3.56 \(\int \frac{(c+d \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^3} \, dx\)
Optimal. Leaf size=320 \[ -\frac{(b c-a d) \left (A b^2-a (b B-a C)\right )}{2 b^2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)}{b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}+\frac{\left (3 a^2 b (A c-B d-c C)+a^3 (-(d (A-C)+B c))+3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b (d (A-C)+B c)+a^3 (A c-B d-c C)-3 a b^2 (A c-B d-c C)-b^3 (d (A-C)+B c)\right )}{\left (a^2+b^2\right )^3} \]
[Out]
((a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) + 3*a^2*b*(B*c + (A - C)*d) - b^3*(B*c + (A - C)*d))*x)/(a
^2 + b^2)^3 + ((3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) - a^3*(B*c + (A - C)*d) + 3*a*b^2*(B*c + (A
- C)*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((A*b^2 - a*(b*B - a*C))*(b*c - a*d))/(2*b^
2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*
c + (A - 3*C)*d))/(b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))
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Rubi [A] time = 0.703237, antiderivative size = 320, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules used =
{3635, 3628, 3531, 3530} \[ -\frac{(b c-a d) \left (A b^2-a (b B-a C)\right )}{2 b^2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{-a^2 b^2 (d (A-3 C)+B c)+a^4 C d+2 a b^3 (A c-B d-c C)+b^4 (A d+B c)}{b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}+\frac{\left (3 a^2 b (A c-B d-c C)+a^3 (-(d (A-C)+B c))+3 a b^2 (d (A-C)+B c)-b^3 (A c-B d-c C)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b (d (A-C)+B c)+a^3 (A c-B d-c C)-3 a b^2 (A c-B d-c C)-b^3 (d (A-C)+B c)\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
Int[((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]
[Out]
((a^3*(A*c - c*C - B*d) - 3*a*b^2*(A*c - c*C - B*d) + 3*a^2*b*(B*c + (A - C)*d) - b^3*(B*c + (A - C)*d))*x)/(a
^2 + b^2)^3 + ((3*a^2*b*(A*c - c*C - B*d) - b^3*(A*c - c*C - B*d) - a^3*(B*c + (A - C)*d) + 3*a*b^2*(B*c + (A
- C)*d))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((A*b^2 - a*(b*B - a*C))*(b*c - a*d))/(2*b^
2*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2) - (a^4*C*d + b^4*(B*c + A*d) + 2*a*b^3*(A*c - c*C - B*d) - a^2*b^2*(B*
c + (A - 3*C)*d))/(b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x]))
Rule 3635
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 - a*d)*(c^2*C - B*c*d + A*d^2)*
(c + d*Tan[e + f*x])^(n + 1))/(d^2*f*(n + 1)*(c^2 + d^2)), x] + Dist[1/(d*(c^2 + d^2)), Int[(c + d*Tan[e + f*x
])^(n + 1)*Simp[a*d*(A*c - c*C + B*d) + b*(c^2*C - B*c*d + A*d^2) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d +
a*C*d)*Tan[e + f*x] + b*C*(c^2 + d^2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] &&
NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && LtQ[n, -1]
Rule 3628
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]
Rule 3531
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*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] && NeQ[a*c + b*d, 0]
Rule 3530
