3.1.51 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [51]

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

[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 = 0.31, 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 = {3718, 3711, 3609, 3606, 3556} \begin {gather*} -\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} \end {gather*}

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 3556

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

Rule 3606

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 3609

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 3711

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 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]

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] Result contains complex when optimal does not.
time = 2.27, size = 243, normalized size = 0.98 \begin {gather*} \frac {\frac {(-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) \left (i \left ((a+i b)^2 \log (i-\tan (e+f x))-(a-i b)^2 \log (i+\tan (e+f x))\right )-2 b^2 \tan (e+f x)\right )+6 (B c+(A-C) d) \left ((i a-b)^3 \log (i-\tan (e+f x))-(i a+b)^3 \log (i+\tan (e+f x))+6 a b^2 \tan (e+f x)+b^3 \tan ^2(e+f x)\right )}{12 b f} \end {gather*}

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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Maple [A]
time = 0.12, size = 386, normalized size = 1.56

method	result	size
norman	\(\left (A \,a^{2} c -2 A a b d -A \,b^{2} c -B \,a^{2} d -2 B a b c +B \,b^{2} d -C \,a^{2} c +2 C a b d +C \,b^{2} c \right ) x +\frac {\left (2 A a b d +A \,b^{2} c +B \,a^{2} d +2 B a b c -B \,b^{2} d +C \,a^{2} c -2 C a b d -C \,b^{2} c \right ) \tan \left (f x +e \right )}{f}+\frac {\left (A \,b^{2} d +2 B a b d +B \,b^{2} c +C \,a^{2} d +2 C a b c -C \,b^{2} d \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {b \left (B b d +2 a C d +C b c \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {C \,b^{2} d \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {\left (A \,a^{2} d +2 A a b c -A \,b^{2} d +B \,a^{2} c -2 B a b d -B \,b^{2} c -C \,a^{2} d -2 C a b c +C \,b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}\)	\(294\)
derivativedivides	\(\frac {\frac {C \,b^{2} d \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {B \,b^{2} d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {2 C a b d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {C \,b^{2} c \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {A \,b^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+B a b d \left (\tan ^{2}\left (f x +e \right )\right )+\frac {B \,b^{2} c \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {C \,a^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+C a b c \left (\tan ^{2}\left (f x +e \right )\right )-\frac {C \,b^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+2 A a b d \tan \left (f x +e \right )+A \,b^{2} c \tan \left (f x +e \right )+B \,a^{2} d \tan \left (f x +e \right )+2 B a b c \tan \left (f x +e \right )-B \,b^{2} d \tan \left (f x +e \right )+C \,a^{2} c \tan \left (f x +e \right )-2 C a b d \tan \left (f x +e \right )-C \,b^{2} c \tan \left (f x +e \right )+\frac {\left (A \,a^{2} d +2 A a b c -A \,b^{2} d +B \,a^{2} c -2 B a b d -B \,b^{2} c -C \,a^{2} d -2 C a b c +C \,b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{2} c -2 A a b d -A \,b^{2} c -B \,a^{2} d -2 B a b c +B \,b^{2} d -C \,a^{2} c +2 C a b d +C \,b^{2} c \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\)	\(386\)
default	\(\frac {\frac {C \,b^{2} d \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {B \,b^{2} d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {2 C a b d \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {C \,b^{2} c \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {A \,b^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+B a b d \left (\tan ^{2}\left (f x +e \right )\right )+\frac {B \,b^{2} c \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {C \,a^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+C a b c \left (\tan ^{2}\left (f x +e \right )\right )-\frac {C \,b^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+2 A a b d \tan \left (f x +e \right )+A \,b^{2} c \tan \left (f x +e \right )+B \,a^{2} d \tan \left (f x +e \right )+2 B a b c \tan \left (f x +e \right )-B \,b^{2} d \tan \left (f x +e \right )+C \,a^{2} c \tan \left (f x +e \right )-2 C a b d \tan \left (f x +e \right )-C \,b^{2} c \tan \left (f x +e \right )+\frac {\left (A \,a^{2} d +2 A a b c -A \,b^{2} d +B \,a^{2} c -2 B a b d -B \,b^{2} c -C \,a^{2} d -2 C a b c +C \,b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (A \,a^{2} c -2 A a b d -A \,b^{2} c -B \,a^{2} d -2 B a b c +B \,b^{2} d -C \,a^{2} c +2 C a b d +C \,b^{2} c \right ) \arctan \left (\tan \left (f x +e \right )\right )}{f}\)	\(386\)
risch	\(\text {Expression too large to display}\)	\(1191\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 280, normalized size = 1.13 \begin {gather*} \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} \end {gather*}

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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Fricas [A]
time = 3.35, size = 278, normalized size = 1.12 \begin {gather*} \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} \end {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (218) = 436\).
time = 0.23, size = 617, normalized size = 2.49 \begin {gather*} \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 {\left (e \right )}\right )^{2} \left (c + d \tan {\left (e \right )}\right ) \left (A + B \tan {\left (e \right )} + C \tan ^{2}{\left (e \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6502 vs. \(2 (248) = 496\).
time = 3.76, size = 6502, normalized size = 26.22 \begin {gather*} \text {Too large to display} \end {gather*}

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*...

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Mupad [B]
time = 8.98, size = 300, normalized size = 1.21 \begin {gather*} \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} \end {gather*}

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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