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3.67 \(\int \frac{(c+d \tan (e+f x))^3 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{a+b \tan (e+f x)} \, dx\)

Optimal. Leaf size=363 \[ -\frac{\log (\cos (e+f x)) \left (A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )+b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f \left (a^2+b^2\right )}-\frac{x \left (a \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )-b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{a^2+b^2}+\frac{(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \log (a+b \tan (e+f x))}{b^4 f \left (a^2+b^2\right )}+\frac{d \tan (e+f x) \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b^3 f}+\frac{(-a C d+b B d+b c C) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f} \]

[Out]

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

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Rubi [A]  time = 1.51166, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3647, 3637, 3626, 3617, 31, 3475} \[ -\frac{\log (\cos (e+f x)) \left (A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )+b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f \left (a^2+b^2\right )}-\frac{x \left (a \left (-A \left (c^3-3 c d^2\right )+3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )-b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )}{a^2+b^2}+\frac{(b c-a d)^3 \left (A b^2-a (b B-a C)\right ) \log (a+b \tan (e+f x))}{b^4 f \left (a^2+b^2\right )}+\frac{d \tan (e+f x) \left ((b c-a d) (-a C d+b B d+b c C)+b^2 d (d (A-C)+B c)\right )}{b^3 f}+\frac{(-a C d+b B d+b c C) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f} \]

Antiderivative was successfully verified.

[In]

Int[((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]

[Out]

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

Rule 3647

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

Rule 3637

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*Tan[e + f*x]*(c + d*Tan[e + f*x])
^(n + 1))/(d*f*(n + 2)), x] - Dist[1/(d*(n + 2)), Int[(c + d*Tan[e + f*x])^n*Simp[b*c*C - a*A*d*(n + 2) - (A*b
 + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C*d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] &&  !LtQ[n, -1]

Rule 3626

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

Rule 3617

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[
A/(b*f), Subst[Int[(a + x)^m, x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && EqQ[A, C]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{(c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx &=\frac{C (c+d \tan (e+f x))^3}{3 b f}+\frac{\int \frac{(c+d \tan (e+f x))^2 \left (3 (A b c-a C d)+3 b (B c+(A-C) d) \tan (e+f x)+3 (b c C+b B d-a C d) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{3 b}\\ &=\frac{(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}+\frac{\int \frac{(c+d \tan (e+f x)) \left (6 \left (A b^2 c^2+a d (a C d-b (2 c C+B d))\right )+6 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x)+6 \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{6 b^2}\\ &=\frac{d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac{(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}-\frac{\int \frac{-6 \left (A b^2 \left (b c^3-a d^3\right )-a d \left (a^2 C d^2-a b d (3 c C+B d)+b^2 \left (3 c^2 C+3 B c d-C d^2\right )\right )\right )-6 b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) \tan (e+f x)+6 \left (a^3 C d^3-a^2 b d^2 (3 c C+B d)+a b^2 d \left (3 c^2 C+3 B c d+(A-C) d^2\right )-b^3 \left (c^3 C+3 B c^2 d+3 c (A-C) d^2-B d^3\right )\right ) \tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{6 b^3}\\ &=-\frac{\left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}+\frac{d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac{(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}+\frac{\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^3\right ) \int \frac{1+\tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b^3 \left (a^2+b^2\right )}+\frac{\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \int \tan (e+f x) \, dx}{a^2+b^2}\\ &=-\frac{\left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}-\frac{\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac{d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac{(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}+\frac{\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (e+f x)\right )}{b^4 \left (a^2+b^2\right ) f}\\ &=-\frac{\left (a \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3-A \left (c^3-3 c d^2\right )\right )-b \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right ) x}{a^2+b^2}-\frac{\left (b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )+A \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac{\left (A b^2-a (b B-a C)\right ) (b c-a d)^3 \log (a+b \tan (e+f x))}{b^4 \left (a^2+b^2\right ) f}+\frac{d \left (b^2 d (B c+(A-C) d)+(b c-a d) (b c C+b B d-a C d)\right ) \tan (e+f x)}{b^3 f}+\frac{(b c C+b B d-a C d) (c+d \tan (e+f x))^2}{2 b^2 f}+\frac{C (c+d \tan (e+f x))^3}{3 b f}\\ \end{align*}

