4/b^2/d+1/5*B*tan(d*x+c)*(a+b*tan(d*x+c))^4/b/d

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3688, 3711, 3609, 3606, 3556} \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=-\frac {b \left (a^2 B+2 a A b-b^2 B\right ) \tan (c+d x)}{d}+\frac {\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}-\frac {(a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}-\frac {B (a+b \tan (c+d x))^3}{3 d} \]

[In]

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

[Out]

-((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Cos[c + d*x]])/d -
(b*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x])/d - ((A*b + a*B)*(a + b*Tan[c + d*x])^2)/(2*d) - (B*(a + b*Tan[c +
d*x])^3)/(3*d) + ((5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/(20*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4)/
(5*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))^4}{5 b d}+\frac {\int (a+b \tan (c+d x))^3 \left (-a B-5 b B \tan (c+d x)+(5 A b-a B) \tan ^2(c+d x)\right ) \, dx}{5 b} \\ & = \frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac {\int (a+b \tan (c+d x))^3 (-5 A b-5 b B \tan (c+d x)) \, dx}{5 b} \\ & = -\frac {B (a+b \tan (c+d x))^3}{3 d}+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac {\int (a+b \tan (c+d x))^2 (-5 b (a A-b B)-5 b (A b+a B) \tan (c+d x)) \, dx}{5 b} \\ & = -\frac {(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac {B (a+b \tan (c+d x))^3}{3 d}+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac {\int (a+b \tan (c+d x)) \left (-5 b \left (a^2 A-A b^2-2 a b B\right )-5 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx}{5 b} \\ & = -\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )-\frac {b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{d}-\frac {(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac {B (a+b \tan (c+d x))^3}{3 d}+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B\right ) \int \tan (c+d x) \, dx \\ & = -\left (\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x\right )+\frac {\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\cos (c+d x))}{d}-\frac {b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{d}-\frac {(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac {B (a+b \tan (c+d x))^3}{3 d}+\frac {(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.37 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.20 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\frac {3 (5 A b-a B) (a+b \tan (c+d x))^4}{b}+12 B \tan (c+d x) (a+b \tan (c+d x))^4-30 (A b-a B) \left ((i a-b)^3 \log (i-\tan (c+d x))-(i a+b)^3 \log (i+\tan (c+d x))+6 a b^2 \tan (c+d x)+b^3 \tan ^2(c+d x)\right )+10 B \left (3 i (a+i b)^4 \log (i-\tan (c+d x))-3 i (a-i b)^4 \log (i+\tan (c+d x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-2 b^4 \tan ^3(c+d x)\right )}{60 b d} \]

[In]

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

[Out]

((3*(5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/b + 12*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4 - 30*(A*b - a*B)*((I*a
- b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2)
+ 10*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))/(60*b*d)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.03

