3.61 \(\int \frac{(c+d \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{a+b \tan (e+f x)} \, dx\)
Optimal. Leaf size=254 \[ -\frac{\log (\cos (e+f x)) \left (A \left (2 a c d-b \left (c^2-d^2\right )\right )+a \left (B c^2-B d^2-2 c C d\right )+b \left (2 B c d+c^2 C-C d^2\right )\right )}{f \left (a^2+b^2\right )}-\frac{x \left (a \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{a^2+b^2}+\frac{(b c-a d)^2 \left (A b^2-a (b B-a C)\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )}+\frac{d \tan (e+f x) (-a C d+b B d+b c C)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f} \]
[Out]
-(((a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2)) - ((a*(B*
c^2 - 2*c*C*d - B*d^2) + b*(c^2*C + 2*B*c*d - C*d^2) + A*(2*a*c*d - b*(c^2 - d^2)))*Log[Cos[e + f*x]])/((a^2 +
b^2)*f) + ((A*b^2 - a*(b*B - a*C))*(b*c - a*d)^2*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^2)*f) + (d*(b*c*C + b
*B*d - a*C*d)*Tan[e + f*x])/(b^2*f) + (C*(c + d*Tan[e + f*x])^2)/(2*b*f)
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Rubi [A] time = 0.82732, antiderivative size = 252, normalized size of antiderivative = 0.99,
number of steps used = 6, 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} \[ -\frac{\log (\cos (e+f x)) \left (2 a A c d+a B \left (c^2-d^2\right )-2 a c C d-A b \left (c^2-d^2\right )+b \left (2 B c d+c^2 C-C d^2\right )\right )}{f \left (a^2+b^2\right )}-\frac{x \left (a \left (-A \left (c^2-d^2\right )+2 B c d+c^2 C-C d^2\right )-b \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )\right )}{a^2+b^2}+\frac{(b c-a d)^2 \left (A b^2-a (b B-a C)\right ) \log (a+b \tan (e+f x))}{b^3 f \left (a^2+b^2\right )}+\frac{d \tan (e+f x) (-a C d+b B d+b c C)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f} \]
Antiderivative was successfully verified.
[In]
Int[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]
[Out]
-(((a*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d + B*(c^2 - d^2)))*x)/(a^2 + b^2)) - ((2*a*A
*c*d - 2*a*c*C*d - A*b*(c^2 - d^2) + a*B*(c^2 - d^2) + b*(c^2*C + 2*B*c*d - 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)^2*Log[a + b*Tan[e + f*x]])/(b^3*(a^2 + b^2)*f) + (d*(b*c*C + b
*B*d - a*C*d)*Tan[e + f*x])/(b^2*f) + (C*(c + d*Tan[e + f*x])^2)/(2*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))^2 \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))^2}{2 b f}+\frac{\int \frac{(c+d \tan (e+f x)) \left (2 (A b c-a C d)+2 b (B c+(A-C) d) \tan (e+f x)+2 (b c C+b B d-a C d) \tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{2 b}\\ &=\frac{d (b c C+b B d-a C d) \tan (e+f x)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f}-\frac{\int \frac{-2 \left (A b^2 c^2-a d (2 b c C+b B d-a C d)\right )-2 b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x)-2 \left (a^2 C d^2-a b d (2 c C+B d)+b^2 \left (c^2 C+2 B c d+(A-C) d^2\right )\right ) \tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{2 b^2}\\ &=-\frac{\left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x}{a^2+b^2}+\frac{d (b c C+b B d-a C d) \tan (e+f x)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f}+\frac{\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^2\right ) \int \frac{1+\tan ^2(e+f x)}{a+b \tan (e+f x)} \, dx}{b^2 \left (a^2+b^2\right )}+\frac{\left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (c^2 C+2 B c d-C d^2\right )\right ) \int \tan (e+f x) \, dx}{a^2+b^2}\\ &=-\frac{\left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x}{a^2+b^2}-\frac{\left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (c^2 C+2 B c d-C d^2\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac{d (b c C+b B d-a C d) \tan (e+f x)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f}+\frac{\left (\left (A b^2-a (b B-a C)\right ) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (e+f x)\right )}{b^3 \left (a^2+b^2\right ) f}\\ &=-\frac{\left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x}{a^2+b^2}-\frac{\left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )+b \left (c^2 C+2 B c d-C d^2\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)^2 \log (a+b \tan (e+f x))}{b^3 \left (a^2+b^2\right ) f}+\frac{d (b c C+b B d-a C d) \tan (e+f x)}{b^2 f}+\frac{C (c+d \tan (e+f x))^2}{2 b f}\\ \end{align*}
Mathematica [C] time = 3.03524, size = 190, normalized size = 0.75 \[ \frac{\frac{2 (b c-a d)^2 \left (a (a C-b B)+A b^2\right ) \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right )}+\frac{b (c-i d)^2 (i A+B-i C) \log (\tan (e+f x)+i)}{a-i b}+\frac{b (c+i d)^2 (-i A+B+i C) \log (-\tan (e+f x)+i)}{a+i b}+\frac{2 d \tan (e+f x) (-a C d+b B d+b c C)}{b}+C (c+d \tan (e+f x))^2}{2 b f} \]
Antiderivative was successfully verified.
