3.291 \(\int \frac{\tan ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx\)

Optimal. Leaf size=298 \[ \frac{a (A b-a B) \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b+2 a^3 B+8 a b^2 B-5 A b^3\right )}{6 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a \left (5 a^2 A b^3+a^4 A b+7 a^3 b^2 B+2 a^5 B+17 a b^4 B-8 A b^5\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]

[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)*Tan[c + d*x]^2)
/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^2*(a^2*A*b - 5*A*b^3 + 2*a^3*B + 8*a*b^2*B))/(6*b^3*(a^2 + b^
2)^2*d*(a + b*Tan[c + d*x])^2) - (a*(a^4*A*b + 5*a^2*A*b^3 - 8*A*b^5 + 2*a^5*B + 7*a^3*b^2*B + 17*a*b^4*B))/(3
*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

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Rubi [A]  time = 0.572276, antiderivative size = 298, 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 = {3605, 3635, 3628, 3531, 3530} \[ \frac{a (A b-a B) \tan ^2(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b+2 a^3 B+8 a b^2 B-5 A b^3\right )}{6 b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{a \left (5 a^2 A b^3+a^4 A b+7 a^3 b^2 B+2 a^5 B+17 a b^4 B-8 A b^5\right )}{3 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\left (-6 a^2 A b^2+a^4 A+4 a^3 b B-4 a b^3 B+A b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}-\frac{x \left (4 a^3 A b+6 a^2 b^2 B+a^4 (-B)-4 a A b^3-b^4 B\right )}{\left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[(Tan[c + d*x]^3*(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)*Tan[c + d*x]^2)
/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) + (a^2*(a^2*A*b - 5*A*b^3 + 2*a^3*B + 8*a*b^2*B))/(6*b^3*(a^2 + b^
2)^2*d*(a + b*Tan[c + d*x])^2) - (a*(a^4*A*b + 5*a^2*A*b^3 - 8*A*b^5 + 2*a^5*B + 7*a^3*b^2*B + 17*a*b^4*B))/(3
*b^3*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3605

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((b*c - a*d)*(B*c - A*d)*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e
+ f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - Dist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m -
2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a*A*d*(b*d*(m - 1) - a*c*(n + 1)) + (b*B*c - (A*b + a*B)*d)*(b*c*(m - 1)
+ a*d*(n + 1)) - d*((a*A - b*B)*(b*c - a*d) + (A*b + a*B)*(a*c + b*d))*(n + 1)*Tan[e + f*x] - b*(d*(A*b*c + a
*B*c - a*A*d)*(m + n) - b*B*(c^2*(m - 1) - d^2*(n + 1)))*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] && GtQ[m, 1] && LtQ[n, -1] && (Inte
gerQ[m] || IntegersQ[2*m, 2*n])

Rule 3635

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e
_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(c^2*C - B*c*d + A*d^2)*
(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x
])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d +
a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &&
NeQ[b*c - a*d, 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 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 ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^4} \, dx &=\frac{a (A b-a B) \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{\int \frac{\tan (c+d x) \left (-2 a (A b-a B)+3 b (A b-a B) \tan (c+d x)+\left (a A b+2 a^2 B+3 b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=\frac{a (A b-a B) \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b-5 A b^3+2 a^3 B+8 a b^2 B\right )}{6 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac{\int \frac{a \left (a^2 A b-5 A b^3+2 a^3 B+8 a b^2 B\right )-3 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+\left (a^2+b^2\right ) \left (a A b+2 a^2 B+3 b^2 B\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=\frac{a (A b-a B) \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b-5 A b^3+2 a^3 B+8 a b^2 B\right )}{6 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a \left (a^4 A b+5 a^2 A b^3-8 A b^5+2 a^5 B+7 a^3 b^2 B+17 a b^4 B\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\int \frac{-3 b^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right )-3 b^2 \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}{3 b^2 \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) \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b-5 A b^3+2 a^3 B+8 a b^2 B\right )}{6 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a \left (a^4 A b+5 a^2 A b^3-8 A b^5+2 a^5 B+7 a^3 b^2 B+17 a b^4 B\right )}{3 b^3 \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) \tan ^2(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac{a^2 \left (a^2 A b-5 A b^3+2 a^3 B+8 a b^2 B\right )}{6 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac{a \left (a^4 A b+5 a^2 A b^3-8 A b^5+2 a^5 B+7 a^3 b^2 B+17 a b^4 B\right )}{3 b^3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.30685, size = 465, normalized size = 1.56 \[ -\frac{B \tan ^2(c+d x)}{b d (a+b \tan (c+d x))^3}-\frac{-\frac{(-2 a B-A b) \tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac{-\frac{2 a^2 B+a A b-2 b^2 B}{3 b d (a+b \tan (c+d x))^3}+\frac{\frac{\left (6 a A b^3+6 b^4 B\right ) \left (-\frac{b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac{a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac{b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}+\frac{4 a b (a-b) (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac{i \log (-\tan (c+d x)+i)}{2 (a+i b)^4}+\frac{i \log (\tan (c+d x)+i)}{2 (a-i b)^4}\right )}{b}-6 A b^2 \left (-\frac{2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac{\log (-\tan (c+d x)+i)}{2 (-b+i a)^3}+\frac{\log (\tan (c+d x)+i)}{2 (b+i a)^3}\right )}{3 b d}}{2 b}}{b} \]

