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

Optimal. Leaf size=250 \[ \frac {a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a^2 A+2 a b B-A b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {a^3 A+3 a^2 b B-3 a A b^2-b^3 B}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {\left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (a^4 (-B)+4 a^3 A b+6 a^2 b^2 B-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]

[Out]

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

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Rubi [A]  time = 0.43, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3591, 3529, 3531, 3530} \[ \frac {a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {a^3 A+3 a^2 b B-3 a A b^2-b^3 B}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {a^2 A+2 a b B-A b^2}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {\left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3591

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

Rubi steps

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

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Mathematica [C]  time = 1.33, size = 248, normalized size = 0.99 \[ \frac {\frac {2 a (A b-a B)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac {3 \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {6 \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {6 \left (a^4 A+4 a^3 b B-6 a^2 A b^2-4 a b^3 B+A b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}+\frac {3 (A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^4}+\frac {3 (A-i B) \log (\tan (c+d x)+i)}{(a-i b)^4}}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.83, size = 838, normalized size = 3.35 \[ -\frac {12 \, B a^{6} b - 27 \, A a^{5} b^{2} - 30 \, B a^{4} b^{3} + 18 \, A a^{3} b^{4} + 2 \, B a^{2} b^{5} + A a b^{6} - {\left (2 \, B a^{5} b^{2} - 11 \, A a^{4} b^{3} - 30 \, B a^{3} b^{4} + 30 \, A a^{2} b^{5} + 12 \, B a b^{6} - 3 \, A b^{7} - 6 \, {\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 )} d x\right )} \tan \left (d x + c\right )^{3} + 6 \, {\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}\right )} d x - 3 \, {\left (2 \, B a^{6} b - 9 \, A a^{5} b^{2} - 24 \, B a^{4} b^{3} + 26 \, A a^{3} b^{4} + 16 \, B a^{2} b^{5} - 9 \, A a b^{6} - 2 \, B b^{7} - 6 \, {\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 )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\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} + {\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 )} \tan \left (d x + c\right )^{3} + 3 \, {\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 )} \tan \left (d x + c\right )^{2} + 3 \, {\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 )} \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 (2 \, B a^{7} - 6 \, A a^{6} b - 16 \, B a^{5} b^{2} + 23 \, A a^{4} b^{3} + 24 \, B a^{3} b^{4} - 16 \, A a^{2} b^{5} - 2 \, B a b^{6} - A b^{7} - 6 \, {\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 )} 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 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(12*B*a^6*b - 27*A*a^5*b^2 - 30*B*a^4*b^3 + 18*A*a^3*b^4 + 2*B*a^2*b^5 + A*a*b^6 - (2*B*a^5*b^2 - 11*A*a^
4*b^3 - 30*B*a^3*b^4 + 30*A*a^2*b^5 + 12*B*a*b^6 - 3*A*b^7 - 6*(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)*d*x)*tan(d*x + c)^3 + 6*(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)*d*x - 3*(2*B*
a^6*b - 9*A*a^5*b^2 - 24*B*a^4*b^3 + 26*A*a^3*b^4 + 16*B*a^2*b^5 - 9*A*a*b^6 - 2*B*b^7 - 6*(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)*d*x)*tan(d*x + c)^2 + 3*(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 + (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)*tan(d*x + c)^3 + 3*(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)*tan(d*x + c)^2 + 3*(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)*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*B*a^7 - 6*A*a^6*b - 16*B*a^5*b^2 + 23*A*a^4*b^3 + 24*B*a^3*b^4 - 16*A*a^2*b^5 - 2*B*a*b^6 - A*b^7 -
6*(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)*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*ta
n(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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giac [B]  time = 0.96, size = 638, normalized size = 2.55 \[ -\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (A a^{4} b + 4 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3} - 4 \, B a b^{4} + A b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac {11 \, A a^{4} b^{4} \tan \left (d x + c\right )^{3} + 44 \, B a^{3} b^{5} \tan \left (d x + c\right )^{3} - 66 \, A a^{2} b^{6} \tan \left (d x + c\right )^{3} - 44 \, B a b^{7} \tan \left (d x + c\right )^{3} + 11 \, A b^{8} \tan \left (d x + c\right )^{3} + 39 \, A a^{5} b^{3} \tan \left (d x + c\right )^{2} + 150 \, B a^{4} b^{4} \tan \left (d x + c\right )^{2} - 210 \, A a^{3} b^{5} \tan \left (d x + c\right )^{2} - 120 \, B a^{2} b^{6} \tan \left (d x + c\right )^{2} + 15 \, A a b^{7} \tan \left (d x + c\right )^{2} - 6 \, B b^{8} \tan \left (d x + c\right )^{2} + 48 \, A a^{6} b^{2} \tan \left (d x + c\right ) + 174 \, B a^{5} b^{3} \tan \left (d x + c\right ) - 219 \, A a^{4} b^{4} \tan \left (d x + c\right ) - 96 \, B a^{3} b^{5} \tan \left (d x + c\right ) - 6 \, A a^{2} b^{6} \tan \left (d x + c\right ) - 6 \, B a b^{7} \tan \left (d x + c\right ) - 3 \, A b^{8} \tan \left (d x + c\right ) - 2 \, B a^{8} + 22 \, A a^{7} b + 62 \, B a^{6} b^{2} - 69 \, A a^{5} b^{3} - 26 \, B a^{4} b^{4} - 4 \, A a^{3} b^{5} - 2 \, B a^{2} b^{6} - A a b^{7}}{{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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*(A*a^4*b + 4*B*a^3*b^2 - 6*A*a^2*b^3 - 4*B*a*b^4 + A*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*A*a^4*b^4*tan(d*x + c)^3 + 44*B*a^3*b^5*tan
(d*x + c)^3 - 66*A*a^2*b^6*tan(d*x + c)^3 - 44*B*a*b^7*tan(d*x + c)^3 + 11*A*b^8*tan(d*x + c)^3 + 39*A*a^5*b^3
*tan(d*x + c)^2 + 150*B*a^4*b^4*tan(d*x + c)^2 - 210*A*a^3*b^5*tan(d*x + c)^2 - 120*B*a^2*b^6*tan(d*x + c)^2 +
15*A*a*b^7*tan(d*x + c)^2 - 6*B*b^8*tan(d*x + c)^2 + 48*A*a^6*b^2*tan(d*x + c) + 174*B*a^5*b^3*tan(d*x + c) -
219*A*a^4*b^4*tan(d*x + c) - 96*B*a^3*b^5*tan(d*x + c) - 6*A*a^2*b^6*tan(d*x + c) - 6*B*a*b^7*tan(d*x + c) -
3*A*b^8*tan(d*x + c) - 2*B*a^8 + 22*A*a^7*b + 62*B*a^6*b^2 - 69*A*a^5*b^3 - 26*B*a^4*b^4 - 4*A*a^3*b^5 - 2*B*a
^2*b^6 - A*a*b^7)/((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*(b*tan(d*x + c) + a)^3))/d

