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3.51 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\)

Optimal. Leaf size=248 \[ -\frac {\log (\cos (e+f x)) \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}+x \left (a^2 (A c-B d-c C)-2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right )+\frac {(d (A-C)+B c) (a+b \tan (e+f x))^2}{2 f}+\frac {b \tan (e+f x) (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac {(a C d-4 b (B d+c C)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f} \]

[Out]

(a^2*(A*c-B*d-C*c)-b^2*(A*c-B*d-C*c)-2*a*b*(B*c+(A-C)*d))*x-(2*a*b*(A*c-B*d-C*c)+a^2*(B*c+(A-C)*d)-b^2*(B*c+(A
-C)*d))*ln(cos(f*x+e))/f+b*(A*a*d+A*b*c+B*a*c-B*b*d-C*a*d-C*b*c)*tan(f*x+e)/f+1/2*(B*c+(A-C)*d)*(a+b*tan(f*x+e
))^2/f-1/12*(a*C*d-4*b*(B*d+C*c))*(a+b*tan(f*x+e))^3/b^2/f+1/4*C*d*tan(f*x+e)*(a+b*tan(f*x+e))^3/b/f

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Rubi [A]  time = 0.45, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3637, 3630, 3528, 3525, 3475} \[ -\frac {\log (\cos (e+f x)) \left (a^2 (d (A-C)+B c)+2 a b (A c-B d-c C)-b^2 (d (A-C)+B c)\right )}{f}+x \left (a^2 (A c-B d-c C)-2 a b (d (A-C)+B c)-b^2 (A c-B d-c C)\right )+\frac {(d (A-C)+B c) (a+b \tan (e+f x))^2}{2 f}+\frac {b \tan (e+f x) (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac {(a C d-4 b (B d+c C)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(A*c - c*C - B*d) - b^2*(A*c - c*C - B*d) - 2*a*b*(B*c + (A - C)*d))*x - ((2*a*b*(A*c - c*C - B*d) + a^2*
(B*c + (A - C)*d) - b^2*(B*c + (A - C)*d))*Log[Cos[e + f*x]])/f + (b*(A*b*c + a*B*c - b*c*C + a*A*d - b*B*d -
a*C*d)*Tan[e + f*x])/f + ((B*c + (A - C)*d)*(a + b*Tan[e + f*x])^2)/(2*f) - ((a*C*d - 4*b*(c*C + B*d))*(a + b*
Tan[e + f*x])^3)/(12*b^2*f) + (C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^3)/(4*b*f)

Rule 3475

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

Rule 3525

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c - b
*d)*x, x] + (Dist[b*c + a*d, Int[Tan[e + f*x], x], x] + Simp[(b*d*Tan[e + f*x])/f, x]) /; FreeQ[{a, b, c, d, e
, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]

Rule 3630

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

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]

Rubi steps

\begin {align*} \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}-\frac {\int (a+b \tan (e+f x))^2 \left (-4 A b c+a C d-4 b (B c+(A-C) d) \tan (e+f x)+(a C d-4 b (c C+B d)) \tan ^2(e+f x)\right ) \, dx}{4 b}\\ &=-\frac {(a C d-4 b (c C+B d)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}-\frac {\int (a+b \tan (e+f x))^2 (-4 b (A c-c C-B d)-4 b (B c+(A-C) d) \tan (e+f x)) \, dx}{4 b}\\ &=\frac {(B c+(A-C) d) (a+b \tan (e+f x))^2}{2 f}-\frac {(a C d-4 b (c C+B d)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}-\frac {\int (a+b \tan (e+f x)) (4 b (b B c+b (A-C) d-a (A c-c C-B d))-4 b (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)) \, dx}{4 b}\\ &=\left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right ) x+\frac {b (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)}{f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^2}{2 f}-\frac {(a C d-4 b (c C+B d)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}-\left (-2 a b (A c-c C-B d)-a^2 (B c+(A-C) d)+b^2 (B c+(A-C) d)\right ) \int \tan (e+f x) \, dx\\ &=\left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)-2 a b (B c+(A-C) d)\right ) x-\frac {\left (2 a b (A c-c C-B d)+a^2 (B c+(A-C) d)-b^2 (B c+(A-C) d)\right ) \log (\cos (e+f x))}{f}+\frac {b (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)}{f}+\frac {(B c+(A-C) d) (a+b \tan (e+f x))^2}{2 f}-\frac {(a C d-4 b (c C+B d)) (a+b \tan (e+f x))^3}{12 b^2 f}+\frac {C d \tan (e+f x) (a+b \tan (e+f x))^3}{4 b f}\\ \end {align*}