method	result	size
parts	\(\frac {\left (A \,b^{3}+3 B a \,b^{2}\right ) \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (3 A a \,b^{2}+3 B \,a^{2} b \right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {A \,a^{3} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B \,b^{3} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\)	\(207\)
norman	\(\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) x +\frac {\left (A \,a^{3}-3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {b \left (3 A a b +3 B \,a^{2}-B \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} \left (A b +3 B a \right ) \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {\left (3 A \,a^{2} b -A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(220\)
derivativedivides	\(\frac {\frac {B \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {A \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {3 B a \,b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+A a \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )+B \,a^{2} b \left (\tan ^{3}\left (d x +c \right )\right )-\frac {B \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {3 A \,a^{2} b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {A \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}}{2}-\frac {3 B a \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+A \tan \left (d x +c \right ) a^{3}-3 A a \,b^{2} \tan \left (d x +c \right )-3 B \,a^{2} b \tan \left (d x +c \right )+B \,b^{3} \tan \left (d x +c \right )+\frac {\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(271\)
default	\(\frac {\frac {B \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {A \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {3 B a \,b^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+A a \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )+B \,a^{2} b \left (\tan ^{3}\left (d x +c \right )\right )-\frac {B \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {3 A \,a^{2} b \left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {A \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {B \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}}{2}-\frac {3 B a \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+A \tan \left (d x +c \right ) a^{3}-3 A a \,b^{2} \tan \left (d x +c \right )-3 B \,a^{2} b \tan \left (d x +c \right )+B \,b^{3} \tan \left (d x +c \right )+\frac {\left (-3 A \,a^{2} b +A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-A \,a^{3}+3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(271\)
parallelrisch	\(-\frac {-12 B \,b^{3} \left (\tan ^{5}\left (d x +c \right )\right )-15 A \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )-45 B a \,b^{2} \left (\tan ^{4}\left (d x +c \right )\right )-60 A a \,b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-60 B \,a^{2} b \left (\tan ^{3}\left (d x +c \right )\right )+20 B \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )+60 A x \,a^{3} d -180 A a \,b^{2} d x -90 A \,a^{2} b \left (\tan ^{2}\left (d x +c \right )\right )+30 A \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )-180 B \,a^{2} b d x +60 B \,b^{3} d x -30 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{3}+90 B a \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+90 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b -30 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{3}-60 A \tan \left (d x +c \right ) a^{3}+180 A a \,b^{2} \tan \left (d x +c \right )+30 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3}-90 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{2}+180 B \,a^{2} b \tan \left (d x +c \right )-60 B \,b^{3} \tan \left (d x +c \right )}{60 d}\)	\(306\)
risch	\(-A \,a^{3} x +3 A a \,b^{2} x +3 B \,a^{2} b x -B \,b^{3} x +\frac {2 i A \,b^{3} c}{d}+\frac {6 i B a \,b^{2} c}{d}-3 i A \,a^{2} b x +i A \,b^{3} x -\frac {6 i A \,a^{2} b c}{d}-\frac {2 i a^{3} B c}{d}+\frac {2 i \left (-60 A a \,b^{2}-60 B \,a^{2} b +15 A \,a^{3}+23 B \,b^{3}-135 i A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+180 i B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-45 i A \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}+90 i B a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+90 i B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-45 i A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-135 i A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+180 i B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+15 A \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+45 B \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+60 A \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+90 B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+90 A \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+140 B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+60 A \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+70 B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+60 i A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-45 i B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+30 i A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+60 i A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-45 i B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-15 i B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-15 i B \,a^{3} {\mathrm e}^{8 i \left (d x +c \right )}+30 i A \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-210 A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-210 B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-90 A a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-90 B \,a^{2} b \,{\mathrm e}^{8 i \left (d x +c \right )}-270 A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-270 B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-330 A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-330 B \,a^{2} b \,{\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}}+3 i B a \,b^{2} x -i B \,a^{3} x +\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,a^{2} b}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}-\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a \,b^{2}}{d}\)	\(755\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.06 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {12 \, B b^{3} \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} d x + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{2} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (185) = 370\).

Time = 0.18 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.91 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\begin {cases} - A a^{3} x + \frac {A a^{3} \tan {\left (c + d x \right )}}{d} - \frac {3 A a^{2} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 A a^{2} b \tan ^{2}{\left (c + d x \right )}}{2 d} + 3 A a b^{2} x + \frac {A a b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac {3 A a b^{2} \tan {\left (c + d x \right )}}{d} + \frac {A b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {A b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {B a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + 3 B a^{2} b x + \frac {B a^{2} b \tan ^{3}{\left (c + d x \right )}}{d} - \frac {3 B a^{2} b \tan {\left (c + d x \right )}}{d} + \frac {3 B a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {3 B a b^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {3 B a b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - B b^{3} x + \frac {B b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {B b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{3} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{3} \tan ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(tan(d*x+c)**2*(a+b*tan(d*x+c))**3*(A+B*tan(d*x+c)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.06 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {12 \, B b^{3} \tan \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{2} - 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} {\left (d x + c\right )} - 30 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3757 vs. \(2 (192) = 384\).

