3.1.61 \(\int \frac {(c+d \tan (e+f x))^2 (A+B \tan (e+f x)+C \tan ^2(e+f x))}{a+b \tan (e+f x)} \, dx\) [61]

Optimal. Leaf size=254 \[ -\frac {\left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) x}{a^2+b^2}-\frac {\left (a \left (B c^2-2 c C d-B d^2\right )+b \left (c^2 C+2 B c d-C d^2\right )+A \left (2 a c d-b \left (c^2-d^2\right )\right )\right ) \log (\cos (e+f x))}{\left (a^2+b^2\right ) f}+\frac {\left (A b^2-a (b B-a C)\right ) (b c-a d)^2 \log (a+b \tan (e+f x))}{b^3 \left (a^2+b^2\right ) f}+\frac {d (b c C+b B d-a C d) \tan (e+f x)}{b^2 f}+\frac {C (c+d \tan (e+f x))^2}{2 b f} \]

[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-B*d^2-2*C*c*d)+b*(2*B
*c*d+C*c^2-C*d^2)+A*(2*a*c*d-b*(c^2-d^2)))*ln(cos(f*x+e))/(a^2+b^2)/f+(A*b^2-a*(B*b-C*a))*(-a*d+b*c)^2*ln(a+b*
tan(f*x+e))/b^3/(a^2+b^2)/f+d*(B*b*d-C*a*d+C*b*c)*tan(f*x+e)/b^2/f+1/2*C*(c+d*tan(f*x+e))^2/b/f

________________________________________________________________________________________

Rubi [A]
time = 0.55, 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 = {3728, 3718, 3707, 3698, 31, 3556} \begin {gather*} -\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} \end {gather*}

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 31

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

Rule 3556

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

Rule 3698

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 3707

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 3718

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 3728

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])))

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 ) \text {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] Result contains complex when optimal does not.
time = 1.96, size = 190, normalized size = 0.75 \begin {gather*} \frac {\frac {b (-i A+B+i C) (c+i d)^2 \log (i-\tan (e+f x))}{a+i b}+\frac {b (i A+B-i C) (c-i d)^2 \log (i+\tan (e+f x))}{a-i b}+\frac {2 \left (A b^2+a (-b B+a C)\right ) (b c-a d)^2 \log (a+b \tan (e+f x))}{b^2 \left (a^2+b^2\right )}+\frac {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} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 317, normalized size = 1.25

method	result	size
derivativedivides	\(\frac {\frac {d \left (\frac {C b d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+b \tan \left (f x +e \right ) B d -\tan \left (f x +e \right ) C a d +2 C b c \tan \left (f x +e \right )\right )}{b^{2}}+\frac {\left (A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (2 A a c d -A b \,c^{2}+A b \,d^{2}+B a \,c^{2}-B a \,d^{2}+2 B b c d -2 C a c d +C b \,c^{2}-C b \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{2}-A a \,d^{2}+2 A b c d -2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}-2 C b c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}}{f}\)	\(317\)
default	\(\frac {\frac {d \left (\frac {C b d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+b \tan \left (f x +e \right ) B d -\tan \left (f x +e \right ) C a d +2 C b c \tan \left (f x +e \right )\right )}{b^{2}}+\frac {\left (A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (2 A a c d -A b \,c^{2}+A b \,d^{2}+B a \,c^{2}-B a \,d^{2}+2 B b c d -2 C a c d +C b \,c^{2}-C b \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A a \,c^{2}-A a \,d^{2}+2 A b c d -2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}-2 C b c d \right ) \arctan \left (\tan \left (f x +e \right )\right )}{a^{2}+b^{2}}}{f}\)	\(317\)
norman	\(\frac {\left (A a \,c^{2}-A a \,d^{2}+2 A b c d -2 B a c d +B b \,c^{2}-B b \,d^{2}-C a \,c^{2}+C a \,d^{2}-2 C b c d \right ) x}{a^{2}+b^{2}}+\frac {d \left (B b d -a C d +2 C b c \right ) \tan \left (f x +e \right )}{b^{2} f}+\frac {C \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{2 b f}+\frac {\left (A \,a^{2} d^{2} b^{2}-2 A a \,b^{3} c d +A \,b^{4} c^{2}-B \,a^{3} d^{2} b +2 B \,a^{2} c d \,b^{2}-B a \,b^{3} c^{2}+a^{4} C \,d^{2}-2 C \,a^{3} c d b +C \,a^{2} c^{2} b^{2}\right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) b^{3} f}+\frac {\left (2 A a c d -A b \,c^{2}+A b \,d^{2}+B a \,c^{2}-B a \,d^{2}+2 B b c d -2 C a c d +C b \,c^{2}-C b \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{2}+b^{2}\right )}\)	\(318\)
risch	\(\text {Expression too large to display}\)	\(1458\)

