\(\int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\) [259]

Optimal result

Integrand size = 23, antiderivative size = 202 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) x-\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \log (\cos (c+d x))}{d}+\frac {b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)}{d}+\frac {\left (2 a A b+a^2 B-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(A b+a B) (a+b \tan (c+d x))^3}{3 d}+\frac {B (a+b \tan (c+d x))^4}{4 d} \]

[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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3609, 3606, 3556} \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 B+2 a A b-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {b \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \tan (c+d x)}{d}-\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \log (\cos (c+d x))}{d}+x \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right )+\frac {(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac {B (a+b \tan (c+d x))^4}{4 d} \]

[In]

Int[(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)

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.99 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.19 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {-2 (A b-a B) \left (3 i (a+i b)^4 \log (i-\tan (c+d x))-3 i (a-i b)^4 \log (i+\tan (c+d x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-2 b^4 \tan ^3(c+d x)\right )+B \left (6 (-i a+b)^5 \log (i-\tan (c+d x))+6 (i a+b)^5 \log (i+\tan (c+d x))+60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)-6 b^3 \left (-10 a^2+b^2\right ) \tan ^2(c+d x)+20 a b^4 \tan ^3(c+d x)+3 b^5 \tan ^4(c+d x)\right )}{12 b d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00

method	result	size
norman	\(\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) x +\frac {b \left (6 A \,a^{2} b -A \,b^{3}+4 B \,a^{3}-4 B a \,b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b^{2} \left (4 A a b +6 B \,a^{2}-B \,b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{3} \left (A b +4 B a \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(201\)
parts	\(A \,a^{4} x +\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,b^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\)	\(204\)
derivativedivides	\(\frac {\frac {B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {4 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+3 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+6 A \,a^{2} b^{2} \tan \left (d x +c \right )-A \,b^{4} \tan \left (d x +c \right )+4 B \,a^{3} b \tan \left (d x +c \right )-4 B a \,b^{3} \tan \left (d x +c \right )+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(237\)
default	\(\frac {\frac {B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+\frac {A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {4 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+3 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+6 A \,a^{2} b^{2} \tan \left (d x +c \right )-A \,b^{4} \tan \left (d x +c \right )+4 B \,a^{3} b \tan \left (d x +c \right )-4 B a \,b^{3} \tan \left (d x +c \right )+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(237\)
parallelrisch	\(\frac {3 B \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )+4 A \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )+16 B a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )+12 A x \,a^{4} d -72 A \,a^{2} b^{2} d x +12 A \,b^{4} d x +24 A a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )-48 B \,a^{3} b d x +48 B a \,b^{3} d x +36 B \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-6 B \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )+24 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b -24 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}+72 A \,a^{2} b^{2} \tan \left (d x +c \right )-12 A \,b^{4} \tan \left (d x +c \right )+6 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}-36 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}+6 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}+48 B \,a^{3} b \tan \left (d x +c \right )-48 B a \,b^{3} \tan \left (d x +c \right )}{12 d}\)	\(284\)
risch	\(-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,b^{4}}{d}+A \,a^{4} x +\frac {8 i A \,a^{3} b c}{d}+\frac {2 i a^{4} B c}{d}+i B \,b^{4} x -\frac {12 i B \,a^{2} b^{2} c}{d}+\frac {2 i B \,b^{4} c}{d}+A \,b^{4} x +i B \,a^{4} x -\frac {8 i A a \,b^{3} c}{d}-6 i B \,a^{2} b^{2} x -4 i A a \,b^{3} x +\frac {4 i b \left (9 A \,a^{2} b -8 B a \,b^{2}+6 B \,a^{3}-2 A \,b^{3}-12 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+9 A \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+27 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 i B \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+27 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}-24 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-20 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i B \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i B \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+18 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-5 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+6 B \,a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+18 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-18 i B \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 i A a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 i B \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-12 i A a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 i A a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-9 i B \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+4 i A \,a^{3} b x +4 B a \,b^{3} x -6 A \,a^{2} b^{2} x -4 B \,a^{3} b x -\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,a^{3} b}{d}+\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A a \,b^{3}}{d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,a^{2} b^{2}}{d}\)	\(638\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {3 \, B b^{4} \tan \left (d x + c\right )^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{3} + 12 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x + 6 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{2} - 6 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 12 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )}{12 \, d} \]

[In]

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

[Out]