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) (b c-a d)}{2 b^2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\int \frac{a^2 C d+b^2 (B c+A d)+a b (A c-c C-B d)-b (A b c-a B c-b c C-a A d-b B d+a C d) \tan (e+f x)+\left (a^2+b^2\right ) C d \tan ^2(e+f x)}{(a+b \tan (e+f x))^2} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) (b c-a d)}{2 b^2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)}{b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\int \frac{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 )-b \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 ) \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{b \left (a^2+b^2\right )^2}\\ &=\frac{\left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)+3 a^2 b (B c+(A-C) d)-b^3 (B c+(A-C) d)\right ) x}{\left (a^2+b^2\right )^3}-\frac{\left (A b^2-a (b B-a C)\right ) (b c-a d)}{2 b^2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)}{b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)-a^3 (B c+(A-C) d)+3 a b^2 (B c+(A-C) d)\right ) \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{\left (a^3 (A c-c C-B d)-3 a b^2 (A c-c C-B d)+3 a^2 b (B c+(A-C) d)-b^3 (B c+(A-C) d)\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b (A c-c C-B d)-b^3 (A c-c C-B d)-a^3 (B c+(A-C) d)+3 a b^2 (B c+(A-C) d)\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac{\left (A b^2-a (b B-a C)\right ) (b c-a d)}{2 b^2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{a^4 C d+b^4 (B c+A d)+2 a b^3 (A c-c C-B d)-a^2 b^2 (B c+(A-3 C) d)}{b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 6.22806, size = 331, normalized size = 1.03 \[ \frac{2 b (d (A-C)+B c) \left (\frac{b \left (2 a \log (a+b \tan (e+f x))-\frac{a^2+b^2}{a+b \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2}-\frac{i \log (-\tan (e+f x)+i)}{2 (a+i b)^2}+\frac{i \log (\tan (e+f x)+i)}{2 (a-i b)^2}\right )-b (a A d+a B c-a C d-A b c+b B d+b c C) \left (\frac{b \left (\left (6 a^2-2 b^2\right ) \log (a+b \tan (e+f x))-\frac{\left (a^2+b^2\right ) \left (5 a^2+4 a b \tan (e+f x)+b^2\right )}{(a+b \tan (e+f x))^2}\right )}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (e+f x)+i)}{(b-i a)^3}+\frac{\log (\tan (e+f x)+i)}{(b+i a)^3}\right )+\frac{-a C d-b B d+b c C}{(a+b \tan (e+f x))^2}-\frac{2 b C (c+d \tan (e+f x))}{(a+b \tan (e+f x))^2}}{2 b^2 f} \]
Antiderivative was successfully verified.
[In]
Integrate[((c + d*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^3,x]
[Out]
((b*c*C - b*B*d - a*C*d)/(a + b*Tan[e + f*x])^2 - (2*b*C*(c + d*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2 + 2*b*(B
*c + (A - C)*d)*(((-I/2)*Log[I - Tan[e + f*x]])/(a + I*b)^2 + ((I/2)*Log[I + Tan[e + f*x]])/(a - I*b)^2 + (b*(
2*a*Log[a + b*Tan[e + f*x]] - (a^2 + b^2)/(a + b*Tan[e + f*x])))/(a^2 + b^2)^2) - b*(-(A*b*c) + a*B*c + b*c*C
+ a*A*d + b*B*d - a*C*d)*(Log[I - Tan[e + f*x]]/((-I)*a + b)^3 + Log[I + Tan[e + f*x]]/(I*a + b)^3 + (b*((6*a^
2 - 2*b^2)*Log[a + b*Tan[e + f*x]] - ((a^2 + b^2)*(5*a^2 + b^2 + 4*a*b*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2))