[In]
Integrate[((c + d*Tan[e + f*x])^2*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x]),x]
[Out]
((b*((-I)*A + B + I*C)*(c + I*d)^2*Log[I - Tan[e + f*x]])/(a + I*b) + (b*(I*A + B - I*C)*(c - I*d)^2*Log[I + T
an[e + f*x]])/(a - I*b) + (2*(A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)^2*Log[a + b*Tan[e + f*x]])/(b^2*(a^2 + b^2
)) + (2*d*(b*c*C + b*B*d - a*C*d)*Tan[e + f*x])/b + C*(c + d*Tan[e + f*x])^2)/(2*b*f)
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Maple [B] time = 0.048, size = 861, normalized size = 3.4 \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))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x)
[Out]
2/f/b/(a^2+b^2)*ln(a+b*tan(f*x+e))*a^2*B*c*d-2/f/b^2/(a^2+b^2)*ln(a+b*tan(f*x+e))*C*a^3*c*d+1/2/f/(a^2+b^2)*ln
(1+tan(f*x+e)^2)*A*b*d^2+1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*a*c^2-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*a*d^2
-1/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*C*a*c*d+2/f/(a^2+b^2)*A*arctan(tan(f*x+e))*b*c*d-2/f/(a^2+b^2)*B*arctan(tan(
f*x+e))*a*c*d-2/f/(a^2+b^2)*C*arctan(tan(f*x+e))*b*c*d+1/f/b/(a^2+b^2)*ln(a+b*tan(f*x+e))*a^2*A*d^2-1/f/b^2/(a
^2+b^2)*ln(a+b*tan(f*x+e))*B*a^3*d^2-1/f*d^2/b^2*a*C*tan(f*x+e)+2/f*d/b*C*c*tan(f*x+e)+1/f*b/(a^2+b^2)*ln(a+b*
tan(f*x+e))*A*c^2-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*b*c^2-1/f/(a^2+b^2)*ln(a+b*tan(f*x+e))*B*a*c^2-2/f/(a^2
+b^2)*ln(a+b*tan(f*x+e))*A*a*c*d+1/f/b^3/(a^2+b^2)*ln(a+b*tan(f*x+e))*C*a^4*d^2+1/f/b/(a^2+b^2)*ln(a+b*tan(f*x
+e))*C*a^2*c^2+1/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*B*b*c*d+1/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*A*a*c*d+1/f*d^2/b*B*t
an(f*x+e)+1/2/f*d^2/b*C*tan(f*x+e)^2+1/2/f/(a^2+b^2)*ln(1+tan(f*x+e)^2)*C*b*c^2-1/2/f/(a^2+b^2)*ln(1+tan(f*x+e
)^2)*C*b*d^2+1/f/(a^2+b^2)*A*arctan(tan(f*x+e))*a*c^2-1/f/(a^2+b^2)*A*arctan(tan(f*x+e))*a*d^2+1/f/(a^2+b^2)*B
*arctan(tan(f*x+e))*b*c^2-1/f/(a^2+b^2)*B*arctan(tan(f*x+e))*b*d^2-1/f/(a^2+b^2)*C*arctan(tan(f*x+e))*a*c^2+1/
f/(a^2+b^2)*C*arctan(tan(f*x+e))*a*d^2
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Maxima [A] time = 1.52073, size = 392, normalized size = 1.54 \begin{align*} \frac{\frac{2 \,{\left ({\left ({\left (A - C\right )} a + B b\right )} c^{2} - 2 \,{\left (B a -{\left (A - C\right )} b\right )} c d -{\left ({\left (A - C\right )} a + B b\right )} d^{2}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} + \frac{2 \,{\left ({\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} c^{2} - 2 \,{\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} c d +{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} b^{3} + b^{5}} + \frac{{\left ({\left (B a -{\left (A - C\right )} b\right )} c^{2} + 2 \,{\left ({\left (A - C\right )} a + B b\right )} c d -{\left (B a -{\left (A - C\right )} b\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{C b d^{2} \tan \left (f x + e\right )^{2} + 2 \,{\left (2 \, C b c d -{\left (C a - B b\right )} d^{2}\right )} \tan \left (f x + e\right )}{b^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="maxima")
[Out]