Time = 3.79 (sec) , antiderivative size = 3757, normalized size of antiderivative = 18.69 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/60*(60*A*a^3*d*x*tan(d*x)^5*tan(c)^5 - 180*B*a^2*b*d*x*tan(d*x)^5*tan(c)^5 - 180*A*a*b^2*d*x*tan(d*x)^5*tan
(c)^5 + 60*B*b^3*d*x*tan(d*x)^5*tan(c)^5 - 30*B*a^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)^5*tan(c)^5 - 90*A*a^2*b*log(4*(tan(d*x)^2*tan(c)^2 - 2*t
an(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 + 90*B*a*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*ta
n(c)^5 + 30*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)^5*tan(c)^5 - 300*A*a^3*d*x*tan(d*x)^4*tan(c)^4 + 900*B*a^2*b*d*x*tan(d*x)^4*tan(c)^4 + 900
*A*a*b^2*d*x*tan(d*x)^4*tan(c)^4 - 300*B*b^3*d*x*tan(d*x)^4*tan(c)^4 - 30*B*a^3*tan(d*x)^5*tan(c)^5 - 90*A*a^2
*b*tan(d*x)^5*tan(c)^5 + 135*B*a*b^2*tan(d*x)^5*tan(c)^5 + 45*A*b^3*tan(d*x)^5*tan(c)^5 + 150*B*a^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*A*a^2*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 - 450*B*a*b^2*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 - 150*A*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*t
an(c) + 1)/(tan(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^4*tan(c)^4 + 60*A*a^3*tan(d*x)^5*tan(c)
^4 - 180*B*a^2*b*tan(d*x)^5*tan(c)^4 - 180*A*a*b^2*tan(d*x)^5*tan(c)^4 + 60*B*b^3*tan(d*x)^5*tan(c)^4 + 60*A*a
^3*tan(d*x)^4*tan(c)^5 - 180*B*a^2*b*tan(d*x)^4*tan(c)^5 - 180*A*a*b^2*tan(d*x)^4*tan(c)^5 + 60*B*b^3*tan(d*x)
^4*tan(c)^5 + 600*A*a^3*d*x*tan(d*x)^3*tan(c)^3 - 1800*B*a^2*b*d*x*tan(d*x)^3*tan(c)^3 - 1800*A*a*b^2*d*x*tan(
d*x)^3*tan(c)^3 + 600*B*b^3*d*x*tan(d*x)^3*tan(c)^3 - 30*B*a^3*tan(d*x)^5*tan(c)^3 - 90*A*a^2*b*tan(d*x)^5*tan
(c)^3 + 90*B*a*b^2*tan(d*x)^5*tan(c)^3 + 30*A*b^3*tan(d*x)^5*tan(c)^3 + 90*B*a^3*tan(d*x)^4*tan(c)^4 + 270*A*a
^2*b*tan(d*x)^4*tan(c)^4 - 495*B*a*b^2*tan(d*x)^4*tan(c)^4 - 165*A*b^3*tan(d*x)^4*tan(c)^4 - 30*B*a^3*tan(d*x)
^3*tan(c)^5 - 90*A*a^2*b*tan(d*x)^3*tan(c)^5 + 90*B*a*b^2*tan(d*x)^3*tan(c)^5 + 30*A*b^3*tan(d*x)^3*tan(c)^5 +
60*B*a^2*b*tan(d*x)^5*tan(c)^2 + 60*A*a*b^2*tan(d*x)^5*tan(c)^2 - 20*B*b^3*tan(d*x)^5*tan(c)^2 - 300*B*a^3*lo
g(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 - 900*A*a^2*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 + 900*B*a*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)^3*tan(c)^3 + 300*A*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*ta
n(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 - 240*A*a^3*tan(d*x)