Mathematica [C]  time = 4.73759, size = 255, normalized size = 0.7 \[ \frac{\frac{6 (b c-a d)^3 \left (a (a C-b B)+A b^2\right ) \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right )}+\frac{3 b^2 (c+i d)^3 (-i A+B+i C) \log (-\tan (e+f x)+i)}{a+i b}-\frac{3 b^2 (d+i c)^3 (A-i B-C) \log (\tan (e+f x)+i)}{a-i b}+3 (-a C d+b B d+b c C) (c+d \tan (e+f x))^2+\frac{6 d (b c-a d) \tan (e+f x) (-a C d+b B d+b c C)}{b}+6 b d^2 \tan (e+f x) (d (A-C)+B c)+2 b C (c+d \tan (e+f x))^3}{6 b^2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[((c + d*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]

[Out]

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

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Maple [B]  time = 0.051, size = 1304, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x)

[Out]

3/f/(a^2+b^2)*C*arctan(tan(f*x+e))*a*c*d^2-3/f/(a^2+b^2)*C*arctan(tan(f*x+e))*b*c^2*d-3/2/f/(a^2+b^2)*ln(1+tan
(f*x+e)^2)*C*a*c^2*d-1/f/b^4/(a^2+b^2)*ln(a+b*tan(f*x+e))*C*a^5*d^3+1/f/b/(a^2+b^2)*ln(a+b*tan(f*x+e))*C*a^2*c
^3-1/f/b^2/(a^2+b^2)*ln(a+b*tan(f*x+e))*A*a^3*d^3+1/f/b^3/(a^2+b^2)*ln(a+b*tan(f*x+e))*B*a^4*d^3-3/f/(a^2+b^2)
*A*arctan(tan(f*x+e))*a*c*d^2-3/f*d^2/b^2*C*a*c*tan(f*x+e)+3/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*a*c^2*d+3/2/f/
(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*b*c*d^2+3/f/b^3/(a^2+b^2)*ln(a+b*tan(f*x+e))*C*a^4*c*d^2+3/f/b/(a^2+b^2)*ln(a+b
*tan(f*x+e))*B*a^2*c^2*d+3/f/b/(a^2+b^2)*ln(a+b*tan(f*x+e))*A*a^2*c*d^2-3/f/b^2/(a^2+b^2)*ln(a+b*tan(f*x+e))*C
*a^3*c^2*d-3/f/b^2/(a^2+b^2)*ln(a+b*tan(f*x+e))*B*a^3*c*d^2-3/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*C*b*c*d^2+1/3/f
*d^3/b*C*tan(f*x+e)^3-3/f/(a^2+b^2)*B*arctan(tan(f*x+e))*a*c^2*d+3/f/(a^2+b^2)*A*arctan(tan(f*x+e))*b*c^2*d-3/
f/(a^2+b^2)*ln(a+b*tan(f*x+e))*A*a*c^2*d-3/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*a*c*d^2+3/2/f/(a^2+b^2)*ln(1+tan
(f*x+e)^2)*B*b*c^2*d-3/f/(a^2+b^2)*B*arctan(tan(f*x+e))*b*c*d^2+3/f*d/b*C*c^2*tan(f*x+e)-1/f/(a^2+b^2)*ln(a+b*
tan(f*x+e))*B*a*c^3-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*a*d^3-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*b*c^3+1/2/
f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*a*c^3-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*b*d^3+1/2/f/(a^2+b^2)*ln(1+tan(f*x
+e)^2)*C*a*d^3+1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*C*b*c^3+1/f/(a^2+b^2)*A*arctan(tan(f*x+e))*a*c^3-1/f/(a^2+b^
2)*A*arctan(tan(f*x+e))*b*d^3+1/f/(a^2+b^2)*B*arctan(tan(f*x+e))*a*d^3+1/f/(a^2+b^2)*B*arctan(tan(f*x+e))*b*c^
3-1/f/(a^2+b^2)*C*arctan(tan(f*x+e))*a*c^3+1/f/(a^2+b^2)*C*arctan(tan(f*x+e))*b*d^3+1/f*b/(a^2+b^2)*ln(a+b*tan
(f*x+e))*A*c^3-1/2/f*d^3/b^2*C*tan(f*x+e)^2*a+3/2/f*d^2/b*C*tan(f*x+e)^2*c-1/f*d^3/b^2*B*a*tan(f*x+e)+3/f*d^2/
b*B*c*tan(f*x+e)+1/f*d^3/b^3*a^2*C*tan(f*x+e)+1/2/f*d^3/b*B*tan(f*x+e)^2+1/f*d^3/b*A*tan(f*x+e)-1/f*d^3/b*C*ta
n(f*x+e)

