3.1208 \(\int \frac{(c+d \tan (e+f x))^3}{(a+b \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=239 \[ \frac{\left (a^2 \left (3 c^2-d^2\right )+8 a b c d-b^2 \left (c^2-3 d^2\right )\right ) (b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x (a c+b d) \left (a^2 \left (c^2-3 d^2\right )+8 a b c d-b^2 \left (3 c^2-d^2\right )\right )}{\left (a^2+b^2\right )^3}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{\left (a^2 d+4 a b c+5 b^2 d\right ) (b c-a d)^2}{2 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))} \]

[Out]

((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(a^2 + b^2)^3 + ((b*c - a*d)*(8*a*b*c*d -
b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((b*c - a*d)^
2*(4*a*b*c + a^2*d + 5*b^2*d))/(2*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((b*c - a*d)^2*(c + d*Tan[e + f*
x]))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)

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Rubi [A]  time = 0.5115, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3565, 3628, 3531, 3530} \[ \frac{\left (a^2 \left (3 c^2-d^2\right )+8 a b c d-b^2 \left (c^2-3 d^2\right )\right ) (b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x (a c+b d) \left (a^2 \left (c^2-3 d^2\right )+8 a b c d-b^2 \left (3 c^2-d^2\right )\right )}{\left (a^2+b^2\right )^3}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{\left (a^2 d+4 a b c+5 b^2 d\right ) (b c-a d)^2}{2 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^3,x]

[Out]

((a*c + b*d)*(8*a*b*c*d + a^2*(c^2 - 3*d^2) - b^2*(3*c^2 - d^2))*x)/(a^2 + b^2)^3 + ((b*c - a*d)*(8*a*b*c*d -
b^2*(c^2 - 3*d^2) + a^2*(3*c^2 - d^2))*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])/((a^2 + b^2)^3*f) - ((b*c - a*d)^
2*(4*a*b*c + a^2*d + 5*b^2*d))/(2*b^2*(a^2 + b^2)^2*f*(a + b*Tan[e + f*x])) - ((b*c - a*d)^2*(c + d*Tan[e + f*
x]))/(2*b*(a^2 + b^2)*f*(a + b*Tan[e + f*x])^2)

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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))^3}{(a+b \tan (e+f x))^3} \, dx &=-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\int \frac{d (2 b c-a d)^2+b c^2 (2 a c+b d)+2 b \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \tan (e+f x)+d \left (\left (a^2+2 b^2\right ) d^2-b c (b c-2 a d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=-\frac{(b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\int \frac{-2 b \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )-a b \left (6 c^2 d-2 d^3\right )\right )-2 b \left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{2 b \left (a^2+b^2\right )^2}\\ &=\frac{(a c+b d) \left (8 a b c d+a^2 \left (c^2-3 d^2\right )-b^2 \left (3 c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac{(b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\left (-2 b^2 \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )-a b \left (6 c^2 d-2 d^3\right )\right )+2 a b \left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right )\right ) \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{2 b \left (a^2+b^2\right )^3}\\ &=\frac{(a c+b d) \left (8 a b c d+a^2 \left (c^2-3 d^2\right )-b^2 \left (3 c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3}+\frac{(b c-a d) \left (3 a^2 c^2-b^2 c^2+8 a b c d-a^2 d^2+3 b^2 d^2\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac{(b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )}{2 b^2 \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac{(b c-a d)^2 (c+d \tan (e+f x))}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}\\ \end{align*}