Antiderivative was successfully verified.

[In]

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

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Maple [B]  time = 0.059, size = 780, normalized size = 2.6 \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)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x)

[Out]

3/d*a/(a^2+b^2)^3*b^2/(a+b*tan(d*x+c))*A-1/d*a^6/(a^2+b^2)^3/b^3/(a+b*tan(d*x+c))*B-3/d*a^4/(a^2+b^2)^3/b/(a+b
*tan(d*x+c))*B-6/d*a^2/(a^2+b^2)^3*b/(a+b*tan(d*x+c))*B-1/3/d*a^4/b^3/(a^2+b^2)/(a+b*tan(d*x+c))^3*B+1/d/(a^2+
b^2)^4*ln(a+b*tan(d*x+c))*A*a^4-1/d*a^3/(a^2+b^2)^3/(a+b*tan(d*x+c))*A+1/d/(a^2+b^2)^4*ln(a+b*tan(d*x+c))*A*b^
4-3/2/d*a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A-1/2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*a^4-1/2/d/(a^2+b^2)^4*ln(1
+tan(d*x+c)^2)*A*b^4+1/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*a^4+1/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*b^4+4/d/(a^
2+b^2)^4*ln(a+b*tan(d*x+c))*B*a^3*b+3/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*A*a^2*b^2-2/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*ln(a+b*tan(d*x+c))*B*a*b^3+1/d*a^5/b^3/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B-6/d/(a^2+b^2)^4*ln(a+b*tan(d*x+c
))*A*a^2*b^2-6/d/(a^2+b^2)^4*B*arctan(tan(d*x+c))*a^2*b^2+4/d/(a^2+b^2)^4*A*arctan(tan(d*x+c))*a*b^3-1/2/d*a^4
/b^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*A+2/d*a^3/b/(a^2+b^2)^2/(a+b*tan(d*x+c))^2*B+1/3/d*a^3/b^2/(a^2+b^2)/(a+b*
tan(d*x+c))^3*A

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Maxima [A]  time = 1.6824, size = 743, normalized size = 2.49 \begin{align*} \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^{8} + A a^{7} b + 4 \, B a^{6} b^{2} + 14 \, A a^{5} b^{3} + 26 \, B a^{4} b^{4} - 11 \, A a^{3} b^{5} + 6 \,{\left (B a^{6} b^{2} + 3 \, B a^{4} b^{4} + A a^{3} b^{5} + 6 \, B a^{2} b^{6} - 3 \, A a b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (2 \, B a^{7} b + A a^{6} b^{2} + 6 \, B a^{5} b^{3} + 8 \, A a^{4} b^{4} + 20 \, B a^{3} b^{5} - 9 \, A a^{2} b^{6}\right )} \tan \left (d x + c\right )}{a^{9} b^{3} + 3 \, a^{7} b^{5} + 3 \, a^{5} b^{7} + a^{3} b^{9} +{\left (a^{6} b^{6} + 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} + b^{12}\right )} \tan \left (d x + c\right )^{3} + 3 \,{\left (a^{7} b^{5} + 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} + a b^{11}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{8} b^{4} + 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} + a^{2} b^{10}\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)^3*(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^8 + A*a^7*b + 4*B*a^6*b^2 + 14*A*a^5*b^3 + 26*B*a^4*
b^4 - 11*A*a^3*b^5 + 6*(B*a^6*b^2 + 3*B*a^4*b^4 + A*a^3*b^5 + 6*B*a^2*b^6 - 3*A*a*b^7)*tan(d*x + c)^2 + 3*(2*B
*a^7*b + A*a^6*b^2 + 6*B*a^5*b^3 + 8*A*a^4*b^4 + 20*B*a^3*b^5 - 9*A*a^2*b^6)*tan(d*x + c))/(a^9*b^3 + 3*a^7*b^
5 + 3*a^5*b^7 + a^3*b^9 + (a^6*b^6 + 3*a^4*b^8 + 3*a^2*b^10 + b^12)*tan(d*x + c)^3 + 3*(a^7*b^5 + 3*a^5*b^7 +
3*a^3*b^9 + a*b^11)*tan(d*x + c)^2 + 3*(a^8*b^4 + 3*a^6*b^6 + 3*a^4*b^8 + a^2*b^10)*tan(d*x + c)))/d