1/2*(2*(((A - C)*a + B*b)*c^2 - 2*(B*a - (A - C)*b)*c*d - ((A - C)*a + B*b)*d^2)*(f*x + e)/(a^2 + b^2) + 2*((C
*a^2*b^2 - B*a*b^3 + A*b^4)*c^2 - 2*(C*a^3*b - B*a^2*b^2 + A*a*b^3)*c*d + (C*a^4 - B*a^3*b + A*a^2*b^2)*d^2)*l
og(b*tan(f*x + e) + a)/(a^2*b^3 + b^5) + ((B*a - (A - C)*b)*c^2 + 2*((A - C)*a + B*b)*c*d - (B*a - (A - C)*b)*
d^2)*log(tan(f*x + e)^2 + 1)/(a^2 + b^2) + (C*b*d^2*tan(f*x + e)^2 + 2*(2*C*b*c*d - (C*a - B*b)*d^2)*tan(f*x +
e))/b^2)/f
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Fricas [A] time = 2.91679, size = 822, normalized size = 3.24 \begin{align*} \frac{{\left (C a^{2} b^{2} + C b^{4}\right )} d^{2} \tan \left (f x + e\right )^{2} + 2 \,{\left ({\left ({\left (A - C\right )} a b^{3} + B b^{4}\right )} c^{2} - 2 \,{\left (B a b^{3} -{\left (A - C\right )} b^{4}\right )} c d -{\left ({\left (A - C\right )} a b^{3} + B b^{4}\right )} d^{2}\right )} f x +{\left ({\left (C a^{2} b^{2} - B a b^{3} + A b^{4}\right )} c^{2} - 2 \,{\left (C a^{3} b - B a^{2} b^{2} + A a b^{3}\right )} c d +{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} d^{2}\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 ) -{\left ({\left (C a^{2} b^{2} + C b^{4}\right )} c^{2} - 2 \,{\left (C a^{3} b - B a^{2} b^{2} + C a b^{3} - B b^{4}\right )} c d +{\left (C a^{4} - B a^{3} b + A a^{2} b^{2} - B a b^{3} +{\left (A - C\right )} b^{4}\right )} d^{2}\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (2 \,{\left (C a^{2} b^{2} + C b^{4}\right )} c d -{\left (C a^{3} b - B a^{2} b^{2} + C a b^{3} - B b^{4}\right )} d^{2}\right )} \tan \left (f x + e\right )}{2 \,{\left (a^{2} b^{3} + b^{5}\right )} f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="fricas")
[Out]
1/2*((C*a^2*b^2 + C*b^4)*d^2*tan(f*x + e)^2 + 2*(((A - C)*a*b^3 + B*b^4)*c^2 - 2*(B*a*b^3 - (A - C)*b^4)*c*d -
((A - C)*a*b^3 + B*b^4)*d^2)*f*x + ((C*a^2*b^2 - B*a*b^3 + A*b^4)*c^2 - 2*(C*a^3*b - B*a^2*b^2 + A*a*b^3)*c*d
+ (C*a^4 - B*a^3*b + A*a^2*b^2)*d^2)*log((b^2*tan(f*x + e)^2 + 2*a*b*tan(f*x + e) + a^2)/(tan(f*x + e)^2 + 1)
) - ((C*a^2*b^2 + C*b^4)*c^2 - 2*(C*a^3*b - B*a^2*b^2 + C*a*b^3 - B*b^4)*c*d + (C*a^4 - B*a^3*b + A*a^2*b^2 -
B*a*b^3 + (A - C)*b^4)*d^2)*log(1/(tan(f*x + e)^2 + 1)) + 2*(2*(C*a^2*b^2 + C*b^4)*c*d - (C*a^3*b - B*a^2*b^2
+ C*a*b^3 - B*b^4)*d^2)*tan(f*x + e))/((a^2*b^3 + b^5)*f)
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Sympy [A] time = 36.782, size = 4444, normalized size = 17.5 \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))**2*(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))**2*(A + B*tan(e) + C*tan(e)**2)/tan(e), Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), ((A*c
**2*x + A*c*d*log(tan(e + f*x)**2 + 1)/f - A*d**2*x + A*d**2*tan(e + f*x)/f + B*c**2*log(tan(e + f*x)**2 + 1)/
(2*f) - 2*B*c*d*x + 2*B*c*d*tan(e + f*x)/f - B*d**2*log(tan(e + f*x)**2 + 1)/(2*f) + B*d**2*tan(e + f*x)**2/(2
*f) - C*c**2*x + C*c**2*tan(e + f*x)/f - C*c*d*log(tan(e + f*x)**2 + 1)/f + C*c*d*tan(e + f*x)**2/f + C*d**2*x
+ C*d**2*tan(e + f*x)**3/(3*f) - C*d**2*tan(e + f*x)/f)/a, Eq(b, 0)), (-I*A*c**2*f*x*tan(e + f*x)/(-2*b*f*tan