1/12*(3*B*b^4*tan(d*x + c)^4 + 4*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^3 + 12*(A*a^4 - 4*B*a^3*b - 6*A*a^2*b^2 + 4*
B*a*b^3 + A*b^4)*d*x + 6*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*tan(d*x + c)^2 - 6*(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)) + 12*(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 [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.72 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} A a^{4} x + \frac {2 A a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - 6 A a^{2} b^{2} x + \frac {6 A a^{2} b^{2} \tan {\left (c + d x \right )}}{d} - \frac {2 A a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 A a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} + A b^{4} x + \frac {A b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {A b^{4} \tan {\left (c + d x \right )}}{d} + \frac {B a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 B a^{3} b x + \frac {4 B a^{3} b \tan {\left (c + d x \right )}}{d} - \frac {3 B a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 B a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 B a b^{3} x + \frac {4 B a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 B a b^{3} \tan {\left (c + d x \right )}}{d} + \frac {B b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {B b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {3 \, B b^{4} \tan \left (d x + c\right )^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{2} + 12 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )} + 6 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )}{12 \, d} \]

[In]

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

[Out]

1/12*(3*B*b^4*tan(d*x + c)^4 + 4*(4*B*a*b^3 + A*b^4)*tan(d*x + c)^3 + 6*(6*B*a^2*b^2 + 4*A*a*b^3 - B*b^4)*tan(
d*x + c)^2 + 12*(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) + 6*(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) + 12*(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. 3133 vs. \(2 (196) = 392\).