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Fricas [B]  time = 2.10378, size = 1777, normalized size = 5.96 \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)^3*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*A*a^7 + 18*B*a^6*b - 30*A*a^5*b^2 - 26*B*a^4*b^3 + 11*A*a^3*b^4 + (2*B*a^7 + A*a^6*b + 6*B*a^5*b^2 + 18
*A*a^4*b^3 + 48*B*a^3*b^4 - 27*A*a^2*b^5 + 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*(A*a^7 - 2*B*a^6*b + 16
*A*a^5*b^2 + 30*B*a^4*b^3 - 23*A*a^3*b^4 - 12*B*a^2*b^5 + 6*A*a*b^6 + 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 -
9*A*a^6*b - 22*B*a^5*b^2 + 26*A*a^4*b^3 + 20*B*a^3*b^4 - 9*A*a^2*b^5 - 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*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)**3*(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 = 1.79057, size = 905, normalized size = 3.04 \begin{align*} \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^{6} \tan \left (d x + c\right )^{3} + 44 \, B a^{3} b^{7} \tan \left (d x + c\right )^{3} - 66 \, A a^{2} b^{8} \tan \left (d x + c\right )^{3} - 44 \, B a b^{9} \tan \left (d x + c\right )^{3} + 11 \, A b^{10} \tan \left (d x + c\right )^{3} + 6 \, B a^{8} b^{2} \tan \left (d x + c\right )^{2} + 24 \, B a^{6} b^{4} \tan \left (d x + c\right )^{2} + 39 \, A a^{5} b^{5} \tan \left (d x + c\right )^{2} + 186 \, B a^{4} b^{6} \tan \left (d x + c\right )^{2} - 210 \, A a^{3} b^{7} \tan \left (d x + c\right )^{2} - 96 \, B a^{2} b^{8} \tan \left (d x + c\right )^{2} + 15 \, A a b^{9} \tan \left (d x + c\right )^{2} + 6 \, B a^{9} b \tan \left (d x + c\right ) + 3 \, A a^{8} b^{2} \tan \left (d x + c\right ) + 24 \, B a^{7} b^{3} \tan \left (d x + c\right ) + 60 \, A a^{6} b^{4} \tan \left (d x + c\right ) + 210 \, B a^{5} b^{5} \tan \left (d x + c\right ) - 201 \, A a^{4} b^{6} \tan \left (d x + c\right ) - 72 \, B a^{3} b^{7} \tan \left (d x + c\right ) + 6 \, A a^{2} b^{8} \tan \left (d x + c\right ) + 2 \, B a^{10} + A a^{9} b + 6 \, B a^{8} b^{2} + 26 \, A a^{7} b^{3} + 74 \, B a^{6} b^{4} - 63 \, A a^{5} b^{5} - 18 \, B a^{4} b^{6}}{{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\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)^3*(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^6*tan(d*x + c)^3 + 44*B*a^3*b^7*tan(
d*x + c)^3 - 66*A*a^2*b^8*tan(d*x + c)^3 - 44*B*a*b^9*tan(d*x + c)^3 + 11*A*b^10*tan(d*x + c)^3 + 6*B*a^8*b^2*
tan(d*x + c)^2 + 24*B*a^6*b^4*tan(d*x + c)^2 + 39*A*a^5*b^5*tan(d*x + c)^2 + 186*B*a^4*b^6*tan(d*x + c)^2 - 21
0*A*a^3*b^7*tan(d*x + c)^2 - 96*B*a^2*b^8*tan(d*x + c)^2 + 15*A*a*b^9*tan(d*x + c)^2 + 6*B*a^9*b*tan(d*x + c)
+ 3*A*a^8*b^2*tan(d*x + c) + 24*B*a^7*b^3*tan(d*x + c) + 60*A*a^6*b^4*tan(d*x + c) + 210*B*a^5*b^5*tan(d*x + c
) - 201*A*a^4*b^6*tan(d*x + c) - 72*B*a^3*b^7*tan(d*x + c) + 6*A*a^2*b^8*tan(d*x + c) + 2*B*a^10 + A*a^9*b + 6
*B*a^8*b^2 + 26*A*a^7*b^3 + 74*B*a^6*b^4 - 63*A*a^5*b^5 - 18*B*a^4*b^6)/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*
a^2*b^9 + b^11)*(b*tan(d*x + c) + a)^3))/d