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Mathematica [C]  time = 3.43, size = 243, normalized size = 0.98 \[ \frac {-6 (-a A d-a B c+a C d+A b c-b B d-b c C) \left (-2 b^2 \tan (e+f x)+i \left ((a+i b)^2 \log (-\tan (e+f x)+i)-(a-i b)^2 \log (\tan (e+f x)+i)\right )\right )+6 (d (A-C)+B c) \left (6 a b^2 \tan (e+f x)+(-b+i a)^3 \log (-\tan (e+f x)+i)-(b+i a)^3 \log (\tan (e+f x)+i)+b^3 \tan ^2(e+f x)\right )+\frac {(4 b (B d+c C)-a C d) (a+b \tan (e+f x))^3}{b}+3 C d \tan (e+f x) (a+b \tan (e+f x))^3}{12 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-(a*C*d) + 4*b*(c*C + B*d))*(a + b*Tan[e + f*x])^3)/b + 3*C*d*Tan[e + f*x]*(a + b*Tan[e + f*x])^3 - 6*(A*b*
c - a*B*c - b*c*C - a*A*d - b*B*d + a*C*d)*(I*((a + I*b)^2*Log[I - Tan[e + f*x]] - (a - I*b)^2*Log[I + Tan[e +
 f*x]]) - 2*b^2*Tan[e + f*x]) + 6*(B*c + (A - C)*d)*((I*a - b)^3*Log[I - Tan[e + f*x]] - (I*a + b)^3*Log[I + T
an[e + f*x]] + 6*a*b^2*Tan[e + f*x] + b^3*Tan[e + f*x]^2))/(12*b*f)

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fricas [A]  time = 0.57, size = 273, normalized size = 1.10 \[ \frac {3 \, C b^{2} d \tan \left (f x + e\right )^{4} + 4 \, {\left (C b^{2} c + {\left (2 \, C a b + B b^{2}\right )} d\right )} \tan \left (f x + e\right )^{3} + 12 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d\right )} f x + 6 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d\right )} \tan \left (f x + e\right )}{12 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*C*b^2*d*tan(f*x + e)^4 + 4*(C*b^2*c + (2*C*a*b + B*b^2)*d)*tan(f*x + e)^3 + 12*(((A - C)*a^2 - 2*B*a*b
 - (A - C)*b^2)*c - (B*a^2 + 2*(A - C)*a*b - B*b^2)*d)*f*x + 6*((2*C*a*b + B*b^2)*c + (C*a^2 + 2*B*a*b + (A -
C)*b^2)*d)*tan(f*x + e)^2 - 6*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*d)*lo
g(1/(tan(f*x + e)^2 + 1)) + 12*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d)*tan(f*x
 + e))/f