Time = 2.81 (sec) , antiderivative size = 3133, normalized size of antiderivative = 15.51 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(12*A*a^4*d*x*tan(d*x)^4*tan(c)^4 - 48*B*a^3*b*d*x*tan(d*x)^4*tan(c)^4 - 72*A*a^2*b^2*d*x*tan(d*x)^4*tan(
c)^4 + 48*B*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 12*A*b^4*d*x*tan(d*x)^4*tan(c)^4 - 6*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 - 24*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 + 36*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 + 24*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 - 6*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 - 48*A*a^4*d*x*ta
n(d*x)^3*tan(c)^3 + 192*B*a^3*b*d*x*tan(d*x)^3*tan(c)^3 + 288*A*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 - 192*B*a*b^3*
d*x*tan(d*x)^3*tan(c)^3 - 48*A*b^4*d*x*tan(d*x)^3*tan(c)^3 + 36*B*a^2*b^2*tan(d*x)^4*tan(c)^4 + 24*A*a*b^3*tan
(d*x)^4*tan(c)^4 - 9*B*b^4*tan(d*x)^4*tan(c)^4 + 24*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 + 96*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 - 144*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)^3*tan(c)^3 - 96*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)^3*tan(c)^3 + 24*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 - 48*B*a^3*b*tan(d*x)^4*tan(c)^3 - 72*A*a^2*
b^2*tan(d*x)^4*tan(c)^3 + 48*B*a*b^3*tan(d*x)^4*tan(c)^3 + 12*A*b^4*tan(d*x)^4*tan(c)^3 - 48*B*a^3*b*tan(d*x)^
3*tan(c)^4 - 72*A*a^2*b^2*tan(d*x)^3*tan(c)^4 + 48*B*a*b^3*tan(d*x)^3*tan(c)^4 + 12*A*b^4*tan(d*x)^3*tan(c)^4
+ 72*A*a^4*d*x*tan(d*x)^2*tan(c)^2 - 288*B*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 432*A*a^2*b^2*d*x*tan(d*x)^2*tan(c)
^2 + 288*B*a*b^3*d*x*tan(d*x)^2*tan(c)^2 + 72*A*b^4*d*x*tan(d*x)^2*tan(c)^2 + 36*B*a^2*b^2*tan(d*x)^4*tan(c)^2
+ 24*A*a*b^3*tan(d*x)^4*tan(c)^2 - 6*B*b^4*tan(d*x)^4*tan(c)^2 - 72*B*a^2*b^2*tan(d*x)^3*tan(c)^3 - 48*A*a*b^
3*tan(d*x)^3*tan(c)^3 + 24*B*b^4*tan(d*x)^3*tan(c)^3 + 36*B*a^2*b^2*tan(d*x)^2*tan(c)^4 + 24*A*a*b^3*tan(d*x)^
2*tan(c)^4 - 6*B*b^4*tan(d*x)^2*tan(c)^4 - 16*B*a*b^3*tan(d*x)^4*tan(c) - 4*A*b^4*tan(d*x)^4*tan(c) - 36*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 - 144*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 + 216*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 + 144*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)^2*tan(c)^2 - 36*B*b^4*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)^
2*tan(c)^2 + 144*B*a^3*b*tan(d*x)^3*tan(c)^2 + 216*A*a^2*b^2*tan(d*x)^3*tan(c)^2 - 192*B*a*b^3*tan(d*x)^3*tan(
c)^2 - 48*A*b^4*tan(d*x)^3*tan(c)^2 + 144*B*a^3*b*tan(d*x)^2*tan(c)^3 + 216*A*a^2*b^2*tan(d*x)^2*tan(c)^3 - 19
2*B*a*b^3*tan(d*x)^2*tan(c)^3 - 48*A*b^4*tan(d*x)^2*tan(c)^3 - 16*B*a*b^3*tan(d*x)*tan(c)^4 - 4*A*b^4*tan(d*x)
*tan(c)^4 + 3*B*b^4*tan(d*x)^4 - 48*A*a^4*d*x*tan(d*x)*tan(c) + 192*B*a^3*b*d*x*tan(d*x)*tan(c) + 288*A*a^2*b^
2*d*x*tan(d*x)*tan(c) - 192*B*a*b^3*d*x*tan(d*x)*tan(c) - 48*A*b^4*d*x*tan(d*x)*tan(c) - 72*B*a^2*b^2*tan(d*x)
^3*tan(c) - 48*A*a*b^3*tan(d*x)^3*tan(c) + 24*B*b^4*tan(d*x)^3*tan(c) + 72*B*a^2*b^2*tan(d*x)^2*tan(c)^2 + 48*
A*a*b^3*tan(d*x)^2*tan(c)^2 - 12*B*b^4*tan(d*x)^2*tan(c)^2 - 72*B*a^2*b^2*tan(d*x)*tan(c)^3 - 48*A*a*b^3*tan(d
*x)*tan(c)^3 + 24*B*b^4*tan(d*x)*tan(c)^3 + 3*B*b^4*tan(c)^4 + 16*B*a*b^3*tan(d*x)^3 + 4*A*b^4*tan(d*x)^3 + 24
*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) + 96*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) - 144*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)*tan(c) - 96*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) + 24*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) - 144
*B*a^3*b*tan(d*x)^2*tan(c) - 216*A*a^2*b^2*tan(d*x)^2*tan(c) + 192*B*a*b^3*tan(d*x)^2*tan(c) + 48*A*b^4*tan(d*
x)^2*tan(c) - 144*B*a^3*b*tan(d*x)*tan(c)^2 - 216*A*a^2*b^2*tan(d*x)*tan(c)^2 + 192*B*a*b^3*tan(d*x)*tan(c)^2
+ 48*A*b^4*tan(d*x)*tan(c)^2 + 16*B*a*b^3*tan(c)^3 + 4*A*b^4*tan(c)^3 + 12*A*a^4*d*x - 48*B*a^3*b*d*x - 72*A*a
^2*b^2*d*x + 48*B*a*b^3*d*x + 12*A*b^4*d*x + 36*B*a^2*b^2*tan(d*x)^2 + 24*A*a*b^3*tan(d*x)^2 - 6*B*b^4*tan(d*x
)^2 - 72*B*a^2*b^2*tan(d*x)*tan(c) - 48*A*a*b^3*tan(d*x)*tan(c) + 24*B*b^4*tan(d*x)*tan(c) + 36*B*a^2*b^2*tan(
c)^2 + 24*A*a*b^3*tan(c)^2 - 6*B*b^4*tan(c)^2 - 6*B*a^4*log(4*(tan(d*x)^2*tan(c)^2 - 2*tan(d*x)*tan(c) + 1)/(t
an(d*x)^2*tan(c)^2 + tan(d*x)^2 + tan(c)^2 + 1)) - 24*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)) + 36*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)) + 24*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)) - 6*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)) + 48*B*a^3*b*tan(d*x) + 72*A*a^2*b^2*t
an(d*x) - 48*B*a*b^3*tan(d*x) - 12*A*b^4*tan(d*x) + 48*B*a^3*b*tan(c) + 72*A*a^2*b^2*tan(c) - 48*B*a*b^3*tan(c
) - 12*A*b^4*tan(c) + 36*B*a^2*b^2 + 24*A*a*b^3 - 9*B*b^4)/(d*tan(d*x)^4*tan(c)^4 - 4*d*tan(d*x)^3*tan(c)^3 +
6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)

Mupad [B] (verification not implemented)

Time = 7.08 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01 \[ \int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=x\,\left (A\,a^4-4\,B\,a^3\,b-6\,A\,a^2\,b^2+4\,B\,a\,b^3+A\,b^4\right )-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A\,b^4+4\,B\,a\,b^3-2\,a^2\,b\,\left (3\,A\,b+2\,B\,a\right )\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {A\,b^4}{3}+\frac {4\,B\,a\,b^3}{3}\right )}{d}+\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {B\,a^4}{2}+2\,A\,a^3\,b-3\,B\,a^2\,b^2-2\,A\,a\,b^3+\frac {B\,b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {B\,b^4}{2}-a\,b^2\,\left (2\,A\,b+3\,B\,a\right )\right )}{d}+\frac {B\,b^4\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d} \]

[In]

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

[Out]

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)*(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))/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)^2*((B*b^4)/2 - a*b^2*(2*A*b + 3*B*a)))/d + (B*b^4*ta
n(c + d*x)^4)/(4*d)