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,method=_RETURNVERBOSE)

[Out]

1/f*(d/b^2*(1/2*C*b*d*tan(f*x+e)^2+b*tan(f*x+e)*B*d-tan(f*x+e)*C*a*d+2*C*b*c*tan(f*x+e))+1/b^3*(A*a^2*b^2*d^2-
2*A*a*b^3*c*d+A*b^4*c^2-B*a^3*b*d^2+2*B*a^2*b^2*c*d-B*a*b^3*c^2+C*a^4*d^2-2*C*a^3*b*c*d+C*a^2*b^2*c^2)/(a^2+b^
2)*ln(a+b*tan(f*x+e))+1/(a^2+b^2)*(1/2*(2*A*a*c*d-A*b*c^2+A*b*d^2+B*a*c^2-B*a*d^2+2*B*b*c*d-2*C*a*c*d+C*b*c^2-
C*b*d^2)*ln(1+tan(f*x+e)^2)+(A*a*c^2-A*a*d^2+2*A*b*c*d-2*B*a*c*d+B*b*c^2-B*b*d^2-C*a*c^2+C*a*d^2-2*C*b*c*d)*ar
ctan(tan(f*x+e))))

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 295, normalized size = 1.16 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 4.50, size = 403, normalized size = 1.59 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 2.64, size = 4444, normalized size = 17.50 \begin {gather*} \text {Too large to display} \end {gather*}

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*l
og(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*lo
g(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*t
an(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...

________________________________________________________________________________________

Giac [A]
time = 0.85, size = 338, normalized size = 1.33 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 11.28, size = 325, normalized size = 1.28 \begin {gather*} \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {B\,d^2+2\,C\,c\,d}{b}-\frac {C\,a\,d^2}{b^2}\right )}{f}+\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (b^2\,\left (C\,a^2\,c^2+2\,B\,a^2\,c\,d+A\,a^2\,d^2\right )-b\,\left (B\,a^3\,d^2+2\,C\,c\,a^3\,d\right )-b^3\,\left (B\,a\,c^2+2\,A\,a\,d\,c\right )+A\,b^4\,c^2+C\,a^4\,d^2\right )}{f\,\left (a^2\,b^3+b^5\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (A\,d^2-A\,c^2+B\,c^2\,1{}\mathrm {i}-B\,d^2\,1{}\mathrm {i}+C\,c^2-C\,d^2+A\,c\,d\,2{}\mathrm {i}+2\,B\,c\,d-C\,c\,d\,2{}\mathrm {i}\right )}{2\,f\,\left (b+a\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (B\,c^2-B\,d^2+2\,A\,c\,d-2\,C\,c\,d-A\,c^2\,1{}\mathrm {i}+A\,d^2\,1{}\mathrm {i}+C\,c^2\,1{}\mathrm {i}-C\,d^2\,1{}\mathrm {i}+B\,c\,d\,2{}\mathrm {i}\right )}{2\,f\,\left (a+b\,1{}\mathrm {i}\right )}+\frac {C\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,b\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

________________________________________________________________________________________