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giac [B]  time = 23.12, size = 6502, normalized size = 26.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*A*a^2*c*f*x*tan(f*x)^4*tan(e)^4 - 12*C*a^2*c*f*x*tan(f*x)^4*tan(e)^4 - 24*B*a*b*c*f*x*tan(f*x)^4*tan(
e)^4 - 12*A*b^2*c*f*x*tan(f*x)^4*tan(e)^4 + 12*C*b^2*c*f*x*tan(f*x)^4*tan(e)^4 - 12*B*a^2*d*f*x*tan(f*x)^4*tan
(e)^4 - 24*A*a*b*d*f*x*tan(f*x)^4*tan(e)^4 + 24*C*a*b*d*f*x*tan(f*x)^4*tan(e)^4 + 12*B*b^2*d*f*x*tan(f*x)^4*ta
n(e)^4 - 6*B*a^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan
(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 12*A*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*t
an(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 12*C*a
*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) +
 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 6*B*b^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x
)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 6*A*a^2*d*log(4*(tan(
f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 +
1))*tan(f*x)^4*tan(e)^4 + 6*C*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + t
an(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 + 12*B*a*b*d*log(4*(tan(f*x)^4*tan(e)^2
 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*
tan(e)^4 + 6*A*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*t
an(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 6*C*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*
tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^4*tan(e)^4 - 48*A*
a^2*c*f*x*tan(f*x)^3*tan(e)^3 + 48*C*a^2*c*f*x*tan(f*x)^3*tan(e)^3 + 96*B*a*b*c*f*x*tan(f*x)^3*tan(e)^3 + 48*A
*b^2*c*f*x*tan(f*x)^3*tan(e)^3 - 48*C*b^2*c*f*x*tan(f*x)^3*tan(e)^3 + 48*B*a^2*d*f*x*tan(f*x)^3*tan(e)^3 + 96*
A*a*b*d*f*x*tan(f*x)^3*tan(e)^3 - 96*C*a*b*d*f*x*tan(f*x)^3*tan(e)^3 - 48*B*b^2*d*f*x*tan(f*x)^3*tan(e)^3 + 12
*C*a*b*c*tan(f*x)^4*tan(e)^4 + 6*B*b^2*c*tan(f*x)^4*tan(e)^4 + 6*C*a^2*d*tan(f*x)^4*tan(e)^4 + 12*B*a*b*d*tan(
f*x)^4*tan(e)^4 + 6*A*b^2*d*tan(f*x)^4*tan(e)^4 - 9*C*b^2*d*tan(f*x)^4*tan(e)^4 + 24*B*a^2*c*log(4*(tan(f*x)^4
*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*ta
n(f*x)^3*tan(e)^3 + 48*A*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*
x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 48*C*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*
tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e
)^3 - 24*B*b^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f
*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 + 24*A*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan
(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 24*C*a^2
*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1