/(a^2 + b^2)^3))/(2*b^2*f)
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Maple [B] time = 0.068, size = 1513, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x)
[Out]
-1/f/(a^2+b^2)^2*b^2/(a+b*tan(f*x+e))*B*c-1/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a^3*C*d-1/2/f/(a^2+b^2)^3*ln(1+
tan(f*x+e)^2)*C*b^3*c+1/f/(a^2+b^2)^3*A*arctan(tan(f*x+e))*a^3*c-1/f/(a^2+b^2)^3*A*arctan(tan(f*x+e))*b^3*d-1/
f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*A*a^3*d-1/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*A*b^3*c-1/f/(a^2+b^2)^3*ln(a+b*tan
(f*x+e))*B*a^3*c+1/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*B*b^3*d+1/2/f/(a^2+b^2)/(a+b*tan(f*x+e))^2*B*a*c+1/f/(a^2+
b^2)^3*ln(a+b*tan(f*x+e))*a^3*C*d+1/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*C*b^3*c+1/f/(a^2+b^2)^2/(a+b*tan(f*x+e))*
A*a^2*d+1/f/(a^2+b^2)^2/(a+b*tan(f*x+e))*B*a^2*c+3/f/(a^2+b^2)^3*C*arctan(tan(f*x+e))*a*b^2*c-3/2/f/(a^2+b^2)^
3*ln(1+tan(f*x+e)^2)*A*a^2*b*c-3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*B*a*b^2*c+3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e
)^2)*C*a^2*b*c-3/f/(a^2+b^2)^2/(a+b*tan(f*x+e))*C*a^2*d+1/2/f/(a^2+b^2)/(a+b*tan(f*x+e))^2*A*a*d-1/f/(a^2+b^2)
^3*B*arctan(tan(f*x+e))*a^3*d-1/f/(a^2+b^2)^3*B*arctan(tan(f*x+e))*b^3*c-1/f/(a^2+b^2)^3*C*arctan(tan(f*x+e))*
a^3*c+1/f/(a^2+b^2)^3*C*arctan(tan(f*x+e))*b^3*d+1/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*A*a^3*d+1/2/f/(a^2+b^2)^
3*ln(1+tan(f*x+e)^2)*A*b^3*c+1/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*B*a^3*c-1/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)
*B*b^3*d-1/2/f*b/(a^2+b^2)/(a+b*tan(f*x+e))^2*A*c-1/f/(a^2+b^2)^2*b^2/(a+b*tan(f*x+e))*A*d+3/2/f/(a^2+b^2)^3*l
n(1+tan(f*x+e)^2)*C*a*b^2*d+3/f/(a^2+b^2)^3*A*arctan(tan(f*x+e))*a^2*b*d-3/f/(a^2+b^2)^3*A*arctan(tan(f*x+e))*
a*b^2*c+3/f/(a^2+b^2)^3*B*arctan(tan(f*x+e))*a^2*b*c-1/2/f/b/(a^2+b^2)/(a+b*tan(f*x+e))^2*B*a^2*d+1/2/f/b^2/(a
^2+b^2)/(a+b*tan(f*x+e))^2*a^3*C*d-1/2/f/b/(a^2+b^2)/(a+b*tan(f*x+e))^2*C*a^2*c-2/f/(a^2+b^2)^2*b/(a+b*tan(f*x
+e))*A*a*c-1/f/(a^2+b^2)^2/b^2/(a+b*tan(f*x+e))*a^4*C*d-3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*B*a^2*b*d+3/f/(a^2+
b^2)^3*ln(a+b*tan(f*x+e))*B*a*b^2*c-3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*C*a^2*b*c-3/f/(a^2+b^2)^3*ln(a+b*tan(f*
x+e))*C*a*b^2*d-3/f/(a^2+b^2)^3*C*arctan(tan(f*x+e))*a^2*b*d+2/f/(a^2+b^2)^2*b/(a+b*tan(f*x+e))*B*a*d+3/f/(a^2
+b^2)^3*B*arctan(tan(f*x+e))*a*b^2*d-3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*A*a*b^2*d+3/2/f/(a^2+b^2)^3*ln(1+tan
(f*x+e)^2)*B*a^2*b*d+2/f/(a^2+b^2)^2*b/(a+b*tan(f*x+e))*C*a*c+3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*A*a^2*b*c+3/f
/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*A*a*b^2*d
