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

Optimal. Leaf size=156 \[ -\frac{(b c-a d) \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^2 f \left (c^2+d^2\right )}-\frac{\log (\cos (e+f x)) (-a A d+a B c+a C d+A b c+b B d-b c C)}{f \left (c^2+d^2\right )}+\frac{x (a (A c+B d-c C)-b (B c-d (A-C)))}{c^2+d^2}+\frac{b C \tan (e+f x)}{d f} \]

[Out]

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

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Rubi [A]  time = 0.341521, antiderivative size = 156, normalized size of antiderivative = 1., 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, 3626, 3617, 31, 3475} \[ -\frac{(b c-a d) \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^2 f \left (c^2+d^2\right )}-\frac{\log (\cos (e+f x)) (-a A d+a B c+a C d+A b c+b B d-b c C)}{f \left (c^2+d^2\right )}+\frac{x (a (A c+B d-c C)-b (B c-d (A-C)))}{c^2+d^2}+\frac{b C \tan (e+f x)}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [C]  time = 1.05483, size = 148, normalized size = 0.95 \[ \frac{\frac{2 (a d-b c) \left (A d^2-B c d+c^2 C\right ) \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )}+\frac{(b-i a) (A+i B-C) \log (-\tan (e+f x)+i)}{c+i d}+\frac{(b+i a) (A-i B-C) \log (\tan (e+f x)+i)}{c-i d}+\frac{2 b C \tan (e+f x)}{d}}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.05, size = 506, normalized size = 3.2 \begin{align*}{\frac{Cb\tan \left ( fx+e \right ) }{fd}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) Aad}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) Abc}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) Bac}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) Bbd}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) aCd}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) Cbc}{2\,f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{A\arctan \left ( \tan \left ( fx+e \right ) \right ) ac}{f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{A\arctan \left ( \tan \left ( fx+e \right ) \right ) bd}{f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{B\arctan \left ( \tan \left ( fx+e \right ) \right ) ad}{f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{B\arctan \left ( \tan \left ( fx+e \right ) \right ) bc}{f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{C\arctan \left ( \tan \left ( fx+e \right ) \right ) ac}{f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{C\arctan \left ( \tan \left ( fx+e \right ) \right ) bd}{f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{d\ln \left ( c+d\tan \left ( fx+e \right ) \right ) Aa}{f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) Abc}{f \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) Bac}{f \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) B{c}^{2}b}{fd \left ({c}^{2}+{d}^{2} \right ) }}+{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) C{c}^{2}a}{fd \left ({c}^{2}+{d}^{2} \right ) }}-{\frac{\ln \left ( c+d\tan \left ( fx+e \right ) \right ) C{c}^{3}b}{{d}^{2}f \left ({c}^{2}+{d}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48502, size = 240, normalized size = 1.54 \begin{align*} \frac{\frac{2 \, C b \tan \left (f x + e\right )}{d} + \frac{2 \,{\left ({\left ({\left (A - C\right )} a - B b\right )} c +{\left (B a +{\left (A - C\right )} b\right )} d\right )}{\left (f x + e\right )}}{c^{2} + d^{2}} - \frac{2 \,{\left (C b c^{3} - A a d^{3} -{\left (C a + B b\right )} c^{2} d +{\left (B a + A b\right )} c d^{2}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{2} d^{2} + d^{4}} + \frac{{\left ({\left (B a +{\left (A - C\right )} b\right )} c -{\left ({\left (A - C\right )} a - B b\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59051, size = 470, normalized size = 3.01 \begin{align*} \frac{2 \,{\left ({\left ({\left (A - C\right )} a - B b\right )} c d^{2} +{\left (B a +{\left (A - C\right )} b\right )} d^{3}\right )} f x -{\left (C b c^{3} - A a d^{3} -{\left (C a + B b\right )} c^{2} d +{\left (B a + A b\right )} c d^{2}\right )} \log \left (\frac{d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) +{\left (C b c^{3} + C b c d^{2} -{\left (C a + B b\right )} c^{2} d -{\left (C a + B b\right )} d^{3}\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (C b c^{2} d + C b d^{3}\right )} \tan \left (f x + e\right )}{2 \,{\left (c^{2} d^{2} + d^{4}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 23.7434, size = 2387, normalized size = 15.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*(a + b*tan(e))*(A + B*tan(e) + C*tan(e)**2)/tan(e), Eq(c, 0) & Eq(d, 0) & Eq(f, 0)), (-I*A*a*
f*x*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) - A*a*f*x/(-2*d*f*tan(e + f*x) + 2*I*d*f) - I*A*a/(-2*d*f*tan
(e + f*x) + 2*I*d*f) - A*b*f*x*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) + I*A*b*f*x/(-2*d*f*tan(e + f*x) +
2*I*d*f) + A*b/(-2*d*f*tan(e + f*x) + 2*I*d*f) - B*a*f*x*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) + I*B*a
*f*x/(-2*d*f*tan(e + f*x) + 2*I*d*f) + B*a/(-2*d*f*tan(e + f*x) + 2*I*d*f) - I*B*b*f*x*tan(e + f*x)/(-2*d*f*ta
n(e + f*x) + 2*I*d*f) - B*b*f*x/(-2*d*f*tan(e + f*x) + 2*I*d*f) - B*b*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-
2*d*f*tan(e + f*x) + 2*I*d*f) + I*B*b*log(tan(e + f*x)**2 + 1)/(-2*d*f*tan(e + f*x) + 2*I*d*f) + I*B*b/(-2*d*f
*tan(e + f*x) + 2*I*d*f) - I*C*a*f*x*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) - C*a*f*x/(-2*d*f*tan(e + f*
x) + 2*I*d*f) - C*a*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) + I*C*a*log(tan(e +
f*x)**2 + 1)/(-2*d*f*tan(e + f*x) + 2*I*d*f) + I*C*a/(-2*d*f*tan(e + f*x) + 2*I*d*f) + 3*C*b*f*x*tan(e + f*x)/
(-2*d*f*tan(e + f*x) + 2*I*d*f) - 3*I*C*b*f*x/(-2*d*f*tan(e + f*x) + 2*I*d*f) - I*C*b*log(tan(e + f*x)**2 + 1)
*tan(e + f*x)/(-2*d*f*tan(e + f*x) + 2*I*d*f) - C*b*log(tan(e + f*x)**2 + 1)/(-2*d*f*tan(e + f*x) + 2*I*d*f) -
2*C*b*tan(e + f*x)**2/(-2*d*f*tan(e + f*x) + 2*I*d*f) - 3*C*b/(-2*d*f*tan(e + f*x) + 2*I*d*f), Eq(c, -I*d)),
(-I*A*a*f*x*tan(e + f*x)/(2*d*f*tan(e + f*x) + 2*I*d*f) + A*a*f*x/(2*d*f*tan(e + f*x) + 2*I*d*f) - I*A*a/(2*d*
f*tan(e + f*x) + 2*I*d*f) + A*b*f*x*tan(e + f*x)/(2*d*f*tan(e + f*x) + 2*I*d*f) + I*A*b*f*x/(2*d*f*tan(e + f*x
) + 2*I*d*f) - A*b/(2*d*f*tan(e + f*x) + 2*I*d*f) + B*a*f*x*tan(e + f*x)/(2*d*f*tan(e + f*x) + 2*I*d*f) + I*B*