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Maxima [A]  time = 1.53438, size = 589, normalized size = 1.62 \begin{align*} \frac{\frac{6 \,{\left ({\left ({\left (A - C\right )} a + B b\right )} c^{3} - 3 \,{\left (B a -{\left (A - C\right )} b\right )} c^{2} d - 3 \,{\left ({\left (A - C\right )} a + B b\right )} c d^{2} +{\left (B a -{\left (A - C\right )} b\right )} d^{3}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} + \frac{6 \,{\left ({\left (C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c^{3} - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} + A a b^{4}\right )} c^{2} d + 3 \,{\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3}\right )} c d^{2} -{\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} b^{4} + b^{6}} + \frac{3 \,{\left ({\left (B a -{\left (A - C\right )} b\right )} c^{3} + 3 \,{\left ({\left (A - C\right )} a + B b\right )} c^{2} d - 3 \,{\left (B a -{\left (A - C\right )} b\right )} c d^{2} -{\left ({\left (A - C\right )} a + B b\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \, C b^{2} d^{3} \tan \left (f x + e\right )^{3} + 3 \,{\left (3 \, C b^{2} c d^{2} -{\left (C a b - B b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 6 \,{\left (3 \, C b^{2} c^{2} d - 3 \,{\left (C a b - B b^{2}\right )} c d^{2} +{\left (C a^{2} - B a b +{\left (A - C\right )} b^{2}\right )} d^{3}\right )} \tan \left (f x + e\right )}{b^{3}}}{6 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="maxima")

[Out]

1/6*(6*(((A - C)*a + B*b)*c^3 - 3*(B*a - (A - C)*b)*c^2*d - 3*((A - C)*a + B*b)*c*d^2 + (B*a - (A - C)*b)*d^3)
*(f*x + e)/(a^2 + b^2) + 6*((C*a^2*b^3 - B*a*b^4 + A*b^5)*c^3 - 3*(C*a^3*b^2 - B*a^2*b^3 + A*a*b^4)*c^2*d + 3*
(C*a^4*b - B*a^3*b^2 + A*a^2*b^3)*c*d^2 - (C*a^5 - B*a^4*b + A*a^3*b^2)*d^3)*log(b*tan(f*x + e) + a)/(a^2*b^4
+ b^6) + 3*((B*a - (A - C)*b)*c^3 + 3*((A - C)*a + B*b)*c^2*d - 3*(B*a - (A - C)*b)*c*d^2 - ((A - C)*a + B*b)*
d^3)*log(tan(f*x + e)^2 + 1)/(a^2 + b^2) + (2*C*b^2*d^3*tan(f*x + e)^3 + 3*(3*C*b^2*c*d^2 - (C*a*b - B*b^2)*d^
3)*tan(f*x + e)^2 + 6*(3*C*b^2*c^2*d - 3*(C*a*b - B*b^2)*c*d^2 + (C*a^2 - B*a*b + (A - C)*b^2)*d^3)*tan(f*x +
e))/b^3)/f

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Fricas [A]  time = 5.57819, size = 1269, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (C a^{2} b^{3} + C b^{5}\right )} d^{3} \tan \left (f x + e\right )^{3} + 6 \,{\left ({\left ({\left (A - C\right )} a b^{4} + B b^{5}\right )} c^{3} - 3 \,{\left (B a b^{4} -{\left (A - C\right )} b^{5}\right )} c^{2} d - 3 \,{\left ({\left (A - C\right )} a b^{4} + B b^{5}\right )} c d^{2} +{\left (B a b^{4} -{\left (A - C\right )} b^{5}\right )} d^{3}\right )} f x + 3 \,{\left (3 \,{\left (C a^{2} b^{3} + C b^{5}\right )} c d^{2} -{\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 3 \,{\left ({\left (C a^{2} b^{3} - B a b^{4} + A b^{5}\right )} c^{3} - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} + A a b^{4}\right )} c^{2} d + 3 \,{\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3}\right )} c d^{2} -{\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} d^{3}\right )} \log \left (\frac{b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \,{\left ({\left (C a^{2} b^{3} + C b^{5}\right )} c^{3} - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} c^{2} d + 3 \,{\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} +{\left (A - C\right )} b^{5}\right )} c d^{2} -{\left (C a^{5} - B a^{4} b + A a^{3} b^{2} +{\left (A - C\right )} a b^{4} + B b^{5}\right )} d^{3}\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \,{\left (3 \,{\left (C a^{2} b^{3} + C b^{5}\right )} c^{2} d - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} + C a b^{4} - B b^{5}\right )} c d^{2} +{\left (C a^{4} b - B a^{3} b^{2} + A a^{2} b^{3} - B a b^{4} +{\left (A - C\right )} b^{5}\right )} d^{3}\right )} \tan \left (f x + e\right )}{6 \,{\left (a^{2} b^{4} + b^{6}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="fricas")

