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

Optimal. Leaf size=247 \[ -\frac{A b-a B}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{a^2 (-B)+2 a A b+b^2 B}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

[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]))

________________________________________________________________________________________

Rubi [A]  time = 0.408388, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3531, 3530} \[ -\frac{A b-a B}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{a^2 (-B)+2 a A b+b^2 B}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[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 3529

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 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3530

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

Rubi steps

\begin{align*} \int \frac{A+B \tan (c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\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]  time = 6.23327, size = 327, normalized size = 1.32 \[ -\frac{(A b-a B) \left (\frac{6 b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{6 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{2 b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac{24 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac{3 i \log (-\tan (c+d x)+i)}{(a+i b)^4}-\frac{3 i \log (\tan (c+d x)+i)}{(a-i b)^4}\right )}{6 b d}-\frac{B \left (\frac{4 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{b}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{2 b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (c+d x)+i)}{(-b+i a)^3}-\frac{\log (\tan (c+d x)+i)}{(b+i a)^3}\right )}{2 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]))))/(6*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*Tan[
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 [B]  time = 0.053, size = 695, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.53686, size = 694, normalized size = 2.81 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 2.06781, size = 1777, normalized size = 7.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

________________________________________________________________________________________

Giac [B]  time = 1.2606, size = 851, normalized size = 3.45 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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