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maple [B]  time = 0.26, size = 702, normalized size = 2.81 \[ \frac {a A}{3 d \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {a^{2} B}{3 d \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {a^{2} A}{2 d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {A \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {B a b}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3} A}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 a \,b^{2} A}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 a^{2} b B}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} B}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\ln \left (a +b \tan \left (d x +c \right )\right ) A \,a^{4}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {6 \ln \left (a +b \tan \left (d x +c \right )\right ) A \,a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {\ln \left (a +b \tan \left (d x +c \right )\right ) A \,b^{4}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {4 \ln \left (a +b \tan \left (d x +c \right )\right ) B \,a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {4 \ln \left (a +b \tan \left (d x +c \right )\right ) B a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{4}}{2 d \left (a^{2}+b^{2}\right )^{4}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,b^{4}}{2 d \left (a^{2}+b^{2}\right )^{4}}+\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B \,a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {2 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {4 A \arctan \left (\tan \left (d x +c \right )\right ) a^{3} b}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {4 A \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{3}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{4}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {6 B \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{4}}{d \left (a^{2}+b^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.96, size = 523, normalized size = 2.09 \[ -\frac {\frac {6 \, {\left (B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {6 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (A a^{4} + 4 \, B a^{3} b - 6 \, A a^{2} b^{2} - 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {2 \, B a^{6} - 11 \, A a^{5} b - 20 \, B a^{4} b^{2} + 14 \, A a^{3} b^{3} + 2 \, B a^{2} b^{4} + A a b^{5} - 6 \, {\left (A a^{3} b^{3} + 3 \, B a^{2} b^{4} - 3 \, A a b^{5} - B b^{6}\right )} \tan \left (d x + c\right )^{2} - 3 \, {\left (5 \, A a^{4} b^{2} + 14 \, B a^{3} b^{3} - 12 \, A a^{2} b^{4} - 2 \, B a b^{5} - A b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7} + {\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{3} + 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{2} + 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} + a^{2} b^{8}\right )} \tan \left (d x + c\right )}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*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*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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*(A*a^4 + 4*B*a^3*b - 6*A*a^2*b^2 - 4*B*a*b^3 + A*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) + (2*B*a^6 - 11*A*a^5*b - 20*B*a^4*b^2 + 14*A*a^3*b^3 + 2*B*
a^2*b^4 + A*a*b^5 - 6*(A*a^3*b^3 + 3*B*a^2*b^4 - 3*A*a*b^5 - B*b^6)*tan(d*x + c)^2 - 3*(5*A*a^4*b^2 + 14*B*a^3
*b^3 - 12*A*a^2*b^4 - 2*B*a*b^5 - A*b^6)*tan(d*x + c))/(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7 + (a^6*b^4 + 3
*a^4*b^6 + 3*a^2*b^8 + b^10)*tan(d*x + c)^3 + 3*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*tan(d*x + c)^2 + 3*(
a^8*b^2 + 3*a^6*b^4 + 3*a^4*b^6 + a^2*b^8)*tan(d*x + c)))/d

