3.1224 \(\int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=240 \[ -\frac {\left (a^2 \left (3 c^2-d^2\right )+8 a b c d-b^2 \left (c^2-3 d^2\right )\right ) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}+\frac {x (a c+b d) \left (a^2 \left (c^2-3 d^2\right )+8 a b c d-b^2 \left (3 c^2-d^2\right )\right )}{\left (c^2+d^2\right )^3}-\frac {\left (4 a c d+b \left (c^2+5 d^2\right )\right ) (b c-a d)^2}{2 d^2 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}-\frac {(b c-a d)^2 (a+b \tan (e+f x))}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2} \]

[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/(c^2+d^2)^3-(-a*d+b*c)*(8*a*b*c*d-b^2*(c^2-3*d^2)+a^2*
(3*c^2-d^2))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(c^2+d^2)^3/f-1/2*(-a*d+b*c)^2*(a+b*tan(f*x+e))/d/(c^2+d^2)/f/(c+d*
tan(f*x+e))^2-1/2*(-a*d+b*c)^2*(4*a*c*d+b*(c^2+5*d^2))/d^2/(c^2+d^2)^2/f/(c+d*tan(f*x+e))

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Rubi [A]  time = 0.52, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3565, 3628, 3531, 3530} \[ -\frac {\left (a^2 \left (3 c^2-d^2\right )+8 a b c d-b^2 \left (c^2-3 d^2\right )\right ) (b c-a d) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}+\frac {x (a c+b d) \left (a^2 \left (c^2-3 d^2\right )+8 a b c d-b^2 \left (3 c^2-d^2\right )\right )}{\left (c^2+d^2\right )^3}-\frac {\left (4 a c d+b \left (c^2+5 d^2\right )\right ) (b c-a d)^2}{2 d^2 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}-\frac {(b c-a d)^2 (a+b \tan (e+f x))}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

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]

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 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]

Rubi steps

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

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Mathematica [C]  time = 5.54, size = 327, normalized size = 1.36 \[ \frac {2 b d \left (3 a^2-b^2\right ) \left (\frac {d \left (2 c \log (c+d \tan (e+f x))-\frac {c^2+d^2}{c+d \tan (e+f x)}\right )}{\left (c^2+d^2\right )^2}-\frac {i \log (-\tan (e+f x)+i)}{2 (c+i d)^2}+\frac {i \log (\tan (e+f x)+i)}{2 (c-i d)^2}\right )+d \left (a^3 d-3 a^2 b c-3 a b^2 d+b^3 c\right ) \left (\frac {d \left (\left (6 c^2-2 d^2\right ) \log (c+d \tan (e+f x))-\frac {\left (c^2+d^2\right ) \left (5 c^2+4 c d \tan (e+f x)+d^2\right )}{(c+d \tan (e+f x))^2}\right )}{\left (c^2+d^2\right )^3}+\frac {\log (-\tan (e+f x)+i)}{(d-i c)^3}+\frac {\log (\tan (e+f x)+i)}{(d+i c)^3}\right )-\frac {b^2 (a d+b c)}{(c+d \tan (e+f x))^2}-\frac {2 b^2 d (a+b \tan (e+f x))}{(c+d \tan (e+f x))^2}}{2 d^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b^2*(b*c + a*d))/(c + d*Tan[e + f*x])^2) - (2*b^2*d*(a + b*Tan[e + f*x]))/(c + d*Tan[e + f*x])^2 + 2*b*(3*
a^2 - b^2)*d*(((-1/2*I)*Log[I - Tan[e + f*x]])/(c + I*d)^2 + ((I/2)*Log[I + Tan[e + f*x]])/(c - I*d)^2 + (d*(2
*c*Log[c + d*Tan[e + f*x]] - (c^2 + d^2)/(c + d*Tan[e + f*x])))/(c^2 + d^2)^2) + d*(-3*a^2*b*c + b^3*c + a^3*d
- 3*a*b^2*d)*(Log[I - Tan[e + f*x]]/((-I)*c + d)^3 + Log[I + Tan[e + f*x]]/(I*c + d)^3 + (d*((6*c^2 - 2*d^2)*
Log[c + d*Tan[e + f*x]] - ((c^2 + d^2)*(5*c^2 + d^2 + 4*c*d*Tan[e + f*x]))/(c + d*Tan[e + f*x])^2))/(c^2 + d^2
)^3))/(2*d^2*f)

