3.292 \(\int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=261 \[ -\frac{a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

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

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Rubi [A]  time = 0.482599, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3604, 3628, 3529, 3531, 3530} \[ -\frac{a^2 (A b-a B)}{3 b^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac{3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{\left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(Tan[c + d*x]^2*(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^2*(A*b - a*B))/(3*b^2*(a^2
+ b^2)*d*(a + b*Tan[c + d*x])^3) + (a*(2*A*b^3 - a*(a^2 + 3*b^2)*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 3604

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.)
+ (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((B*c - A*d)*(b*c - a*d)^2*(c + d*Tan[e + f*x])^(n + 1))/(f*d^2*(n +
1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x])^(n + 1)*Simp[B*(b*c - a*d)^2 + A*d*(a^2
*c - b^2*c + 2*a*b*d) + d*(B*(a^2*c - b^2*c + 2*a*b*d) + A*(2*a*b*c - a^2*d + b^2*d))*Tan[e + f*x] + b^2*B*(c^
2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^
2, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]

Rule 3628

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

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{\tan ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=-\frac{a^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{-a (A b-a B)+b (A b-a B) \tan (c+d x)+\left (a^2+b^2\right ) B \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{a^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{-b \left (a^2 A-A b^2+2 a b B\right )+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}{b \left (a^2+b^2\right )^2}\\ &=-\frac{a^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{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{-b \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )+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}{b \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^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{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^2 (A b-a B)}{3 b^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{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 = 6.26573, size = 411, normalized size = 1.57 \[ -\frac{B \tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{\frac{a B+2 A b}{3 b d (a+b \tan (c+d x))^3}+\frac{\frac{\left (6 A b^3-6 a b^2 B\right ) \left (-\frac{b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right )}{b}+6 b B \left (-\frac{2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}\right )}{3 b d}}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.059, size = 709, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^2*(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+t
an(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(ta
n(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^3-1/3/d*a^2/
b/(a^2+b^2)/(a+b*tan(d*x+c))^3*A+1/3/d*a^3/b^2/(a^2+b^2)/(a+b*tan(d*x+c))^3*B+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*ta
n(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/d*a/(a^2+b^2)^2*b/(a+b*tan(d*x+c))^2*A-1/2/d*a^4/b^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B-3/2/d*a^2/(
a^2+b^2)^2/(a+b*tan(d*x+c))^2*B

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Maxima [B]  time = 1.61439, size = 710, normalized size = 2.72 \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{B a^{7} + 2 \, A a^{6} b + 14 \, B a^{5} b^{2} - 20 \, A a^{4} b^{3} - 11 \, B a^{3} b^{4} + 2 \, A a^{2} b^{5} + 6 \,{\left (B a^{3} b^{4} - 3 \, A a^{2} b^{5} - 3 \, B a b^{6} + A b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (B a^{6} b + 8 \, B a^{4} b^{3} - 14 \, A a^{3} b^{4} - 9 \, B a^{2} b^{5} + 2 \, A a b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} +{\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(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) + (B*a^7 + 2*A*a^6*b + 14*B*a^5*b^2 - 20*A*a^4*b^3 - 11*B*a^
3*b^4 + 2*A*a^2*b^5 + 6*(B*a^3*b^4 - 3*A*a^2*b^5 - 3*B*a*b^6 + A*b^7)*tan(d*x + c)^2 + 3*(B*a^6*b + 8*B*a^4*b^
3 - 14*A*a^3*b^4 - 9*B*a^2*b^5 + 2*A*a*b^6)*tan(d*x + c))/(a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 + a^3*b^8 + (a^6*b^
5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^10)*tan(d*x + c)^2
+ 3*(a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*tan(d*x + c)))/d

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Fricas [B]  time = 2.12452, size = 1829, normalized size = 7.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*B*a^7 - 12*A*a^6*b - 30*B*a^5*b^2 + 30*A*a^4*b^3 + 11*B*a^3*b^4 - 2*A*a^2*b^5 + (B*a^6*b + 2*A*a^5*b^2
+ 18*B*a^4*b^3 - 30*A*a^3*b^4 - 27*B*a^2*b^5 + 12*A*a*b^6 - 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*(B*a^7
+ 2*A*a^6*b + 16*B*a^5*b^2 - 24*A*a^4*b^3 - 23*B*a^3*b^4 + 16*A*a^2*b^5 + 6*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*(2*A*a^7 + 9*B*a^6*b - 16*A*a^5*b^2 - 26*B*a^4*b^3 + 24*A*a^3*b^4 + 9*B*a^2*b^5 - 2*A*a*b^6 -
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)

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.4937, size = 853, normalized size = 3.27 \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^{5} \tan \left (d x + c\right )^{3} - 44 \, A a^{3} b^{6} \tan \left (d x + c\right )^{3} - 66 \, B a^{2} b^{7} \tan \left (d x + c\right )^{3} + 44 \, A a b^{8} \tan \left (d x + c\right )^{3} + 11 \, B b^{9} \tan \left (d x + c\right )^{3} + 39 \, B a^{5} b^{4} \tan \left (d x + c\right )^{2} - 150 \, A a^{4} b^{5} \tan \left (d x + c\right )^{2} - 210 \, B a^{3} b^{6} \tan \left (d x + c\right )^{2} + 120 \, A a^{2} b^{7} \tan \left (d x + c\right )^{2} + 15 \, B a b^{8} \tan \left (d x + c\right )^{2} + 6 \, A b^{9} \tan \left (d x + c\right )^{2} + 3 \, B a^{8} b \tan \left (d x + c\right ) + 60 \, B a^{6} b^{3} \tan \left (d x + c\right ) - 174 \, A a^{5} b^{4} \tan \left (d x + c\right ) - 201 \, B a^{4} b^{5} \tan \left (d x + c\right ) + 96 \, A a^{3} b^{6} \tan \left (d x + c\right ) + 6 \, B a^{2} b^{7} \tan \left (d x + c\right ) + 6 \, A a b^{8} \tan \left (d x + c\right ) + B a^{9} + 2 \, A a^{8} b + 26 \, B a^{7} b^{2} - 62 \, A a^{6} b^{3} - 63 \, B a^{5} b^{4} + 26 \, A a^{4} b^{5} + 2 \, A a^{2} b^{7}}{{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\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(tan(d*x+c)^2*(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^5*tan(d*x + c)^3 - 44*A*a^3*b^6*tan
(d*x + c)^3 - 66*B*a^2*b^7*tan(d*x + c)^3 + 44*A*a*b^8*tan(d*x + c)^3 + 11*B*b^9*tan(d*x + c)^3 + 39*B*a^5*b^4
*tan(d*x + c)^2 - 150*A*a^4*b^5*tan(d*x + c)^2 - 210*B*a^3*b^6*tan(d*x + c)^2 + 120*A*a^2*b^7*tan(d*x + c)^2 +
15*B*a*b^8*tan(d*x + c)^2 + 6*A*b^9*tan(d*x + c)^2 + 3*B*a^8*b*tan(d*x + c) + 60*B*a^6*b^3*tan(d*x + c) - 174
*A*a^5*b^4*tan(d*x + c) - 201*B*a^4*b^5*tan(d*x + c) + 96*A*a^3*b^6*tan(d*x + c) + 6*B*a^2*b^7*tan(d*x + c) +
6*A*a*b^8*tan(d*x + c) + B*a^9 + 2*A*a^8*b + 26*B*a^7*b^2 - 62*A*a^6*b^3 - 63*B*a^5*b^4 + 26*A*a^4*b^5 + 2*A*a
^2*b^7)/((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)^3))/d