^4*tan(c)^3 + 900*B*a^2*b*tan(d*x)^4*tan(c)^3 + 900*A*a*b^2*tan(d*x)^4*tan(c)^3 - 300*B*b^3*tan(d*x)^4*tan(c)^
3 - 240*A*a^3*tan(d*x)^3*tan(c)^4 + 900*B*a^2*b*tan(d*x)^3*tan(c)^4 + 900*A*a*b^2*tan(d*x)^3*tan(c)^4 - 300*B*
b^3*tan(d*x)^3*tan(c)^4 + 60*B*a^2*b*tan(d*x)^2*tan(c)^5 + 60*A*a*b^2*tan(d*x)^2*tan(c)^5 - 20*B*b^3*tan(d*x)^
2*tan(c)^5 - 45*B*a*b^2*tan(d*x)^5*tan(c) - 15*A*b^3*tan(d*x)^5*tan(c) - 600*A*a^3*d*x*tan(d*x)^2*tan(c)^2 + 1
800*B*a^2*b*d*x*tan(d*x)^2*tan(c)^2 + 1800*A*a*b^2*d*x*tan(d*x)^2*tan(c)^2 - 600*B*b^3*d*x*tan(d*x)^2*tan(c)^2
+ 90*B*a^3*tan(d*x)^4*tan(c)^2 + 270*A*a^2*b*tan(d*x)^4*tan(c)^2 - 450*B*a*b^2*tan(d*x)^4*tan(c)^2 - 150*A*b^
3*tan(d*x)^4*tan(c)^2 - 120*B*a^3*tan(d*x)^3*tan(c)^3 - 360*A*a^2*b*tan(d*x)^3*tan(c)^3 + 540*B*a*b^2*tan(d*x)
^3*tan(c)^3 + 180*A*b^3*tan(d*x)^3*tan(c)^3 + 90*B*a^3*tan(d*x)^2*tan(c)^4 + 270*A*a^2*b*tan(d*x)^2*tan(c)^4 -
450*B*a*b^2*tan(d*x)^2*tan(c)^4 - 150*A*b^3*tan(d*x)^2*tan(c)^4 - 45*B*a*b^2*tan(d*x)*tan(c)^5 - 15*A*b^3*tan
(d*x)*tan(c)^5 + 12*B*b^3*tan(d*x)^5 - 120*B*a^2*b*tan(d*x)^4*tan(c) - 120*A*a*b^2*tan(d*x)^4*tan(c) + 100*B*b
^3*tan(d*x)^4*tan(c) + 300*B*a^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*tan(c)^2 + ta
n(d*x)^2 + tan(c)^2 + 1))*tan(d*x)^2*tan(c)^2 + 900*A*a^2*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 - 900*B*a*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 - 300*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)^2*tan(c)^2 + 360*A*a^3*tan(d*x)^3*tan(c)^2 - 1440*B*a^2*b*tan(d*x)^3*tan(c)^2 - 1440*A*a*b^2*tan(d*x)^3*ta
n(c)^2 + 600*B*b^3*tan(d*x)^3*tan(c)^2 + 360*A*a^3*tan(d*x)^2*tan(c)^3 - 1440*B*a^2*b*tan(d*x)^2*tan(c)^3 - 14
40*A*a*b^2*tan(d*x)^2*tan(c)^3 + 600*B*b^3*tan(d*x)^2*tan(c)^3 - 120*B*a^2*b*tan(d*x)*tan(c)^4 - 120*A*a*b^2*t
an(d*x)*tan(c)^4 + 100*B*b^3*tan(d*x)*tan(c)^4 + 12*B*b^3*tan(c)^5 + 45*B*a*b^2*tan(d*x)^4 + 15*A*b^3*tan(d*x)
^4 + 300*A*a^3*d*x*tan(d*x)*tan(c) - 900*B*a^2*b*d*x*tan(d*x)*tan(c) - 900*A*a*b^2*d*x*tan(d*x)*tan(c) + 300*B
*b^3*d*x*tan(d*x)*tan(c) - 90*B*a^3*tan(d*x)^3*tan(c) - 270*A*a^2*b*tan(d*x)^3*tan(c) + 450*B*a*b^2*tan(d*x)^3
*tan(c) + 150*A*b^3*tan(d*x)^3*tan(c) + 120*B*a^3*tan(d*x)^2*tan(c)^2 + 360*A*a^2*b*tan(d*x)^2*tan(c)^2 - 540*
B*a*b^2*tan(d*x)^2*tan(c)^2 - 180*A*b^3*tan(d*x)^2*tan(c)^2 - 90*B*a^3*tan(d*x)*tan(c)^3 - 270*A*a^2*b*tan(d*x
)*tan(c)^3 + 450*B*a*b^2*tan(d*x)*tan(c)^3 + 150*A*b^3*tan(d*x)*tan(c)^3 + 45*B*a*b^2*tan(c)^4 + 15*A*b^3*tan(
