\(\int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [294]

Optimal result

Integrand size = 23, antiderivative size = 247 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (a^4 A-6 a^2 A b^2+A b^4+4 a^3 b B-4 a b^3 B\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (4 a^3 A b-4 a A b^3-a^4 B+6 a^2 b^2 B-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {A b-a B}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {2 a A b-a^2 B+b^2 B}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {3 a^2 A b-A b^3-a^3 B+3 a b^2 B}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 247, 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 = {3610, 3612, 3611} \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {A b-a B}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a^2 (-B)+2 a A b+b^2 B}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {\left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

[In]

Int[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^4,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)/(a^2 + b^2)^4 + ((4*a^3*A*b - 4*a*A*b^3 - a^4*B + 6*
a^2*b^2*B - b^4*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - (A*b - a*B)/(3*(a^2 + b^2)*d*(a +
b*Tan[c + d*x])^3) - (2*a*A*b - a^2*B + b^2*B)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (3*a^2*A*b - A*b^
3 - a^3*B + 3*a*b^2*B)/((a^2 + b^2)^3*d*(a + b*Tan[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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

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]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.28 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {(A b-a B) \left (\frac {3 i \log (i-\tan (c+d x))}{(a+i b)^4}-\frac {3 i \log (i+\tan (c+d x))}{(a-i b)^4}-\frac {24 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {6 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{6 b d}-\frac {B \left (\frac {\log (i-\tan (c+d x))}{(i a-b)^3}-\frac {\log (i+\tan (c+d x))}{(i a+b)^3}-\frac {2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )}{2 b d} \]

[In]

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

[Out]

-1/6*((A*b - a*B)*(((3*I)*Log[I - Tan[c + d*x]])/(a + I*b)^4 - ((3*I)*Log[I + Tan[c + d*x]])/(a - I*b)^4 - (24
*a*(a - b)*b*(a + b)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^4 + (2*b)/((a^2 + b^2)*(a + b*Tan[c + d*x])^3) + (6*
a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])^2) + (6*b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/(b*d)
- (B*(Log[I - Tan[c + d*x]]/(I*a - b)^3 - Log[I + Tan[c + d*x]]/(I*a + b)^3 - (2*b*(3*a^2 - b^2)*Log[a + b*Ta
n[c + d*x]])/(a^2 + b^2)^3 + b/((a^2 + b^2)*(a + b*Tan[c + d*x])^2) + (4*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*
x]))))/(2*b*d)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A b -B a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {2 A a b -B \,a^{2}+B \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(284\)
default	\(\frac {\frac {\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {A b -B a}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {2 A a b -B \,a^{2}+B \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\)	\(284\)
norman	\(\frac {\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{3} x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {b^{3} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {20 A \,a^{4} b^{3}+6 A \,a^{2} b^{5}+2 A \,b^{7}-9 B \,a^{5} b^{2}+8 B \,a^{3} b^{4}+B a \,b^{6}}{6 b^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (8 A \,a^{3} b^{3}-3 B \,a^{4} b^{2}+6 B \,a^{2} b^{4}+B \,b^{6}\right ) \tan \left (d x +c \right )}{2 b d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (3 A \,a^{2} b^{3}-A \,b^{5}-B \,a^{3} b^{2}+3 B a \,b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 b \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a^{2} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 b^{2} \left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}+4 B \,a^{3} b -4 B a \,b^{3}\right ) a x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(731\)
risch	\(\text {Expression too large to display}\)	\(1363\)
parallelrisch	\(\text {Expression too large to display}\)	\(1648\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 815 vs. \(2 (239) = 478\).

Time = 0.30 (sec) , antiderivative size = 815, normalized size of antiderivative = 3.30 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {27 \, B a^{5} b^{2} - 48 \, A a^{4} b^{3} - 18 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - B a b^{6} - 2 \, A b^{7} - {\left (11 \, B a^{4} b^{3} - 26 \, A a^{3} b^{4} - 30 \, B a^{2} b^{5} + 18 \, A a b^{6} + 3 \, B b^{7} - 6 \, {\left (A a^{4} b^{3} + 4 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - 4 \, B a b^{6} + A b^{7}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 6 \, {\left (A a^{7} + 4 \, B a^{6} b - 6 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + A a^{3} b^{4}\right )} d x - 3 \, {\left (9 \, B a^{5} b^{2} - 20 \, A a^{4} b^{3} - 26 \, B a^{3} b^{4} + 22 \, A a^{2} b^{5} + 9 \, B a b^{6} - 2 \, A b^{7} - 6 \, {\left (A a^{5} b^{2} + 4 \, B a^{4} b^{3} - 6 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + A a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (B a^{7} - 4 \, A a^{6} b - 6 \, B a^{5} b^{2} + 4 \, A a^{4} b^{3} + B a^{3} b^{4} + {\left (B a^{4} b^{3} - 4 \, A a^{3} b^{4} - 6 \, B a^{2} b^{5} + 4 \, A a b^{6} + B b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (B a^{5} b^{2} - 4 \, A a^{4} b^{3} - 6 \, B a^{3} b^{4} + 4 \, A a^{2} b^{5} + B a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (B a^{6} b - 4 \, A a^{5} b^{2} - 6 \, B a^{4} b^{3} + 4 \, A a^{3} b^{4} + B a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 3 \, {\left (6 \, B a^{6} b - 12 \, A a^{5} b^{2} - 23 \, B a^{4} b^{3} + 30 \, A a^{3} b^{4} + 16 \, B a^{2} b^{5} - 2 \, A a b^{6} + B b^{7} - 6 \, {\left (A a^{6} b + 4 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \]