[Out]

1/6*(2*(C*a^2*b^3 + C*b^5)*d^3*tan(f*x + e)^3 + 6*(((A - C)*a*b^4 + B*b^5)*c^3 - 3*(B*a*b^4 - (A - C)*b^5)*c^2
*d - 3*((A - C)*a*b^4 + B*b^5)*c*d^2 + (B*a*b^4 - (A - C)*b^5)*d^3)*f*x + 3*(3*(C*a^2*b^3 + C*b^5)*c*d^2 - (C*
a^3*b^2 - B*a^2*b^3 + C*a*b^4 - B*b^5)*d^3)*tan(f*x + e)^2 + 3*((C*a^2*b^3 - B*a*b^4 + A*b^5)*c^3 - 3*(C*a^3*b
^2 - B*a^2*b^3 + A*a*b^4)*c^2*d + 3*(C*a^4*b - B*a^3*b^2 + A*a^2*b^3)*c*d^2 - (C*a^5 - B*a^4*b + A*a^3*b^2)*d^
3)*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)/(tan(f*x + e)^2 + 1)) - 3*((C*a^2*b^3 + C*b^5)*c^3 - 3*
(C*a^3*b^2 - B*a^2*b^3 + C*a*b^4 - B*b^5)*c^2*d + 3*(C*a^4*b - B*a^3*b^2 + A*a^2*b^3 - B*a*b^4 + (A - C)*b^5)*
c*d^2 - (C*a^5 - B*a^4*b + A*a^3*b^2 + (A - C)*a*b^4 + B*b^5)*d^3)*log(1/(tan(f*x + e)^2 + 1)) + 6*(3*(C*a^2*b
^3 + C*b^5)*c^2*d - 3*(C*a^3*b^2 - B*a^2*b^3 + C*a*b^4 - B*b^5)*c*d^2 + (C*a^4*b - B*a^3*b^2 + A*a^2*b^3 - B*a
*b^4 + (A - C)*b^5)*d^3)*tan(f*x + e))/((a^2*b^4 + b^6)*f)

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Sympy [A]  time = 46.9782, size = 7096, normalized size = 19.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e)),x)

[Out]