(e + f*x) + 2*I*b*f) - A*c**2*f*x/(-2*b*f*tan(e + f*x) + 2*I*b*f) - I*A*c**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) -
2*A*c*d*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + 2*I*A*c*d*f*x/(-2*b*f*tan(e + f*x) + 2*I*b*f) + 2*
A*c*d/(-2*b*f*tan(e + f*x) + 2*I*b*f) - I*A*d**2*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - A*d**2*f*x
/(-2*b*f*tan(e + f*x) + 2*I*b*f) - A*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f
) + I*A*d**2*log(tan(e + f*x)**2 + 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + I*A*d**2/(-2*b*f*tan(e + f*x) + 2*I*b*
f) - B*c**2*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + I*B*c**2*f*x/(-2*b*f*tan(e + f*x) + 2*I*b*f) +
B*c**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 2*I*B*c*d*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 2*B*c*d*
f*x/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 2*B*c*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I
*b*f) + 2*I*B*c*d*log(tan(e + f*x)**2 + 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + 2*I*B*c*d/(-2*b*f*tan(e + f*x) +
2*I*b*f) + 3*B*d**2*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 3*I*B*d**2*f*x/(-2*b*f*tan(e + f*x) + 2
*I*b*f) - I*B*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - B*d**2*log(tan(e +
f*x)**2 + 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 2*B*d**2*tan(e + f*x)**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 3*B*
d**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) - I*C*c**2*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - C*c**2*f*x/
(-2*b*f*tan(e + f*x) + 2*I*b*f) - C*c**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f)
+ I*C*c**2*log(tan(e + f*x)**2 + 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + I*C*c**2/(-2*b*f*tan(e + f*x) + 2*I*b*f
) + 6*C*c*d*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 6*I*C*c*d*f*x/(-2*b*f*tan(e + f*x) + 2*I*b*f) -
2*I*C*c*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 2*C*c*d*log(tan(e + f*x)**2
+ 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 4*C*c*d*tan(e + f*x)**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 6*C*c*d/(-2*
b*f*tan(e + f*x) + 2*I*b*f) + 3*I*C*d**2*f*x*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) + 3*C*d**2*f*x/(-2*b
*f*tan(e + f*x) + 2*I*b*f) + 2*C*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*b*f*tan(e + f*x) + 2*I*b*f) -
2*I*C*d**2*log(tan(e + f*x)**2 + 1)/(-2*b*f*tan(e + f*x) + 2*I*b*f) - C*d**2*tan(e + f*x)**3/(-2*b*f*tan(e + f
*x) + 2*I*b*f) - I*C*d**2*tan(e + f*x)**2/(-2*b*f*tan(e + f*x) + 2*I*b*f) - 3*I*C*d**2/(-2*b*f*tan(e + f*x) +
2*I*b*f), Eq(a, -I*b)), (-I*A*c**2*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + A*c**2*f*x/(2*b*f*tan(e +
f*x) + 2*I*b*f) - I*A*c**2/(2*b*f*tan(e + f*x) + 2*I*b*f) + 2*A*c*d*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*
I*b*f) + 2*I*A*c*d*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) - 2*A*c*d/(2*b*f*tan(e + f*x) + 2*I*b*f) - I*A*d**2*f*x*
tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + A*d**2*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) + A*d**2*log(tan(e + f