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mupad [B]  time = 6.80, size = 447, normalized size = 1.79 \[ -\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-5\,A\,a^4\,b-14\,B\,a^3\,b^2+12\,A\,a^2\,b^3+2\,B\,a\,b^4+A\,b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,B\,a^6-11\,A\,a^5\,b-20\,B\,a^4\,b^2+14\,A\,a^3\,b^3+2\,B\,a^2\,b^4+A\,a\,b^5}{6\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-A\,a^3\,b^2-3\,B\,a^2\,b^3+3\,A\,a\,b^4+B\,b^5\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{{\left (a^2+b^2\right )}^2}-\frac {4\,b\,\left (2\,A\,b-B\,a\right )}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^3\,\left (A\,b-B\,a\right )}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) - (log(a + b*tan(
c + d*x))*(A/(a^2 + b^2)^2 - (4*b*(2*A*b - B*a))/(a^2 + b^2)^3 + (8*b^3*(A*b - B*a))/(a^2 + b^2)^4))/d - ((tan
(c + d*x)*(A*b^5 + 12*A*a^2*b^3 - 14*B*a^3*b^2 - 5*A*a^4*b + 2*B*a*b^4))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2
)) + (2*B*a^6 + 14*A*a^3*b^3 + 2*B*a^2*b^4 - 20*B*a^4*b^2 + A*a*b^5 - 11*A*a^5*b)/(6*b*(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2)) + (tan(c + d*x)^2*(B*b^5 - A*a^3*b^2 - 3*B*a^2*b^3 + 3*A*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 + B*1i))/(2*d*(a^3*b*4i - a*b^3*4i + a^4 + b^4 - 6*a^2*b^2))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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