Piecewise((zoo*x*(c + d*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/tan(e), Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), (-3*I
*A*c**3*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*A*c**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*
A*c**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*c**2*d*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*A*c
**2*d*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*A*c**2*d/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*A*c*d**2*f*x*tan(
e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*c*d**2*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*c*d**2*log(tan
(e + f*x)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*A*c*d**2*log(tan(e + f*x)**2 + 1)/(-6*b*f
*tan(e + f*x) + 6*I*b*f) + 9*I*A*c*d**2/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*A*d**3*f*x*tan(e + f*x)/(-6*b*f*ta
n(e + f*x) + 6*I*b*f) - 9*I*A*d**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*A*d**3*log(tan(e + f*x)**2 + 1)*t
an(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*A*d**3*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f
) - 6*A*d**3*tan(e + f*x)**2/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*d**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*B*
c**3*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 3*I*B*c**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 3*B*c
**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*B*c**2*d*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*B*c**2
*d*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*B*c**2*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x)
 + 6*I*b*f) + 9*I*B*c**2*d*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*B*c**2*d/(-6*b*f*tan
(e + f*x) + 6*I*b*f) + 27*B*c*d**2*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 27*I*B*c*d**2*f*x/(-6*b*
f*tan(e + f*x) + 6*I*b*f) - 9*I*B*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f)
 - 9*B*c*d**2*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 18*B*c*d**2*tan(e + f*x)**2/(-6*b*f*t
an(e + f*x) + 6*I*b*f) - 27*B*c*d**2/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*B*d**3*f*x*tan(e + f*x)/(-6*b*f*tan
(e + f*x) + 6*I*b*f) + 9*B*d**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 6*B*d**3*log(tan(e + f*x)**2 + 1)*tan(e
+ f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 6*I*B*d**3*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f) -
 3*B*d**3*tan(e + f*x)**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*B*d**3*tan(e + f*x)**2/(-6*b*f*tan(e + f*x) +
6*I*b*f) - 9*I*B*d**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*C*c**3*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I
*b*f) - 3*C*c**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 3*C*c**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*b*f*
tan(e + f*x) + 6*I*b*f) + 3*I*C*c**3*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 3*I*C*c**3/(-6
*b*f*tan(e + f*x) + 6*I*b*f) + 27*C*c**2*d*f*x*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 27*I*C*c**2*d*f*
x/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*C*c**2*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x) +
6*I*b*f) - 9*C*c**2*d*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 18*C*c**2*d*tan(e + f*x)**2/(
-6*b*f*tan(e + f*x) + 6*I*b*f) - 27*C*c**2*d/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 27*I*C*c*d**2*f*x*tan(e + f*x)/
(-6*b*f*tan(e + f*x) + 6*I*b*f) + 27*C*c*d**2*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 18*C*c*d**2*log(tan(e + f*
x)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 18*I*C*c*d**2*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e
 + f*x) + 6*I*b*f) - 9*C*c*d**2*tan(e + f*x)**3/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*C*c*d**2*tan(e + f*x)**2
/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 27*I*C*c*d**2/(-6*b*f*tan(e + f*x) + 6*I*b*f) - 15*C*d**3*f*x*tan(e + f*x)/
(-6*b*f*tan(e + f*x) + 6*I*b*f) + 15*I*C*d**3*f*x/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 6*I*C*d**3*log(tan(e + f*x
)**2 + 1)*tan(e + f*x)/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 6*C*d**3*log(tan(e + f*x)**2 + 1)/(-6*b*f*tan(e + f*x
) + 6*I*b*f) - 2*C*d**3*tan(e + f*x)**4/(-6*b*f*tan(e + f*x) + 6*I*b*f) - I*C*d**3*tan(e + f*x)**3/(-6*b*f*tan