Mathematica [C]  time = 5.01496, size = 327, normalized size = 1.37 \[ \frac{2 b d \left (3 c^2-d^2\right ) \left (\frac{b \left (2 a \log (a+b \tan (e+f x))-\frac{a^2+b^2}{a+b \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2}-\frac{i \log (-\tan (e+f x)+i)}{2 (a+i b)^2}+\frac{i \log (\tan (e+f x)+i)}{2 (a-i b)^2}\right )+b \left (a d \left (d^2-3 c^2\right )+b \left (c^3-3 c d^2\right )\right ) \left (\frac{b \left (\left (6 a^2-2 b^2\right ) \log (a+b \tan (e+f x))-\frac{\left (a^2+b^2\right ) \left (5 a^2+4 a b \tan (e+f x)+b^2\right )}{(a+b \tan (e+f x))^2}\right )}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (e+f x)+i)}{(b-i a)^3}+\frac{\log (\tan (e+f x)+i)}{(b+i a)^3}\right )-\frac{2 b d^2 (c+d \tan (e+f x))}{(a+b \tan (e+f x))^2}-\frac{d^2 (a d+b c)}{(a+b \tan (e+f x))^2}}{2 b^2 f} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^3/(a + b*Tan[e + f*x])^3,x]

[Out]

(-((d^2*(b*c + a*d))/(a + b*Tan[e + f*x])^2) - (2*b*d^2*(c + d*Tan[e + f*x]))/(a + b*Tan[e + f*x])^2 + 2*b*d*(
3*c^2 - d^2)*(((-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*d*(-3*c^2 + d^2) + b*(c^3
- 3*c*d^2))*(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)*Lo
g[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 = 0.044, size = 1063, normalized size = 4.5 \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))^3/(a+b*tan(f*x+e))^3,x)

[Out]

-3/f/(a^2+b^2)^3*arctan(tan(f*x+e))*a*b^2*c^3+3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*a^2*b*c^3-3/f/(a^2+b^2)^3*ln(
a+b*tan(f*x+e))*a*b^2*d^3-3/f/(a^2+b^2)^2*b^2/(a+b*tan(f*x+e))*c^2*d+3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a^3*
c^2*d-3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a^2*b*c^3+3/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a*b^2*d^3-3/2/f/(a^2
+b^2)^3*ln(1+tan(f*x+e)^2)*b^3*c*d^2-3/f/(a^2+b^2)^3*arctan(tan(f*x+e))*a^3*c*d^2-3/f/(a^2+b^2)^3*arctan(tan(f
*x+e))*a^2*b*d^3-1/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a^3*d^3-3/f/(a^2+b^2)^2/(a+b*tan(f*x+e))*a^2*d^3+1/2/f/(
a^2+b^2)^3*ln(1+tan(f*x+e)^2)*b^3*c^3+1/f/(a^2+b^2)^3*arctan(tan(f*x+e))*a^3*c^3+1/f/(a^2+b^2)^3*arctan(tan(f*
x+e))*b^3*d^3+1/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*a^3*d^3-1/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*b^3*c^3-1/2/f*b/(a
^2+b^2)/(a+b*tan(f*x+e))^2*c^3-3/f/(a^2+b^2)^3*arctan(tan(f*x+e))*b^3*c^2*d+3/2/f/(a^2+b^2)/(a+b*tan(f*x+e))^2
*a*c^2*d+3/f/(a^2+b^2)^2/(a+b*tan(f*x+e))*a^2*c^2*d+3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*b^3*c*d^2+1/2/f/b^2/(a^
2+b^2)/(a+b*tan(f*x+e))^2*a^3*d^3-1/f/(a^2+b^2)^2/b^2/(a+b*tan(f*x+e))*a^4*d^3-2/f/(a^2+b^2)^2*b/(a+b*tan(f*x+
e))*a*c^3-3/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*a^3*c^2*d+9/f/(a^2+b^2)^3*arctan(tan(f*x+e))*a*b^2*c*d^2+9/2/f/(a
^2+b^2)^3*ln(1+tan(f*x+e)^2)*a^2*b*c*d^2-3/2/f/b/(a^2+b^2)/(a+b*tan(f*x+e))^2*a^2*c*d^2+6/f/(a^2+b^2)^2*b/(a+b
*tan(f*x+e))*a*c*d^2-9/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*a^2*b*c*d^2+9/f/(a^2+b^2)^3*ln(a+b*tan(f*x+e))*a*b^2*c
^2*d-9/2/f/(a^2+b^2)^3*ln(1+tan(f*x+e)^2)*a*b^2*c^2*d+9/f/(a^2+b^2)^3*arctan(tan(f*x+e))*a^2*b*c^2*d