[In]

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

[Out]

1/6*(27*B*a^5*b^2 - 48*A*a^4*b^3 - 18*B*a^3*b^4 - 6*A*a^2*b^5 - B*a*b^6 - 2*A*b^7 - (11*B*a^4*b^3 - 26*A*a^3*b
^4 - 30*B*a^2*b^5 + 18*A*a*b^6 + 3*B*b^7 - 6*(A*a^4*b^3 + 4*B*a^3*b^4 - 6*A*a^2*b^5 - 4*B*a*b^6 + A*b^7)*d*x)*
tan(d*x + c)^3 + 6*(A*a^7 + 4*B*a^6*b - 6*A*a^5*b^2 - 4*B*a^4*b^3 + A*a^3*b^4)*d*x - 3*(9*B*a^5*b^2 - 20*A*a^4
*b^3 - 26*B*a^3*b^4 + 22*A*a^2*b^5 + 9*B*a*b^6 - 2*A*b^7 - 6*(A*a^5*b^2 + 4*B*a^4*b^3 - 6*A*a^3*b^4 - 4*B*a^2*
b^5 + A*a*b^6)*d*x)*tan(d*x + c)^2 - 3*(B*a^7 - 4*A*a^6*b - 6*B*a^5*b^2 + 4*A*a^4*b^3 + B*a^3*b^4 + (B*a^4*b^3
- 4*A*a^3*b^4 - 6*B*a^2*b^5 + 4*A*a*b^6 + B*b^7)*tan(d*x + c)^3 + 3*(B*a^5*b^2 - 4*A*a^4*b^3 - 6*B*a^3*b^4 +
4*A*a^2*b^5 + B*a*b^6)*tan(d*x + c)^2 + 3*(B*a^6*b - 4*A*a^5*b^2 - 6*B*a^4*b^3 + 4*A*a^3*b^4 + B*a^2*b^5)*tan(
d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 3*(6*B*a^6*b - 12*A*a^5*
b^2 - 23*B*a^4*b^3 + 30*A*a^3*b^4 + 16*B*a^2*b^5 - 2*A*a*b^6 + B*b^7 - 6*(A*a^6*b + 4*B*a^5*b^2 - 6*A*a^4*b^3
- 4*B*a^3*b^4 + A*a^2*b^5)*d*x)*tan(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*tan(d*x
+ c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*tan(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6
*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*d*tan(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8)*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (239) = 478\).

Time = 0.53 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.08 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {11 \, B a^{5} - 26 \, A a^{4} b - 14 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3} - B a b^{4} - 2 \, A b^{5} + 6 \, {\left (B a^{3} b^{2} - 3 \, A a^{2} b^{3} - 3 \, B a b^{4} + A b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (5 \, B a^{4} b - 14 \, A a^{3} b^{2} - 12 \, B a^{2} b^{3} + 2 \, A a b^{4} - B b^{5}\right )} \tan \left (d x + c\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6} + {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b + 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} + a^{2} b^{7}\right )} \tan \left (d x + c\right )}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(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)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) - 6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(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)
/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (11*B*a^5 - 26*A*a^4*b - 14*B*a^3*b^2 - 4*A*a^2*b^3 - B*a*b
^4 - 2*A*b^5 + 6*(B*a^3*b^2 - 3*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*tan(d*x + c)^2 + 3*(5*B*a^4*b - 14*A*a^3*b^2 -
12*B*a^2*b^3 + 2*A*a*b^4 - B*b^5)*tan(d*x + c))/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6 + (a^6*b^3 + 3*a^4*b^5
+ 3*a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*tan(d*x + c)^2 + 3*(a^8*b + 3*
a^6*b^3 + 3*a^4*b^5 + a^2*b^7)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (239) = 478\).