(e + f*x) + 6*I*b*f) + 9*C*d**3*tan(e + f*x)**2/(-6*b*f*tan(e + f*x) + 6*I*b*f) + 15*C*d**3/(-6*b*f*tan(e + f*
x) + 6*I*b*f), Eq(a, -I*b)), (-3*I*A*c**3*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 3*A*c**3*f*x/(6*b*
f*tan(e + f*x) + 6*I*b*f) - 3*I*A*c**3/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*A*c**2*d*f*x*tan(e + f*x)/(6*b*f*tan
(e + f*x) + 6*I*b*f) + 9*I*A*c**2*d*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*c**2*d/(6*b*f*tan(e + f*x) + 6*I*
b*f) - 9*I*A*c*d**2*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*A*c*d**2*f*x/(6*b*f*tan(e + f*x) + 6*I
*b*f) + 9*A*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*A*c*d**2*log(tan
(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*A*c*d**2/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*A*d**3*f*x*
tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*A*d**3*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*A*d**3*log(t
an(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 3*A*d**3*log(tan(e + f*x)**2 + 1)/(6*b*f*tan
(e + f*x) + 6*I*b*f) + 6*A*d**3*tan(e + f*x)**2/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*A*d**3/(6*b*f*tan(e + f*x)
+ 6*I*b*f) + 3*B*c**3*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 3*I*B*c**3*f*x/(6*b*f*tan(e + f*x) + 6
*I*b*f) - 3*B*c**3/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*B*c**2*d*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*
f) + 9*B*c**2*d*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*B*c**2*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*t
an(e + f*x) + 6*I*b*f) + 9*I*B*c**2*d*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*B*c**2*d/(
6*b*f*tan(e + f*x) + 6*I*b*f) - 27*B*c*d**2*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) - 27*I*B*c*d**2*f*
x/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*B*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*
I*b*f) + 9*B*c*d**2*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 18*B*c*d**2*tan(e + f*x)**2/(6*b
*f*tan(e + f*x) + 6*I*b*f) + 27*B*c*d**2/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9*I*B*d**3*f*x*tan(e + f*x)/(6*b*f*t
an(e + f*x) + 6*I*b*f) - 9*B*d**3*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) - 6*B*d**3*log(tan(e + f*x)**2 + 1)*tan(e
 + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) - 6*I*B*d**3*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) +
3*B*d**3*tan(e + f*x)**3/(6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*B*d**3*tan(e + f*x)**2/(6*b*f*tan(e + f*x) + 6*I
*b*f) - 9*I*B*d**3/(6*b*f*tan(e + f*x) + 6*I*b*f) - 3*I*C*c**3*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f)
 + 3*C*c**3*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) + 3*C*c**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*tan(e +
 f*x) + 6*I*b*f) + 3*I*C*c**3*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 3*I*C*c**3/(6*b*f*tan(
e + f*x) + 6*I*b*f) - 27*C*c**2*d*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) - 27*I*C*c**2*d*f*x/(6*b*f*t
an(e + f*x) + 6*I*b*f) - 9*I*C*c**2*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 9
*C*c**2*d*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 18*C*c**2*d*tan(e + f*x)**2/(6*b*f*tan(e +
 f*x) + 6*I*b*f) + 27*C*c**2*d/(6*b*f*tan(e + f*x) + 6*I*b*f) + 27*I*C*c*d**2*f*x*tan(e + f*x)/(6*b*f*tan(e +
f*x) + 6*I*b*f) - 27*C*c*d**2*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) - 18*C*c*d**2*log(tan(e + f*x)**2 + 1)*tan(e
+ f*x)/(6*b*f*tan(e + f*x) + 6*I*b*f) - 18*I*C*c*d**2*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f)
+ 9*C*c*d**2*tan(e + f*x)**3/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*I*C*c*d**2*tan(e + f*x)**2/(6*b*f*tan(e + f*x)
 + 6*I*b*f) - 27*I*C*c*d**2/(6*b*f*tan(e + f*x) + 6*I*b*f) + 15*C*d**3*f*x*tan(e + f*x)/(6*b*f*tan(e + f*x) +
6*I*b*f) + 15*I*C*d**3*f*x/(6*b*f*tan(e + f*x) + 6*I*b*f) + 6*I*C*d**3*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(
6*b*f*tan(e + f*x) + 6*I*b*f) - 6*C*d**3*log(tan(e + f*x)**2 + 1)/(6*b*f*tan(e + f*x) + 6*I*b*f) + 2*C*d**3*ta
n(e + f*x)**4/(6*b*f*tan(e + f*x) + 6*I*b*f) - I*C*d**3*tan(e + f*x)**3/(6*b*f*tan(e + f*x) + 6*I*b*f) - 9*C*d
**3*tan(e + f*x)**2/(6*b*f*tan(e + f*x) + 6*I*b*f) - 15*C*d**3/(6*b*f*tan(e + f*x) + 6*I*b*f), Eq(a, I*b)), ((
A*c**3*x + 3*A*c**2*d*log(tan(e + f*x)**2 + 1)/(2*f) - 3*A*c*d**2*x + 3*A*c*d**2*tan(e + f*x)/f - A*d**3*log(t