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Maxima [B]  time = 1.84844, size = 720, normalized size = 3.01 \begin{align*} \frac{\frac{2 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{3} + 3 \,{\left (3 \, a^{2} b - b^{3}\right )} c^{2} d - 3 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d^{2} -{\left (3 \, a^{2} b - b^{3}\right )} d^{3}\right )}{\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \,{\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \,{\left (3 \, a^{2} b - b^{3}\right )} c d^{2} +{\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\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 (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \,{\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \,{\left (3 \, a^{2} b - b^{3}\right )} c d^{2} +{\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\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 (5 \, a^{2} b^{3} + b^{5}\right )} c^{3} - 3 \,{\left (3 \, a^{3} b^{2} - a b^{4}\right )} c^{2} d + 3 \,{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c d^{2} +{\left (a^{5} + 5 \, a^{3} b^{2}\right )} d^{3} + 2 \,{\left (2 \, a b^{4} c^{3} - 6 \, a b^{4} c d^{2} - 3 \,{\left (a^{2} b^{3} - b^{5}\right )} c^{2} d +{\left (a^{4} b + 3 \, a^{2} b^{3}\right )} d^{3}\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))^3/(a+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

1/2*(2*((a^3 - 3*a*b^2)*c^3 + 3*(3*a^2*b - b^3)*c^2*d - 3*(a^3 - 3*a*b^2)*c*d^2 - (3*a^2*b - b^3)*d^3)*(f*x +
e)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*((3*a^2*b - b^3)*c^3 - 3*(a^3 - 3*a*b^2)*c^2*d - 3*(3*a^2*b - b^3)*
c*d^2 + (a^3 - 3*a*b^2)*d^3)*log(b*tan(f*x + e) + a)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - ((3*a^2*b - b^3)*c^
3 - 3*(a^3 - 3*a*b^2)*c^2*d - 3*(3*a^2*b - b^3)*c*d^2 + (a^3 - 3*a*b^2)*d^3)*log(tan(f*x + e)^2 + 1)/(a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6) - ((5*a^2*b^3 + b^5)*c^3 - 3*(3*a^3*b^2 - a*b^4)*c^2*d + 3*(a^4*b - 3*a^2*b^3)*c*d^
2 + (a^5 + 5*a^3*b^2)*d^3 + 2*(2*a*b^4*c^3 - 6*a*b^4*c*d^2 - 3*(a^2*b^3 - b^5)*c^2*d + (a^4*b + 3*a^2*b^3)*d^3
)*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 = 1.63124, size = 1798, normalized size = 7.52 \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))^3/(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/2*((7*a^2*b^3 + b^5)*c^3 - 3*(5*a^3*b^2 - a*b^4)*c^2*d + 9*(a^4*b - a^2*b^3)*c*d^2 - (a^5 - 5*a^3*b^2)*d^3
- 2*((a^5 - 3*a^3*b^2)*c^3 + 3*(3*a^4*b - a^2*b^3)*c^2*d - 3*(a^5 - 3*a^3*b^2)*c*d^2 - (3*a^4*b - a^2*b^3)*d^3