a*f*x/(2*d*f*tan(e + f*x) + 2*I*d*f) - B*a/(2*d*f*tan(e + f*x) + 2*I*d*f) - I*B*b*f*x*tan(e + f*x)/(2*d*f*tan(
e + f*x) + 2*I*d*f) + B*b*f*x/(2*d*f*tan(e + f*x) + 2*I*d*f) + B*b*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*d*
f*tan(e + f*x) + 2*I*d*f) + I*B*b*log(tan(e + f*x)**2 + 1)/(2*d*f*tan(e + f*x) + 2*I*d*f) + I*B*b/(2*d*f*tan(e
+ f*x) + 2*I*d*f) - I*C*a*f*x*tan(e + f*x)/(2*d*f*tan(e + f*x) + 2*I*d*f) + C*a*f*x/(2*d*f*tan(e + f*x) + 2*I
*d*f) + C*a*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*d*f*tan(e + f*x) + 2*I*d*f) + I*C*a*log(tan(e + f*x)**2 +
1)/(2*d*f*tan(e + f*x) + 2*I*d*f) + I*C*a/(2*d*f*tan(e + f*x) + 2*I*d*f) - 3*C*b*f*x*tan(e + f*x)/(2*d*f*tan(
e + f*x) + 2*I*d*f) - 3*I*C*b*f*x/(2*d*f*tan(e + f*x) + 2*I*d*f) - I*C*b*log(tan(e + f*x)**2 + 1)*tan(e + f*x)
/(2*d*f*tan(e + f*x) + 2*I*d*f) + C*b*log(tan(e + f*x)**2 + 1)/(2*d*f*tan(e + f*x) + 2*I*d*f) + 2*C*b*tan(e +
f*x)**2/(2*d*f*tan(e + f*x) + 2*I*d*f) + 3*C*b/(2*d*f*tan(e + f*x) + 2*I*d*f), Eq(c, I*d)), ((A*a*x + A*b*log(
tan(e + f*x)**2 + 1)/(2*f) + B*a*log(tan(e + f*x)**2 + 1)/(2*f) - B*b*x + B*b*tan(e + f*x)/f - C*a*x + C*a*tan
(e + f*x)/f - C*b*log(tan(e + f*x)**2 + 1)/(2*f) + C*b*tan(e + f*x)**2/(2*f))/c, Eq(d, 0)), (x*(a + b*tan(e))*
(A + B*tan(e) + C*tan(e)**2)/(c + d*tan(e)), Eq(f, 0)), (2*A*a*c*d**2*f*x/(2*c**2*d**2*f + 2*d**4*f) + 2*A*a*d
**3*log(c/d + tan(e + f*x))/(2*c**2*d**2*f + 2*d**4*f) - A*a*d**3*log(tan(e + f*x)**2 + 1)/(2*c**2*d**2*f + 2*
d**4*f) - 2*A*b*c*d**2*log(c/d + tan(e + f*x))/(2*c**2*d**2*f + 2*d**4*f) + A*b*c*d**2*log(tan(e + f*x)**2 + 1
)/(2*c**2*d**2*f + 2*d**4*f) + 2*A*b*d**3*f*x/(2*c**2*d**2*f + 2*d**4*f) - 2*B*a*c*d**2*log(c/d + tan(e + f*x)
)/(2*c**2*d**2*f + 2*d**4*f) + B*a*c*d**2*log(tan(e + f*x)**2 + 1)/(2*c**2*d**2*f + 2*d**4*f) + 2*B*a*d**3*f*x
/(2*c**2*d**2*f + 2*d**4*f) + 2*B*b*c**2*d*log(c/d + tan(e + f*x))/(2*c**2*d**2*f + 2*d**4*f) - 2*B*b*c*d**2*f
*x/(2*c**2*d**2*f + 2*d**4*f) + B*b*d**3*log(tan(e + f*x)**2 + 1)/(2*c**2*d**2*f + 2*d**4*f) + 2*C*a*c**2*d*lo
g(c/d + tan(e + f*x))/(2*c**2*d**2*f + 2*d**4*f) - 2*C*a*c*d**2*f*x/(2*c**2*d**2*f + 2*d**4*f) + C*a*d**3*log(
tan(e + f*x)**2 + 1)/(2*c**2*d**2*f + 2*d**4*f) - 2*C*b*c**3*log(c/d + tan(e + f*x))/(2*c**2*d**2*f + 2*d**4*f
) + 2*C*b*c**2*d*tan(e + f*x)/(2*c**2*d**2*f + 2*d**4*f) - C*b*c*d**2*log(tan(e + f*x)**2 + 1)/(2*c**2*d**2*f
+ 2*d**4*f) - 2*C*b*d**3*f*x/(2*c**2*d**2*f + 2*d**4*f) + 2*C*b*d**3*tan(e + f*x)/(2*c**2*d**2*f + 2*d**4*f),
True))

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Giac [A]  time = 1.68285, size = 251, normalized size = 1.61 \begin{align*} \frac{\frac{2 \, C b \tan \left (f x + e\right )}{d} + \frac{2 \,{\left (A a c - C a c - B b c + B a d + A b d - C b d\right )}{\left (f x + e\right )}}{c^{2} + d^{2}} + \frac{{\left (B a c + A b c - C b c - A a d + C a d + B b d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{2} + d^{2}} - \frac{2 \,{\left (C b c^{3} - C a c^{2} d - B b c^{2} d + B a c d^{2} + A b c d^{2} - A a d^{3}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{2} d^{2} + d^{4}}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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