an(e + f*x)**2 + 1)/(2*f) + A*d**3*tan(e + f*x)**2/(2*f) + B*c**3*log(tan(e + f*x)**2 + 1)/(2*f) - 3*B*c**2*d*
x + 3*B*c**2*d*tan(e + f*x)/f - 3*B*c*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + 3*B*c*d**2*tan(e + f*x)**2/(2*f) +
 B*d**3*x + B*d**3*tan(e + f*x)**3/(3*f) - B*d**3*tan(e + f*x)/f - C*c**3*x + C*c**3*tan(e + f*x)/f - 3*C*c**2
*d*log(tan(e + f*x)**2 + 1)/(2*f) + 3*C*c**2*d*tan(e + f*x)**2/(2*f) + 3*C*c*d**2*x + C*c*d**2*tan(e + f*x)**3
/f - 3*C*c*d**2*tan(e + f*x)/f + C*d**3*log(tan(e + f*x)**2 + 1)/(2*f) + C*d**3*tan(e + f*x)**4/(4*f) - C*d**3
*tan(e + f*x)**2/(2*f))/a, Eq(b, 0)), (x*(c + d*tan(e))**3*(A + B*tan(e) + C*tan(e)**2)/(a + b*tan(e)), Eq(f,
0)), (-6*A*a**3*b**2*d**3*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 18*A*a**2*b**3*c*d**2*log(a/b +
 tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 6*A*a**2*b**3*d**3*tan(e + f*x)/(6*a**2*b**4*f + 6*b**6*f) + 6*A*a
*b**4*c**3*f*x/(6*a**2*b**4*f + 6*b**6*f) - 18*A*a*b**4*c**2*d*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6
*f) + 9*A*a*b**4*c**2*d*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) - 18*A*a*b**4*c*d**2*f*x/(6*a**2*b
**4*f + 6*b**6*f) - 3*A*a*b**4*d**3*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) + 6*A*b**5*c**3*log(a/
b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) - 3*A*b**5*c**3*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*
f) + 18*A*b**5*c**2*d*f*x/(6*a**2*b**4*f + 6*b**6*f) + 9*A*b**5*c*d**2*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f
 + 6*b**6*f) - 6*A*b**5*d**3*f*x/(6*a**2*b**4*f + 6*b**6*f) + 6*A*b**5*d**3*tan(e + f*x)/(6*a**2*b**4*f + 6*b*
*6*f) + 6*B*a**4*b*d**3*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) - 18*B*a**3*b**2*c*d**2*log(a/b + t
an(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) - 6*B*a**3*b**2*d**3*tan(e + f*x)/(6*a**2*b**4*f + 6*b**6*f) + 18*B*a*
*2*b**3*c**2*d*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 18*B*a**2*b**3*c*d**2*tan(e + f*x)/(6*a**2
*b**4*f + 6*b**6*f) + 3*B*a**2*b**3*d**3*tan(e + f*x)**2/(6*a**2*b**4*f + 6*b**6*f) - 6*B*a*b**4*c**3*log(a/b
+ tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 3*B*a*b**4*c**3*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*
f) - 18*B*a*b**4*c**2*d*f*x/(6*a**2*b**4*f + 6*b**6*f) - 9*B*a*b**4*c*d**2*log(tan(e + f*x)**2 + 1)/(6*a**2*b*
*4*f + 6*b**6*f) + 6*B*a*b**4*d**3*f*x/(6*a**2*b**4*f + 6*b**6*f) - 6*B*a*b**4*d**3*tan(e + f*x)/(6*a**2*b**4*
f + 6*b**6*f) + 6*B*b**5*c**3*f*x/(6*a**2*b**4*f + 6*b**6*f) + 9*B*b**5*c**2*d*log(tan(e + f*x)**2 + 1)/(6*a**
2*b**4*f + 6*b**6*f) - 18*B*b**5*c*d**2*f*x/(6*a**2*b**4*f + 6*b**6*f) + 18*B*b**5*c*d**2*tan(e + f*x)/(6*a**2
*b**4*f + 6*b**6*f) - 3*B*b**5*d**3*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) + 3*B*b**5*d**3*tan(e
+ f*x)**2/(6*a**2*b**4*f + 6*b**6*f) - 6*C*a**5*d**3*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 18*C
*a**4*b*c*d**2*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 6*C*a**4*b*d**3*tan(e + f*x)/(6*a**2*b**4*
f + 6*b**6*f) - 18*C*a**3*b**2*c**2*d*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) - 18*C*a**3*b**2*c*d*
*2*tan(e + f*x)/(6*a**2*b**4*f + 6*b**6*f) - 3*C*a**3*b**2*d**3*tan(e + f*x)**2/(6*a**2*b**4*f + 6*b**6*f) + 6
*C*a**2*b**3*c**3*log(a/b + tan(e + f*x))/(6*a**2*b**4*f + 6*b**6*f) + 18*C*a**2*b**3*c**2*d*tan(e + f*x)/(6*a
**2*b**4*f + 6*b**6*f) + 9*C*a**2*b**3*c*d**2*tan(e + f*x)**2/(6*a**2*b**4*f + 6*b**6*f) + 2*C*a**2*b**3*d**3*
tan(e + f*x)**3/(6*a**2*b**4*f + 6*b**6*f) - 6*C*a*b**4*c**3*f*x/(6*a**2*b**4*f + 6*b**6*f) - 9*C*a*b**4*c**2*
d*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) + 18*C*a*b**4*c*d**2*f*x/(6*a**2*b**4*f + 6*b**6*f) - 18
*C*a*b**4*c*d**2*tan(e + f*x)/(6*a**2*b**4*f + 6*b**6*f) + 3*C*a*b**4*d**3*log(tan(e + f*x)**2 + 1)/(6*a**2*b*
*4*f + 6*b**6*f) - 3*C*a*b**4*d**3*tan(e + f*x)**2/(6*a**2*b**4*f + 6*b**6*f) + 3*C*b**5*c**3*log(tan(e + f*x)
**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) - 18*C*b**5*c**2*d*f*x/(6*a**2*b**4*f + 6*b**6*f) + 18*C*b**5*c**2*d*tan(e
 + f*x)/(6*a**2*b**4*f + 6*b**6*f) - 9*C*b**5*c*d**2*log(tan(e + f*x)**2 + 1)/(6*a**2*b**4*f + 6*b**6*f) + 9*C
*b**5*c*d**2*tan(e + f*x)**2/(6*a**2*b**4*f + 6*b**6*f) + 6*C*b**5*d**3*f*x/(6*a**2*b**4*f + 6*b**6*f) + 2*C*b
**5*d**3*tan(e + f*x)**3/(6*a**2*b**4*f + 6*b**6*f) - 6*C*b**5*d**3*tan(e + f*x)/(6*a**2*b**4*f + 6*b**6*f), T
rue))