)*f*x - ((5*a^2*b^3 - b^5)*c^3 - 9*(a^3*b^2 - a*b^4)*c^2*d + 3*(a^4*b - 5*a^2*b^3)*c*d^2 + (a^5 + 7*a^3*b^2)*d
^3 + 2*((a^3*b^2 - 3*a*b^4)*c^3 + 3*(3*a^2*b^3 - b^5)*c^2*d - 3*(a^3*b^2 - 3*a*b^4)*c*d^2 - (3*a^2*b^3 - b^5)*
d^3)*f*x)*tan(f*x + e)^2 - ((3*a^4*b - a^2*b^3)*c^3 - 3*(a^5 - 3*a^3*b^2)*c^2*d - 3*(3*a^4*b - a^2*b^3)*c*d^2
+ (a^5 - 3*a^3*b^2)*d^3 + ((3*a^2*b^3 - b^5)*c^3 - 3*(a^3*b^2 - 3*a*b^4)*c^2*d - 3*(3*a^2*b^3 - b^5)*c*d^2 + (
a^3*b^2 - 3*a*b^4)*d^3)*tan(f*x + e)^2 + 2*((3*a^3*b^2 - a*b^4)*c^3 - 3*(a^4*b - 3*a^2*b^3)*c^2*d - 3*(3*a^3*b
^2 - a*b^4)*c*d^2 + (a^4*b - 3*a^2*b^3)*d^3)*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*(3*(a^3*b^2 - a*b^4)*c^3 - 3*(2*a^4*b - 3*a^2*b^3 + b^5)*c^2*d + 3*(a^5 - 3*a^3*b^2
+ 2*a*b^4)*c*d^2 + 3*(a^4*b - a^2*b^3)*d^3 + 2*((a^4*b - 3*a^2*b^3)*c^3 + 3*(3*a^3*b^2 - a*b^4)*c^2*d - 3*(a^
4*b - 3*a^2*b^3)*c*d^2 - (3*a^3*b^2 - a*b^4)*d^3)*f*x)*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))**3/(a+b*tan(f*x+e))**3,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 2.01402, size = 1121, normalized size = 4.69 \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))^3/(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*(a^3*c^3 - 3*a*b^2*c^3 + 9*a^2*b*c^2*d - 3*b^3*c^2*d - 3*a^3*c*d^2 + 9*a*b^2*c*d^2 - 3*a^2*b*d^3 + b^3*
d^3)*(f*x + e)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b*c^3 - b^3*c^3 - 3*a^3*c^2*d + 9*a*b^2*c^2*d - 9*
a^2*b*c*d^2 + 3*b^3*c*d^2 + a^3*d^3 - 3*a*b^2*d^3)*log(tan(f*x + e)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)
+ 2*(3*a^2*b^2*c^3 - b^4*c^3 - 3*a^3*b*c^2*d + 9*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 3*b^4*c*d^2 + a^3*b*d^3 - 3*
a*b^3*d^3)*log(abs(b*tan(f*x + e) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - (9*a^2*b^5*c^3*tan(f*x + e)^2
- 3*b^7*c^3*tan(f*x + e)^2 - 9*a^3*b^4*c^2*d*tan(f*x + e)^2 + 27*a*b^6*c^2*d*tan(f*x + e)^2 - 27*a^2*b^5*c*d^2
*tan(f*x + e)^2 + 9*b^7*c*d^2*tan(f*x + e)^2 + 3*a^3*b^4*d^3*tan(f*x + e)^2 - 9*a*b^6*d^3*tan(f*x + e)^2 + 22*
a^3*b^4*c^3*tan(f*x + e) - 2*a*b^6*c^3*tan(f*x + e) - 24*a^4*b^3*c^2*d*tan(f*x + e) + 54*a^2*b^5*c^2*d*tan(f*x
+ e) + 6*b^7*c^2*d*tan(f*x + e) - 66*a^3*b^4*c*d^2*tan(f*x + e) + 6*a*b^6*c*d^2*tan(f*x + e) + 2*a^6*b*d^3*ta
n(f*x + e) + 14*a^4*b^3*d^3*tan(f*x + e) - 12*a^2*b^5*d^3*tan(f*x + e) + 14*a^4*b^3*c^3 + 3*a^2*b^5*c^3 + b^7*
c^3 - 18*a^5*b^2*c^2*d + 21*a^3*b^4*c^2*d + 3*a*b^6*c^2*d + 3*a^6*b*c*d^2 - 33*a^4*b^3*c*d^2 + a^7*d^3 + 9*a^5
*b^2*d^3 - 4*a^3*b^4*d^3)/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan(f*x + e) + a)^2))/f