)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 48*B*a*b*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)
^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 24*A*b^2*d*log(4*(tan(
f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 +
1))*tan(f*x)^3*tan(e)^3 + 24*C*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 +
tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^3*tan(e)^3 - 12*C*a^2*c*tan(f*x)^4*tan(e)^3 - 24*
B*a*b*c*tan(f*x)^4*tan(e)^3 - 12*A*b^2*c*tan(f*x)^4*tan(e)^3 + 12*C*b^2*c*tan(f*x)^4*tan(e)^3 - 12*B*a^2*d*tan
(f*x)^4*tan(e)^3 - 24*A*a*b*d*tan(f*x)^4*tan(e)^3 + 24*C*a*b*d*tan(f*x)^4*tan(e)^3 + 12*B*b^2*d*tan(f*x)^4*tan
(e)^3 - 12*C*a^2*c*tan(f*x)^3*tan(e)^4 - 24*B*a*b*c*tan(f*x)^3*tan(e)^4 - 12*A*b^2*c*tan(f*x)^3*tan(e)^4 + 12*
C*b^2*c*tan(f*x)^3*tan(e)^4 - 12*B*a^2*d*tan(f*x)^3*tan(e)^4 - 24*A*a*b*d*tan(f*x)^3*tan(e)^4 + 24*C*a*b*d*tan
(f*x)^3*tan(e)^4 + 12*B*b^2*d*tan(f*x)^3*tan(e)^4 + 72*A*a^2*c*f*x*tan(f*x)^2*tan(e)^2 - 72*C*a^2*c*f*x*tan(f*
x)^2*tan(e)^2 - 144*B*a*b*c*f*x*tan(f*x)^2*tan(e)^2 - 72*A*b^2*c*f*x*tan(f*x)^2*tan(e)^2 + 72*C*b^2*c*f*x*tan(
f*x)^2*tan(e)^2 - 72*B*a^2*d*f*x*tan(f*x)^2*tan(e)^2 - 144*A*a*b*d*f*x*tan(f*x)^2*tan(e)^2 + 144*C*a*b*d*f*x*t
an(f*x)^2*tan(e)^2 + 72*B*b^2*d*f*x*tan(f*x)^2*tan(e)^2 + 12*C*a*b*c*tan(f*x)^4*tan(e)^2 + 6*B*b^2*c*tan(f*x)^
4*tan(e)^2 + 6*C*a^2*d*tan(f*x)^4*tan(e)^2 + 12*B*a*b*d*tan(f*x)^4*tan(e)^2 + 6*A*b^2*d*tan(f*x)^4*tan(e)^2 -
6*C*b^2*d*tan(f*x)^4*tan(e)^2 - 24*C*a*b*c*tan(f*x)^3*tan(e)^3 - 12*B*b^2*c*tan(f*x)^3*tan(e)^3 - 12*C*a^2*d*t
an(f*x)^3*tan(e)^3 - 24*B*a*b*d*tan(f*x)^3*tan(e)^3 - 12*A*b^2*d*tan(f*x)^3*tan(e)^3 + 24*C*b^2*d*tan(f*x)^3*t
an(e)^3 + 12*C*a*b*c*tan(f*x)^2*tan(e)^4 + 6*B*b^2*c*tan(f*x)^2*tan(e)^4 + 6*C*a^2*d*tan(f*x)^2*tan(e)^4 + 12*
B*a*b*d*tan(f*x)^2*tan(e)^4 + 6*A*b^2*d*tan(f*x)^2*tan(e)^4 - 6*C*b^2*d*tan(f*x)^2*tan(e)^4 - 4*C*b^2*c*tan(f*
x)^4*tan(e) - 8*C*a*b*d*tan(f*x)^4*tan(e) - 4*B*b^2*d*tan(f*x)^4*tan(e) - 36*B*a^2*c*log(4*(tan(f*x)^4*tan(e)^
2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2
*tan(e)^2 - 72*A*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2
*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 72*C*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)
^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 36
*B*b^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(
e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 - 36*A*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + ta
n(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 36*C*a^2*d*log(4
*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e
)^2 + 1))*tan(f*x)^2*tan(e)^2 + 72*B*a*b*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e
)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 36*A*b^2*d*log(4*(tan(f*x)^4*t