*x)**2 + 1)*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + I*A*d**2*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x
) + 2*I*b*f) + I*A*d**2/(2*b*f*tan(e + f*x) + 2*I*b*f) + B*c**2*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f
) + I*B*c**2*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) - B*c**2/(2*b*f*tan(e + f*x) + 2*I*b*f) - 2*I*B*c*d*f*x*tan(e
+ f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + 2*B*c*d*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) + 2*B*c*d*log(tan(e + f*x)*
*2 + 1)*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + 2*I*B*c*d*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x) +
2*I*b*f) + 2*I*B*c*d/(2*b*f*tan(e + f*x) + 2*I*b*f) - 3*B*d**2*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f
) - 3*I*B*d**2*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) - I*B*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*b*f*tan(
e + f*x) + 2*I*b*f) + B*d**2*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x) + 2*I*b*f) + 2*B*d**2*tan(e + f*x)**
2/(2*b*f*tan(e + f*x) + 2*I*b*f) + 3*B*d**2/(2*b*f*tan(e + f*x) + 2*I*b*f) - I*C*c**2*f*x*tan(e + f*x)/(2*b*f*
tan(e + f*x) + 2*I*b*f) + C*c**2*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) + C*c**2*log(tan(e + f*x)**2 + 1)*tan(e +
f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) + I*C*c**2*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x) + 2*I*b*f) + I*C*c
**2/(2*b*f*tan(e + f*x) + 2*I*b*f) - 6*C*c*d*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f) - 6*I*C*c*d*f*x/(
2*b*f*tan(e + f*x) + 2*I*b*f) - 2*I*C*c*d*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I*b*f)
+ 2*C*c*d*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x) + 2*I*b*f) + 4*C*c*d*tan(e + f*x)**2/(2*b*f*tan(e + f*
x) + 2*I*b*f) + 6*C*c*d/(2*b*f*tan(e + f*x) + 2*I*b*f) + 3*I*C*d**2*f*x*tan(e + f*x)/(2*b*f*tan(e + f*x) + 2*I
*b*f) - 3*C*d**2*f*x/(2*b*f*tan(e + f*x) + 2*I*b*f) - 2*C*d**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*b*f*ta
n(e + f*x) + 2*I*b*f) - 2*I*C*d**2*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x) + 2*I*b*f) + C*d**2*tan(e + f*
x)**3/(2*b*f*tan(e + f*x) + 2*I*b*f) - I*C*d**2*tan(e + f*x)**2/(2*b*f*tan(e + f*x) + 2*I*b*f) - 3*I*C*d**2/(2
*b*f*tan(e + f*x) + 2*I*b*f), Eq(a, I*b)), (x*(c + d*tan(e))**2*(A + B*tan(e) + C*tan(e)**2)/(a + b*tan(e)), E
q(f, 0)), (2*A*a**2*b**2*d**2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + 2*A*a*b**3*c**2*f*x/(2*a**2
*b**3*f + 2*b**5*f) - 4*A*a*b**3*c*d*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + 2*A*a*b**3*c*d*log(t
an(e + f*x)**2 + 1)/(2*a**2*b**3*f + 2*b**5*f) - 2*A*a*b**3*d**2*f*x/(2*a**2*b**3*f + 2*b**5*f) + 2*A*b**4*c**
2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) - A*b**4*c**2*log(tan(e + f*x)**2 + 1)/(2*a**2*b**3*f + 2
*b**5*f) + 4*A*b**4*c*d*f*x/(2*a**2*b**3*f + 2*b**5*f) + A*b**4*d**2*log(tan(e + f*x)**2 + 1)/(2*a**2*b**3*f +
2*b**5*f) - 2*B*a**3*b*d**2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + 4*B*a**2*b**2*c*d*log(a/b +
tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + 2*B*a**2*b**2*d**2*tan(e + f*x)/(2*a**2*b**3*f + 2*b**5*f) - 2*B*a*