________________________________________________________________________________________

Giac [A]  time = 2.25161, size = 774, normalized size = 2.13 \begin{align*} \frac{\frac{6 \,{\left (A a c^{3} - C a c^{3} + B b c^{3} - 3 \, B a c^{2} d + 3 \, A b c^{2} d - 3 \, C b c^{2} d - 3 \, A a c d^{2} + 3 \, C a c d^{2} - 3 \, B b c d^{2} + B a d^{3} - A b d^{3} + C b d^{3}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} + \frac{3 \,{\left (B a c^{3} - A b c^{3} + C b c^{3} + 3 \, A a c^{2} d - 3 \, C a c^{2} d + 3 \, B b c^{2} d - 3 \, B a c d^{2} + 3 \, A b c d^{2} - 3 \, C b c d^{2} - A a d^{3} + C a d^{3} - B b d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{6 \,{\left (C a^{2} b^{3} c^{3} - B a b^{4} c^{3} + A b^{5} c^{3} - 3 \, C a^{3} b^{2} c^{2} d + 3 \, B a^{2} b^{3} c^{2} d - 3 \, A a b^{4} c^{2} d + 3 \, C a^{4} b c d^{2} - 3 \, B a^{3} b^{2} c d^{2} + 3 \, A a^{2} b^{3} c d^{2} - C a^{5} d^{3} + B a^{4} b d^{3} - A a^{3} b^{2} d^{3}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{4} + b^{6}} + \frac{2 \, C b^{2} d^{3} \tan \left (f x + e\right )^{3} + 9 \, C b^{2} c d^{2} \tan \left (f x + e\right )^{2} - 3 \, C a b d^{3} \tan \left (f x + e\right )^{2} + 3 \, B b^{2} d^{3} \tan \left (f x + e\right )^{2} + 18 \, C b^{2} c^{2} d \tan \left (f x + e\right ) - 18 \, C a b c d^{2} \tan \left (f x + e\right ) + 18 \, B b^{2} c d^{2} \tan \left (f x + e\right ) + 6 \, C a^{2} d^{3} \tan \left (f x + e\right ) - 6 \, B a b d^{3} \tan \left (f x + e\right ) + 6 \, A b^{2} d^{3} \tan \left (f x + e\right ) - 6 \, C b^{2} d^{3} \tan \left (f x + e\right )}{b^{3}}}{6 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="giac")

[Out]

1/6*(6*(A*a*c^3 - C*a*c^3 + B*b*c^3 - 3*B*a*c^2*d + 3*A*b*c^2*d - 3*C*b*c^2*d - 3*A*a*c*d^2 + 3*C*a*c*d^2 - 3*
B*b*c*d^2 + B*a*d^3 - A*b*d^3 + C*b*d^3)*(f*x + e)/(a^2 + b^2) + 3*(B*a*c^3 - A*b*c^3 + C*b*c^3 + 3*A*a*c^2*d
- 3*C*a*c^2*d + 3*B*b*c^2*d - 3*B*a*c*d^2 + 3*A*b*c*d^2 - 3*C*b*c*d^2 - A*a*d^3 + C*a*d^3 - B*b*d^3)*log(tan(f
*x + e)^2 + 1)/(a^2 + b^2) + 6*(C*a^2*b^3*c^3 - B*a*b^4*c^3 + A*b^5*c^3 - 3*C*a^3*b^2*c^2*d + 3*B*a^2*b^3*c^2*
d - 3*A*a*b^4*c^2*d + 3*C*a^4*b*c*d^2 - 3*B*a^3*b^2*c*d^2 + 3*A*a^2*b^3*c*d^2 - C*a^5*d^3 + B*a^4*b*d^3 - A*a^
3*b^2*d^3)*log(abs(b*tan(f*x + e) + a))/(a^2*b^4 + b^6) + (2*C*b^2*d^3*tan(f*x + e)^3 + 9*C*b^2*c*d^2*tan(f*x
+ e)^2 - 3*C*a*b*d^3*tan(f*x + e)^2 + 3*B*b^2*d^3*tan(f*x + e)^2 + 18*C*b^2*c^2*d*tan(f*x + e) - 18*C*a*b*c*d^
2*tan(f*x + e) + 18*B*b^2*c*d^2*tan(f*x + e) + 6*C*a^2*d^3*tan(f*x + e) - 6*B*a*b*d^3*tan(f*x + e) + 6*A*b^2*d
^3*tan(f*x + e) - 6*C*b^2*d^3*tan(f*x + e))/b^3)/f

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