an(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(
f*x)^2*tan(e)^2 - 36*C*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)
^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)^2*tan(e)^2 + 36*C*a^2*c*tan(f*x)^3*tan(e)^2 + 72*B*a*b*c*
tan(f*x)^3*tan(e)^2 + 36*A*b^2*c*tan(f*x)^3*tan(e)^2 - 48*C*b^2*c*tan(f*x)^3*tan(e)^2 + 36*B*a^2*d*tan(f*x)^3*
tan(e)^2 + 72*A*a*b*d*tan(f*x)^3*tan(e)^2 - 96*C*a*b*d*tan(f*x)^3*tan(e)^2 - 48*B*b^2*d*tan(f*x)^3*tan(e)^2 +
36*C*a^2*c*tan(f*x)^2*tan(e)^3 + 72*B*a*b*c*tan(f*x)^2*tan(e)^3 + 36*A*b^2*c*tan(f*x)^2*tan(e)^3 - 48*C*b^2*c*
tan(f*x)^2*tan(e)^3 + 36*B*a^2*d*tan(f*x)^2*tan(e)^3 + 72*A*a*b*d*tan(f*x)^2*tan(e)^3 - 96*C*a*b*d*tan(f*x)^2*
tan(e)^3 - 48*B*b^2*d*tan(f*x)^2*tan(e)^3 - 4*C*b^2*c*tan(f*x)*tan(e)^4 - 8*C*a*b*d*tan(f*x)*tan(e)^4 - 4*B*b^
2*d*tan(f*x)*tan(e)^4 + 3*C*b^2*d*tan(f*x)^4 - 48*A*a^2*c*f*x*tan(f*x)*tan(e) + 48*C*a^2*c*f*x*tan(f*x)*tan(e)
 + 96*B*a*b*c*f*x*tan(f*x)*tan(e) + 48*A*b^2*c*f*x*tan(f*x)*tan(e) - 48*C*b^2*c*f*x*tan(f*x)*tan(e) + 48*B*a^2
*d*f*x*tan(f*x)*tan(e) + 96*A*a*b*d*f*x*tan(f*x)*tan(e) - 96*C*a*b*d*f*x*tan(f*x)*tan(e) - 48*B*b^2*d*f*x*tan(
f*x)*tan(e) - 24*C*a*b*c*tan(f*x)^3*tan(e) - 12*B*b^2*c*tan(f*x)^3*tan(e) - 12*C*a^2*d*tan(f*x)^3*tan(e) - 24*
B*a*b*d*tan(f*x)^3*tan(e) - 12*A*b^2*d*tan(f*x)^3*tan(e) + 24*C*b^2*d*tan(f*x)^3*tan(e) + 24*C*a*b*c*tan(f*x)^
2*tan(e)^2 + 12*B*b^2*c*tan(f*x)^2*tan(e)^2 + 12*C*a^2*d*tan(f*x)^2*tan(e)^2 + 24*B*a*b*d*tan(f*x)^2*tan(e)^2
+ 12*A*b^2*d*tan(f*x)^2*tan(e)^2 - 12*C*b^2*d*tan(f*x)^2*tan(e)^2 - 24*C*a*b*c*tan(f*x)*tan(e)^3 - 12*B*b^2*c*
tan(f*x)*tan(e)^3 - 12*C*a^2*d*tan(f*x)*tan(e)^3 - 24*B*a*b*d*tan(f*x)*tan(e)^3 - 12*A*b^2*d*tan(f*x)*tan(e)^3
 + 24*C*b^2*d*tan(f*x)*tan(e)^3 + 3*C*b^2*d*tan(e)^4 + 4*C*b^2*c*tan(f*x)^3 + 8*C*a*b*d*tan(f*x)^3 + 4*B*b^2*d
*tan(f*x)^3 + 24*B*a^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 -
 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) + 48*A*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3
*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) - 48*C*a*b
*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1
)/(tan(e)^2 + 1))*tan(f*x)*tan(e) - 24*B*b^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*t
an(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) + 24*A*a^2*d*log(4*(tan(f*x)^4*t
an(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(
f*x)*tan(e) - 24*C*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 -
 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) - 48*B*a*b*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3
*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) - 24*A*b^2
*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1
)/(tan(e)^2 + 1))*tan(f*x)*tan(e) + 24*C*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*t