Time = 0.67 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.55 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (B a^{4} b - 4 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + 4 \, A a b^{4} + B b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, B a^{4} b^{3} \tan \left (d x + c\right )^{3} - 44 \, A a^{3} b^{4} \tan \left (d x + c\right )^{3} - 66 \, B a^{2} b^{5} \tan \left (d x + c\right )^{3} + 44 \, A a b^{6} \tan \left (d x + c\right )^{3} + 11 \, B b^{7} \tan \left (d x + c\right )^{3} + 39 \, B a^{5} b^{2} \tan \left (d x + c\right )^{2} - 150 \, A a^{4} b^{3} \tan \left (d x + c\right )^{2} - 210 \, B a^{3} b^{4} \tan \left (d x + c\right )^{2} + 120 \, A a^{2} b^{5} \tan \left (d x + c\right )^{2} + 15 \, B a b^{6} \tan \left (d x + c\right )^{2} + 6 \, A b^{7} \tan \left (d x + c\right )^{2} + 48 \, B a^{6} b \tan \left (d x + c\right ) - 174 \, A a^{5} b^{2} \tan \left (d x + c\right ) - 219 \, B a^{4} b^{3} \tan \left (d x + c\right ) + 96 \, A a^{3} b^{4} \tan \left (d x + c\right ) - 6 \, B a^{2} b^{5} \tan \left (d x + c\right ) + 6 \, A a b^{6} \tan \left (d x + c\right ) - 3 \, B b^{7} \tan \left (d x + c\right ) + 22 \, B a^{7} - 70 \, A a^{6} b - 69 \, B a^{5} b^{2} + 14 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - B a b^{6} - 2 \, A b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

[In]

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

[Out]

1/6*(6*(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)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^
6 + b^8) + 3*(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)/(a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(B*a^4*b - 4*A*a^3*b^2 - 6*B*a^2*b^3 + 4*A*a*b^4 + B*b^5)*log(abs(b*tan(d*x +
c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (11*B*a^4*b^3*tan(d*x + c)^3 - 44*A*a^3*b^4*tan(
d*x + c)^3 - 66*B*a^2*b^5*tan(d*x + c)^3 + 44*A*a*b^6*tan(d*x + c)^3 + 11*B*b^7*tan(d*x + c)^3 + 39*B*a^5*b^2*
tan(d*x + c)^2 - 150*A*a^4*b^3*tan(d*x + c)^2 - 210*B*a^3*b^4*tan(d*x + c)^2 + 120*A*a^2*b^5*tan(d*x + c)^2 +
15*B*a*b^6*tan(d*x + c)^2 + 6*A*b^7*tan(d*x + c)^2 + 48*B*a^6*b*tan(d*x + c) - 174*A*a^5*b^2*tan(d*x + c) - 21
9*B*a^4*b^3*tan(d*x + c) + 96*A*a^3*b^4*tan(d*x + c) - 6*B*a^2*b^5*tan(d*x + c) + 6*A*a*b^6*tan(d*x + c) - 3*B
*b^7*tan(d*x + c) + 22*B*a^7 - 70*A*a^6*b - 69*B*a^5*b^2 + 14*A*a^4*b^3 - 4*B*a^3*b^4 - 6*A*a^2*b^5 - B*a*b^6
- 2*A*b^7)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 8.43 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.79 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {-11\,B\,a^5+26\,A\,a^4\,b+14\,B\,a^3\,b^2+4\,A\,a^2\,b^3+B\,a\,b^4+2\,A\,b^5}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,B\,a^4\,b+14\,A\,a^3\,b^2+12\,B\,a^2\,b^3-2\,A\,a\,b^4+B\,b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (B\,a^3\,b^2-3\,A\,a^2\,b^3-3\,B\,a\,b^4+A\,b^5\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {B}{{\left (a^2+b^2\right )}^2}-\frac {4\,b\,\left (A\,a+2\,B\,b\right )}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^3\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )} \]

[In]

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

[Out]

(log(tan(c + d*x) - 1i)*(A + B*1i))/(2*d*(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)) - (log(a + b*tan(
c + d*x))*(B/(a^2 + b^2)^2 - (4*b*(A*a + 2*B*b))/(a^2 + b^2)^3 + (8*b^3*(A*a + B*b))/(a^2 + b^2)^4))/d - ((2*A
*b^5 - 11*B*a^5 + 4*A*a^2*b^3 + 14*B*a^3*b^2 + 26*A*a^4*b + B*a*b^4)/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) +
(tan(c + d*x)*(B*b^5 + 14*A*a^3*b^2 + 12*B*a^2*b^3 - 2*A*a*b^4 - 5*B*a^4*b))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^
4*b^2)) - (tan(c + d*x)^2*(A*b^5 - 3*A*a^2*b^3 + B*a^3*b^2 - 3*B*a*b^4))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/
(d*(a^3 + b^3*tan(c + d*x)^3 + 3*a*b^2*tan(c + d*x)^2 + 3*a^2*b*tan(c + d*x))) + (log(tan(c + d*x) + 1i)*(A*1i
+ B))/(2*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2))