c)^4 + 60*B*a^2*b*tan(d*x)^3 + 60*A*a*b^2*tan(d*x)^3 - 20*B*b^3*tan(d*x)^3 - 150*B*a^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) - 450*A*a^2*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) + 450*B*a*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)*tan(c) + 150*A*b^3*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 + tan(c)^2 + 1))*tan(d*x)*tan(c) - 240*A*a^3*tan(d*x)^2*tan(c) + 900*B*a^2*b*tan(d*x)^2*ta
n(c) + 900*A*a*b^2*tan(d*x)^2*tan(c) - 300*B*b^3*tan(d*x)^2*tan(c) - 240*A*a^3*tan(d*x)*tan(c)^2 + 900*B*a^2*b
*tan(d*x)*tan(c)^2 + 900*A*a*b^2*tan(d*x)*tan(c)^2 - 300*B*b^3*tan(d*x)*tan(c)^2 + 60*B*a^2*b*tan(c)^3 + 60*A*
a*b^2*tan(c)^3 - 20*B*b^3*tan(c)^3 - 60*A*a^3*d*x + 180*B*a^2*b*d*x + 180*A*a*b^2*d*x - 60*B*b^3*d*x + 30*B*a^
3*tan(d*x)^2 + 90*A*a^2*b*tan(d*x)^2 - 90*B*a*b^2*tan(d*x)^2 - 30*A*b^3*tan(d*x)^2 - 90*B*a^3*tan(d*x)*tan(c)
- 270*A*a^2*b*tan(d*x)*tan(c) + 495*B*a*b^2*tan(d*x)*tan(c) + 165*A*b^3*tan(d*x)*tan(c) + 30*B*a^3*tan(c)^2 +
90*A*a^2*b*tan(c)^2 - 90*B*a*b^2*tan(c)^2 - 30*A*b^3*tan(c)^2 + 30*B*a^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)) + 90*A*a^2*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)) - 90*B*a*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)) - 30*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)) + 60*A*a^3*tan(d*x) - 180*B*
a^2*b*tan(d*x) - 180*A*a*b^2*tan(d*x) + 60*B*b^3*tan(d*x) + 60*A*a^3*tan(c) - 180*B*a^2*b*tan(c) - 180*A*a*b^2
*tan(c) + 60*B*b^3*tan(c) + 30*B*a^3 + 90*A*a^2*b - 135*B*a*b^2 - 45*A*b^3)/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d
*x)^4*tan(c)^4 + 10*d*tan(d*x)^3*tan(c)^3 - 10*d*tan(d*x)^2*tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)

Mupad [B] (verification not implemented)

Time = 7.31 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08 \[ \int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,a^3+B\,b^3-3\,a\,b\,\left (A\,b+B\,a\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,b^3}{3}-a\,b\,\left (A\,b+B\,a\right )\right )}{d}-x\,\left (A\,a^3-3\,B\,a^2\,b-3\,A\,a\,b^2+B\,b^3\right )+\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (-\frac {B\,a^3}{2}-\frac {3\,A\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}+\frac {A\,b^3}{2}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {A\,b^3}{4}+\frac {3\,B\,a\,b^2}{4}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-\frac {B\,a^3}{2}-\frac {3\,A\,a^2\,b}{2}+\frac {3\,B\,a\,b^2}{2}+\frac {A\,b^3}{2}\right )}{d}+\frac {B\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d} \]

[In]

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

[Out]

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