an(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1))*tan(f*x)*tan(e) - 36*C*a^2*c*tan(f*x)^2*tan(e) -
 72*B*a*b*c*tan(f*x)^2*tan(e) - 36*A*b^2*c*tan(f*x)^2*tan(e) + 48*C*b^2*c*tan(f*x)^2*tan(e) - 36*B*a^2*d*tan(f
*x)^2*tan(e) - 72*A*a*b*d*tan(f*x)^2*tan(e) + 96*C*a*b*d*tan(f*x)^2*tan(e) + 48*B*b^2*d*tan(f*x)^2*tan(e) - 36
*C*a^2*c*tan(f*x)*tan(e)^2 - 72*B*a*b*c*tan(f*x)*tan(e)^2 - 36*A*b^2*c*tan(f*x)*tan(e)^2 + 48*C*b^2*c*tan(f*x)
*tan(e)^2 - 36*B*a^2*d*tan(f*x)*tan(e)^2 - 72*A*a*b*d*tan(f*x)*tan(e)^2 + 96*C*a*b*d*tan(f*x)*tan(e)^2 + 48*B*
b^2*d*tan(f*x)*tan(e)^2 + 4*C*b^2*c*tan(e)^3 + 8*C*a*b*d*tan(e)^3 + 4*B*b^2*d*tan(e)^3 + 12*A*a^2*c*f*x - 12*C
*a^2*c*f*x - 24*B*a*b*c*f*x - 12*A*b^2*c*f*x + 12*C*b^2*c*f*x - 12*B*a^2*d*f*x - 24*A*a*b*d*f*x + 24*C*a*b*d*f
*x + 12*B*b^2*d*f*x + 12*C*a*b*c*tan(f*x)^2 + 6*B*b^2*c*tan(f*x)^2 + 6*C*a^2*d*tan(f*x)^2 + 12*B*a*b*d*tan(f*x
)^2 + 6*A*b^2*d*tan(f*x)^2 - 6*C*b^2*d*tan(f*x)^2 - 24*C*a*b*c*tan(f*x)*tan(e) - 12*B*b^2*c*tan(f*x)*tan(e) -
12*C*a^2*d*tan(f*x)*tan(e) - 24*B*a*b*d*tan(f*x)*tan(e) - 12*A*b^2*d*tan(f*x)*tan(e) + 24*C*b^2*d*tan(f*x)*tan
(e) + 12*C*a*b*c*tan(e)^2 + 6*B*b^2*c*tan(e)^2 + 6*C*a^2*d*tan(e)^2 + 12*B*a*b*d*tan(e)^2 + 6*A*b^2*d*tan(e)^2
 - 6*C*b^2*d*tan(e)^2 - 6*B*a^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan
(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) - 12*A*a*b*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e)
 + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 12*C*a*b*c*log(4*(tan(f*x)^4*ta
n(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 6*B
*b^2*c*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e)
 + 1)/(tan(e)^2 + 1)) - 6*A*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan
(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 6*C*a^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e)
+ tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 12*B*a*b*d*log(4*(tan(f*x)^4*tan
(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 6*A*
b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(f*x)^2 - 2*tan(f*x)*tan(e)
+ 1)/(tan(e)^2 + 1)) - 6*C*b^2*d*log(4*(tan(f*x)^4*tan(e)^2 - 2*tan(f*x)^3*tan(e) + tan(f*x)^2*tan(e)^2 + tan(
f*x)^2 - 2*tan(f*x)*tan(e) + 1)/(tan(e)^2 + 1)) + 12*C*a^2*c*tan(f*x) + 24*B*a*b*c*tan(f*x) + 12*A*b^2*c*tan(f
*x) - 12*C*b^2*c*tan(f*x) + 12*B*a^2*d*tan(f*x) + 24*A*a*b*d*tan(f*x) - 24*C*a*b*d*tan(f*x) - 12*B*b^2*d*tan(f
*x) + 12*C*a^2*c*tan(e) + 24*B*a*b*c*tan(e) + 12*A*b^2*c*tan(e) - 12*C*b^2*c*tan(e) + 12*B*a^2*d*tan(e) + 24*A
*a*b*d*tan(e) - 24*C*a*b*d*tan(e) - 12*B*b^2*d*tan(e) + 12*C*a*b*c + 6*B*b^2*c + 6*C*a^2*d + 12*B*a*b*d + 6*A*
b^2*d - 9*C*b^2*d)/(f*tan(f*x)^4*tan(e)^4 - 4*f*tan(f*x)^3*tan(e)^3 + 6*f*tan(f*x)^2*tan(e)^2 - 4*f*tan(f*x)*t
an(e) + f)