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Maxima [A] time = 1.52346, size = 775, normalized size = 2.42 \begin{align*} \frac{\frac{2 \,{\left ({\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \,{\left (A - C\right )} a b^{2} - B b^{3}\right )} c -{\left (B a^{3} - 3 \,{\left (A - C\right )} a^{2} b - 3 \, B a b^{2} +{\left (A - C\right )} b^{3}\right )} d\right )}{\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left ({\left (B a^{3} - 3 \,{\left (A - C\right )} a^{2} b - 3 \, B a b^{2} +{\left (A - C\right )} b^{3}\right )} c +{\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \,{\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left ({\left (B a^{3} - 3 \,{\left (A - C\right )} a^{2} b - 3 \, B a b^{2} +{\left (A - C\right )} b^{3}\right )} c +{\left ({\left (A - C\right )} a^{3} + 3 \, B a^{2} b - 3 \,{\left (A - C\right )} a b^{2} - B b^{3}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (C a^{4} b - 3 \, B a^{3} b^{2} +{\left (5 \, A - 3 \, C\right )} a^{2} b^{3} + B a b^{4} + A b^{5}\right )} c +{\left (C a^{5} + B a^{4} b -{\left (3 \, A - 5 \, C\right )} a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right )} d - 2 \,{\left ({\left (B a^{2} b^{3} - 2 \,{\left (A - C\right )} a b^{4} - B b^{5}\right )} c -{\left (C a^{4} b -{\left (A - 3 \, C\right )} a^{2} b^{3} - 2 \, B a b^{4} + A b^{5}\right )} d\right )} \tan \left (f x + e\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (f x + e\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="maxima")
[Out]
1/2*(2*(((A - C)*a^3 + 3*B*a^2*b - 3*(A - C)*a*b^2 - B*b^3)*c - (B*a^3 - 3*(A - C)*a^2*b - 3*B*a*b^2 + (A - C)
*b^3)*d)*(f*x + e)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*((B*a^3 - 3*(A - C)*a^2*b - 3*B*a*b^2 + (A - C)*b^3
)*c + ((A - C)*a^3 + 3*B*a^2*b - 3*(A - C)*a*b^2 - B*b^3)*d)*log(b*tan(f*x + e) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*
b^4 + b^6) + ((B*a^3 - 3*(A - C)*a^2*b - 3*B*a*b^2 + (A - C)*b^3)*c + ((A - C)*a^3 + 3*B*a^2*b - 3*(A - C)*a*b
^2 - B*b^3)*d)*log(tan(f*x + e)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((C*a^4*b - 3*B*a^3*b^2 + (5*A -
3*C)*a^2*b^3 + B*a*b^4 + A*b^5)*c + (C*a^5 + B*a^4*b - (3*A - 5*C)*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*d - 2*((B*
a^2*b^3 - 2*(A - C)*a*b^4 - B*b^5)*c - (C*a^4*b - (A - 3*C)*a^2*b^3 - 2*B*a*b^4 + A*b^5)*d)*tan(f*x + e))/(a^6
*b^2 + 2*a^4*b^4 + a^2*b^6 + (a^4*b^4 + 2*a^2*b^6 + b^8)*tan(f*x + e)^2 + 2*(a^5*b^3 + 2*a^3*b^5 + a*b^7)*tan(
f*x + e)))/f
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Fricas [B] time = 1.32655, size = 2072, normalized size = 6.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="fricas")
[Out]
1/2*(2*(((A - C)*a^5 + 3*B*a^4*b - 3*(A - C)*a^3*b^2 - B*a^2*b^3)*c - (B*a^5 - 3*(A - C)*a^4*b - 3*B*a^3*b^2 +
(A - C)*a^2*b^3)*d)*f*x + (2*(((A - C)*a^3*b^2 + 3*B*a^2*b^3 - 3*(A - C)*a*b^4 - B*b^5)*c - (B*a^3*b^2 - 3*(A
- C)*a^2*b^3 - 3*B*a*b^4 + (A - C)*b^5)*d)*f*x + (C*a^4*b - 3*B*a^3*b^2 + 5*(A - C)*a^2*b^3 + 3*B*a*b^4 - A*b
^5)*c + (C*a^5 + B*a^4*b - (3*A - 7*C)*a^3*b^2 - 5*B*a^2*b^3 + 3*A*a*b^4)*d)*tan(f*x + e)^2 - (3*C*a^4*b - 5*B
*a^3*b^2 + (7*A - 3*C)*a^2*b^3 + B*a*b^4 + A*b^5)*c + (C*a^5 - 3*B*a^4*b + 5*(A - C)*a^3*b^2 + 3*B*a^2*b^3 - A
*a*b^4)*d - (((B*a^3*b^2 - 3*(A - C)*a^2*b^3 - 3*B*a*b^4 + (A - C)*b^5)*c + ((A - C)*a^3*b^2 + 3*B*a^2*b^3 - 3