b**3*c**2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + B*a*b**3*c**2*log(tan(e + f*x)**2 + 1)/(2*a**2*
b**3*f + 2*b**5*f) - 4*B*a*b**3*c*d*f*x/(2*a**2*b**3*f + 2*b**5*f) - B*a*b**3*d**2*log(tan(e + f*x)**2 + 1)/(2
*a**2*b**3*f + 2*b**5*f) + 2*B*b**4*c**2*f*x/(2*a**2*b**3*f + 2*b**5*f) + 2*B*b**4*c*d*log(tan(e + f*x)**2 + 1
)/(2*a**2*b**3*f + 2*b**5*f) - 2*B*b**4*d**2*f*x/(2*a**2*b**3*f + 2*b**5*f) + 2*B*b**4*d**2*tan(e + f*x)/(2*a*
*2*b**3*f + 2*b**5*f) + 2*C*a**4*d**2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) - 4*C*a**3*b*c*d*log(
a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) - 2*C*a**3*b*d**2*tan(e + f*x)/(2*a**2*b**3*f + 2*b**5*f) + 2*C
*a**2*b**2*c**2*log(a/b + tan(e + f*x))/(2*a**2*b**3*f + 2*b**5*f) + 4*C*a**2*b**2*c*d*tan(e + f*x)/(2*a**2*b*
*3*f + 2*b**5*f) + C*a**2*b**2*d**2*tan(e + f*x)**2/(2*a**2*b**3*f + 2*b**5*f) - 2*C*a*b**3*c**2*f*x/(2*a**2*b
**3*f + 2*b**5*f) - 2*C*a*b**3*c*d*log(tan(e + f*x)**2 + 1)/(2*a**2*b**3*f + 2*b**5*f) + 2*C*a*b**3*d**2*f*x/(
2*a**2*b**3*f + 2*b**5*f) - 2*C*a*b**3*d**2*tan(e + f*x)/(2*a**2*b**3*f + 2*b**5*f) + C*b**4*c**2*log(tan(e +
f*x)**2 + 1)/(2*a**2*b**3*f + 2*b**5*f) - 4*C*b**4*c*d*f*x/(2*a**2*b**3*f + 2*b**5*f) + 4*C*b**4*c*d*tan(e + f
*x)/(2*a**2*b**3*f + 2*b**5*f) - C*b**4*d**2*log(tan(e + f*x)**2 + 1)/(2*a**2*b**3*f + 2*b**5*f) + C*b**4*d**2
*tan(e + f*x)**2/(2*a**2*b**3*f + 2*b**5*f), True))
________________________________________________________________________________________
Giac [A] time = 1.7387, size = 456, normalized size = 1.8 \begin{align*} \frac{\frac{2 \,{\left (A a c^{2} - C a c^{2} + B b c^{2} - 2 \, B a c d + 2 \, A b c d - 2 \, C b c d - A a d^{2} + C a d^{2} - B b d^{2}\right )}{\left (f x + e\right )}}{a^{2} + b^{2}} + \frac{{\left (B a c^{2} - A b c^{2} + C b c^{2} + 2 \, A a c d - 2 \, C a c d + 2 \, B b c d - B a d^{2} + A b d^{2} - C b d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (C a^{2} b^{2} c^{2} - B a b^{3} c^{2} + A b^{4} c^{2} - 2 \, C a^{3} b c d + 2 \, B a^{2} b^{2} c d - 2 \, A a b^{3} c d + C a^{4} d^{2} - B a^{3} b d^{2} + A a^{2} b^{2} d^{2}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{3} + b^{5}} + \frac{C b d^{2} \tan \left (f x + e\right )^{2} + 4 \, C b c d \tan \left (f x + e\right ) - 2 \, C a d^{2} \tan \left (f x + e\right ) + 2 \, B b d^{2} \tan \left (f x + e\right )}{b^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^2*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),x, algorithm="giac")
[Out]
1/2*(2*(A*a*c^2 - C*a*c^2 + B*b*c^2 - 2*B*a*c*d + 2*A*b*c*d - 2*C*b*c*d - A*a*d^2 + C*a*d^2 - B*b*d^2)*(f*x +
e)/(a^2 + b^2) + (B*a*c^2 - A*b*c^2 + C*b*c^2 + 2*A*a*c*d - 2*C*a*c*d + 2*B*b*c*d - B*a*d^2 + A*b*d^2 - C*b*d^
2)*log(tan(f*x + e)^2 + 1)/(a^2 + b^2) + 2*(C*a^2*b^2*c^2 - B*a*b^3*c^2 + A*b^4*c^2 - 2*C*a^3*b*c*d + 2*B*a^2*
b^2*c*d - 2*A*a*b^3*c*d + C*a^4*d^2 - B*a^3*b*d^2 + A*a^2*b^2*d^2)*log(abs(b*tan(f*x + e) + a))/(a^2*b^3 + b^5
) + (C*b*d^2*tan(f*x + e)^2 + 4*C*b*c*d*tan(f*x + e) - 2*C*a*d^2*tan(f*x + e) + 2*B*b*d^2*tan(f*x + e))/b^2)/f