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maple [B]  time = 0.03, size = 631, normalized size = 2.54 \[ -\frac {C \arctan \left (\tan \left (f x +e \right )\right ) a^{2} c}{f}+\frac {C \arctan \left (\tan \left (f x +e \right )\right ) b^{2} c}{f}+\frac {2 A a b d \tan \left (f x +e \right )}{f}+\frac {B \left (\tan ^{3}\left (f x +e \right )\right ) b^{2} d}{3 f}+\frac {C \left (\tan ^{3}\left (f x +e \right )\right ) b^{2} c}{3 f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) C \,a^{2} d}{2 f}-\frac {B \,b^{2} d \tan \left (f x +e \right )}{f}+\frac {B \,a^{2} d \tan \left (f x +e \right )}{f}+\frac {A \,b^{2} c \tan \left (f x +e \right )}{f}-\frac {B \arctan \left (\tan \left (f x +e \right )\right ) a^{2} d}{f}+\frac {B \arctan \left (\tan \left (f x +e \right )\right ) b^{2} d}{f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) B \,b^{2} c}{2 f}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) B \,a^{2} c}{2 f}+\frac {C \left (\tan ^{2}\left (f x +e \right )\right ) a^{2} d}{2 f}-\frac {C \,b^{2} c \tan \left (f x +e \right )}{f}+\frac {C \,a^{2} c \tan \left (f x +e \right )}{f}+\frac {A \arctan \left (\tan \left (f x +e \right )\right ) a^{2} c}{f}+\frac {A \left (\tan ^{2}\left (f x +e \right )\right ) b^{2} d}{2 f}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right ) b^{2} c}{2 f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) A \,b^{2} d}{2 f}-\frac {A \arctan \left (\tan \left (f x +e \right )\right ) b^{2} c}{f}+\frac {C \,b^{2} d \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) A \,a^{2} d}{2 f}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) C \,b^{2} d}{2 f}-\frac {C \left (\tan ^{2}\left (f x +e \right )\right ) b^{2} d}{2 f}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right ) a b d}{f}-\frac {2 A \arctan \left (\tan \left (f x +e \right )\right ) a b d}{f}-\frac {2 C a b d \tan \left (f x +e \right )}{f}+\frac {2 C \left (\tan ^{3}\left (f x +e \right )\right ) a b d}{3 f}+\frac {2 C \arctan \left (\tan \left (f x +e \right )\right ) a b d}{f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) C a b c}{f}+\frac {C \left (\tan ^{2}\left (f x +e \right )\right ) a b c}{f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) B a b d}{f}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right ) A a b c}{f}+\frac {2 B a b c \tan \left (f x +e \right )}{f}-\frac {2 B \arctan \left (\tan \left (f x +e \right )\right ) a b c}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.57, size = 274, normalized size = 1.10 \[ \frac {3 \, C b^{2} d \tan \left (f x + e\right )^{4} + 4 \, {\left (C b^{2} c + {\left (2 \, C a b + B b^{2}\right )} d\right )} \tan \left (f x + e\right )^{3} + 6 \, {\left ({\left (2 \, C a b + B b^{2}\right )} c + {\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} d\right )} \tan \left (f x + e\right )^{2} + 12 \, {\left ({\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} c - {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d\right )} {\left (f x + e\right )} + 6 \, {\left ({\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} c + {\left ({\left (A - C\right )} a^{2} - 2 \, B a b - {\left (A - C\right )} b^{2}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 12 \, {\left ({\left (C a^{2} + 2 \, B a b + {\left (A - C\right )} b^{2}\right )} c + {\left (B a^{2} + 2 \, {\left (A - C\right )} a b - B b^{2}\right )} d\right )} \tan \left (f x + e\right )}{12 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*C*b^2*d*tan(f*x + e)^4 + 4*(C*b^2*c + (2*C*a*b + B*b^2)*d)*tan(f*x + e)^3 + 6*((2*C*a*b + B*b^2)*c + (
C*a^2 + 2*B*a*b + (A - C)*b^2)*d)*tan(f*x + e)^2 + 12*(((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)*c - (B*a^2 + 2*(A
 - C)*a*b - B*b^2)*d)*(f*x + e) + 6*((B*a^2 + 2*(A - C)*a*b - B*b^2)*c + ((A - C)*a^2 - 2*B*a*b - (A - C)*b^2)
*d)*log(tan(f*x + e)^2 + 1) + 12*((C*a^2 + 2*B*a*b + (A - C)*b^2)*c + (B*a^2 + 2*(A - C)*a*b - B*b^2)*d)*tan(f
*x + e))/f