*(A - C)*a*b^4 - B*b^5)*d)*tan(f*x + e)^2 + (B*a^5 - 3*(A - C)*a^4*b - 3*B*a^3*b^2 + (A - C)*a^2*b^3)*c + ((A
- C)*a^5 + 3*B*a^4*b - 3*(A - C)*a^3*b^2 - B*a^2*b^3)*d + 2*((B*a^4*b - 3*(A - C)*a^3*b^2 - 3*B*a^2*b^3 + (A -
C)*a*b^4)*c + ((A - C)*a^4*b + 3*B*a^3*b^2 - 3*(A - C)*a^2*b^3 - B*a*b^4)*d)*tan(f*x + e))*log((b^2*tan(f*x +
e)^2 + 2*a*b*tan(f*x + e) + a^2)/(tan(f*x + e)^2 + 1)) + 2*(2*(((A - C)*a^4*b + 3*B*a^3*b^2 - 3*(A - C)*a^2*b
^3 - B*a*b^4)*c - (B*a^4*b - 3*(A - C)*a^3*b^2 - 3*B*a^2*b^3 + (A - C)*a*b^4)*d)*f*x + (C*a^5 - 2*B*a^4*b + 3*
(A - C)*a^3*b^2 + 3*B*a^2*b^3 - (3*A - 2*C)*a*b^4 - B*b^5)*c + (B*a^5 - (2*A - 3*C)*a^4*b - 3*B*a^3*b^2 + 3*(A
- C)*a^2*b^3 + 2*B*a*b^4 - A*b^5)*d)*tan(f*x + e))/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*f*tan(f*x + e)^2
+ 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*f*tan(f*x + e) + (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*f)
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**3,x)
[Out]
Exception raised: AttributeError
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Giac [B] time = 1.49648, size = 1400, normalized size = 4.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^3,x, algorithm="giac")
[Out]
1/2*(2*(A*a^3*c - C*a^3*c + 3*B*a^2*b*c - 3*A*a*b^2*c + 3*C*a*b^2*c - B*b^3*c - B*a^3*d + 3*A*a^2*b*d - 3*C*a^
2*b*d + 3*B*a*b^2*d - A*b^3*d + C*b^3*d)*(f*x + e)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (B*a^3*c - 3*A*a^2*b*
c + 3*C*a^2*b*c - 3*B*a*b^2*c + A*b^3*c - C*b^3*c + A*a^3*d - C*a^3*d + 3*B*a^2*b*d - 3*A*a*b^2*d + 3*C*a*b^2*
d - B*b^3*d)*log(tan(f*x + e)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 2*(B*a^3*b*c - 3*A*a^2*b^2*c + 3*C*
a^2*b^2*c - 3*B*a*b^3*c + A*b^4*c - C*b^4*c + A*a^3*b*d - C*a^3*b*d + 3*B*a^2*b^2*d - 3*A*a*b^3*d + 3*C*a*b^3*
d - B*b^4*d)*log(abs(b*tan(f*x + e) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) + (3*B*a^3*b^4*c*tan(f*x + e)^
2 - 9*A*a^2*b^5*c*tan(f*x + e)^2 + 9*C*a^2*b^5*c*tan(f*x + e)^2 - 9*B*a*b^6*c*tan(f*x + e)^2 + 3*A*b^7*c*tan(f
*x + e)^2 - 3*C*b^7*c*tan(f*x + e)^2 + 3*A*a^3*b^4*d*tan(f*x + e)^2 - 3*C*a^3*b^4*d*tan(f*x + e)^2 + 9*B*a^2*b
^5*d*tan(f*x + e)^2 - 9*A*a*b^6*d*tan(f*x + e)^2 + 9*C*a*b^6*d*tan(f*x + e)^2 - 3*B*b^7*d*tan(f*x + e)^2 + 8*B
*a^4*b^3*c*tan(f*x + e) - 22*A*a^3*b^4*c*tan(f*x + e) + 22*C*a^3*b^4*c*tan(f*x + e) - 18*B*a^2*b^5*c*tan(f*x +
e) + 2*A*a*b^6*c*tan(f*x + e) - 2*C*a*b^6*c*tan(f*x + e) - 2*B*b^7*c*tan(f*x + e) - 2*C*a^6*b*d*tan(f*x + e)
+ 8*A*a^4*b^3*d*tan(f*x + e) - 14*C*a^4*b^3*d*tan(f*x + e) + 22*B*a^3*b^4*d*tan(f*x + e) - 18*A*a^2*b^5*d*tan(
f*x + e) + 12*C*a^2*b^5*d*tan(f*x + e) - 2*B*a*b^6*d*tan(f*x + e) - 2*A*b^7*d*tan(f*x + e) - C*a^6*b*c + 6*B*a
^5*b^2*c - 14*A*a^4*b^3*c + 11*C*a^4*b^3*c - 7*B*a^3*b^4*c - 3*A*a^2*b^5*c - B*a*b^6*c - A*b^7*c - C*a^7*d - B
*a^6*b*d + 6*A*a^5*b^2*d - 9*C*a^5*b^2*d + 11*B*a^4*b^3*d - 7*A*a^3*b^4*d + 4*C*a^3*b^4*d - A*a*b^6*d)/((a^6*b
^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan(f*x + e) + a)^2))/f