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mupad [B]  time = 8.98, size = 300, normalized size = 1.21 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,b^2\,d}{2}+\frac {B\,b^2\,c}{2}+\frac {C\,a^2\,d}{2}-\frac {C\,b^2\,d}{2}+B\,a\,b\,d+C\,a\,b\,c\right )}{f}-x\,\left (A\,b^2\,c-A\,a^2\,c+B\,a^2\,d+C\,a^2\,c-B\,b^2\,d-C\,b^2\,c+2\,A\,a\,b\,d+2\,B\,a\,b\,c-2\,C\,a\,b\,d\right )-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,b^2\,d}{2}-\frac {B\,a^2\,c}{2}-\frac {A\,a^2\,d}{2}+\frac {B\,b^2\,c}{2}+\frac {C\,a^2\,d}{2}-\frac {C\,b^2\,d}{2}-A\,a\,b\,c+B\,a\,b\,d+C\,a\,b\,c\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (A\,b^2\,c+B\,a^2\,d+C\,a^2\,c-B\,b^2\,d-C\,b^2\,c+2\,A\,a\,b\,d+2\,B\,a\,b\,c-2\,C\,a\,b\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {B\,b^2\,d}{3}+\frac {C\,b^2\,c}{3}+\frac {2\,C\,a\,b\,d}{3}\right )}{f}+\frac {C\,b^2\,d\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4\,f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.98, size = 617, normalized size = 2.49 \[ \begin {cases} A a^{2} c x + \frac {A a^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {A a b c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - 2 A a b d x + \frac {2 A a b d \tan {\left (e + f x \right )}}{f} - A b^{2} c x + \frac {A b^{2} c \tan {\left (e + f x \right )}}{f} - \frac {A b^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {A b^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac {B a^{2} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} - B a^{2} d x + \frac {B a^{2} d \tan {\left (e + f x \right )}}{f} - 2 B a b c x + \frac {2 B a b c \tan {\left (e + f x \right )}}{f} - \frac {B a b d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {B a b d \tan ^{2}{\left (e + f x \right )}}{f} - \frac {B b^{2} c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {B b^{2} c \tan ^{2}{\left (e + f x \right )}}{2 f} + B b^{2} d x + \frac {B b^{2} d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {B b^{2} d \tan {\left (e + f x \right )}}{f} - C a^{2} c x + \frac {C a^{2} c \tan {\left (e + f x \right )}}{f} - \frac {C a^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C a^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} - \frac {C a b c \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} + \frac {C a b c \tan ^{2}{\left (e + f x \right )}}{f} + 2 C a b d x + \frac {2 C a b d \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 C a b d \tan {\left (e + f x \right )}}{f} + C b^{2} c x + \frac {C b^{2} c \tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {C b^{2} c \tan {\left (e + f x \right )}}{f} + \frac {C b^{2} d \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {C b^{2} d \tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {C b^{2} d \tan ^{2}{\left (e + f x \right )}}{2 f} & \text {for}\: f \neq 0 \\x \left (a + b \tan {\relax (e )}\right )^{2} \left (c + d \tan {\relax (e )}\right ) \left (A + B \tan {\relax (e )} + C \tan ^{2}{\relax (e )}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*a**2*c*x + A*a**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + A*a*b*c*log(tan(e + f*x)**2 + 1)/f - 2*A*a*b
*d*x + 2*A*a*b*d*tan(e + f*x)/f - A*b**2*c*x + A*b**2*c*tan(e + f*x)/f - A*b**2*d*log(tan(e + f*x)**2 + 1)/(2*
f) + A*b**2*d*tan(e + f*x)**2/(2*f) + B*a**2*c*log(tan(e + f*x)**2 + 1)/(2*f) - B*a**2*d*x + B*a**2*d*tan(e +
f*x)/f - 2*B*a*b*c*x + 2*B*a*b*c*tan(e + f*x)/f - B*a*b*d*log(tan(e + f*x)**2 + 1)/f + B*a*b*d*tan(e + f*x)**2
/f - B*b**2*c*log(tan(e + f*x)**2 + 1)/(2*f) + B*b**2*c*tan(e + f*x)**2/(2*f) + B*b**2*d*x + B*b**2*d*tan(e +
f*x)**3/(3*f) - B*b**2*d*tan(e + f*x)/f - C*a**2*c*x + C*a**2*c*tan(e + f*x)/f - C*a**2*d*log(tan(e + f*x)**2
+ 1)/(2*f) + C*a**2*d*tan(e + f*x)**2/(2*f) - C*a*b*c*log(tan(e + f*x)**2 + 1)/f + C*a*b*c*tan(e + f*x)**2/f +
 2*C*a*b*d*x + 2*C*a*b*d*tan(e + f*x)**3/(3*f) - 2*C*a*b*d*tan(e + f*x)/f + C*b**2*c*x + C*b**2*c*tan(e + f*x)
**3/(3*f) - C*b**2*c*tan(e + f*x)/f + C*b**2*d*log(tan(e + f*x)**2 + 1)/(2*f) + C*b**2*d*tan(e + f*x)**4/(4*f)
 - C*b**2*d*tan(e + f*x)**2/(2*f), Ne(f, 0)), (x*(a + b*tan(e))**2*(c + d*tan(e))*(A + B*tan(e) + C*tan(e)**2)
, True))

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