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fricas [B]  time = 0.79, size = 861, normalized size = 3.59 \[ \frac {b^{3} c^{5} - 9 \, a b^{2} c^{4} d - 3 \, a^{2} b c d^{4} - a^{3} d^{5} + 5 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{3} d^{2} - {\left (7 \, a^{3} - 9 \, a b^{2}\right )} c^{2} d^{3} + 2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{5} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{4} d - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{3} d^{2} - {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d^{3}\right )} f x + {\left (b^{3} c^{5} + 3 \, a b^{2} c^{4} d + 9 \, a^{2} b c d^{4} - a^{3} d^{5} - {\left (9 \, a^{2} b - 7 \, b^{3}\right )} c^{3} d^{2} + 5 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} + 2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{3} d^{2} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d^{3} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{4} - {\left (3 \, a^{2} b - b^{3}\right )} d^{5}\right )} f x\right )} \tan \left (f x + e\right )^{2} - {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{5} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{4} d - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{3} d^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} + {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} d^{2} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c d^{4} + {\left (a^{3} - 3 \, a b^{2}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{4} d - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{3} d^{2} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d^{3} + {\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )} \tan \left (f x + e\right )\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 ) + 2 \, {\left (3 \, a b^{2} c^{5} - 3 \, a^{2} b d^{5} - 3 \, {\left (2 \, a^{2} b - b^{3}\right )} c^{4} d + 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{3} d^{2} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d^{3} - 3 \, {\left (a^{3} - 2 \, a b^{2}\right )} c d^{4} + 2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{4} d + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{3} d^{2} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} - {\left (3 \, a^{2} b - b^{3}\right )} c d^{4}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{7} d + 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} + c d^{7}\right )} f \tan \left (f x + e\right ) + {\left (c^{8} + 3 \, c^{6} d^{2} + 3 \, c^{4} d^{4} + c^{2} d^{6}\right )} f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/2*(b^3*c^5 - 9*a*b^2*c^4*d - 3*a^2*b*c*d^4 - a^3*d^5 + 5*(3*a^2*b - b^3)*c^3*d^2 - (7*a^3 - 9*a*b^2)*c^2*d^3
+ 2*((a^3 - 3*a*b^2)*c^5 + 3*(3*a^2*b - b^3)*c^4*d - 3*(a^3 - 3*a*b^2)*c^3*d^2 - (3*a^2*b - b^3)*c^2*d^3)*f*x
+ (b^3*c^5 + 3*a*b^2*c^4*d + 9*a^2*b*c*d^4 - a^3*d^5 - (9*a^2*b - 7*b^3)*c^3*d^2 + 5*(a^3 - 3*a*b^2)*c^2*d^3
+ 2*((a^3 - 3*a*b^2)*c^3*d^2 + 3*(3*a^2*b - b^3)*c^2*d^3 - 3*(a^3 - 3*a*b^2)*c*d^4 - (3*a^2*b - b^3)*d^5)*f*x)
*tan(f*x + e)^2 - ((3*a^2*b - b^3)*c^5 - 3*(a^3 - 3*a*b^2)*c^4*d - 3*(3*a^2*b - b^3)*c^3*d^2 + (a^3 - 3*a*b^2)
*c^2*d^3 + ((3*a^2*b - b^3)*c^3*d^2 - 3*(a^3 - 3*a*b^2)*c^2*d^3 - 3*(3*a^2*b - b^3)*c*d^4 + (a^3 - 3*a*b^2)*d^
5)*tan(f*x + e)^2 + 2*((3*a^2*b - b^3)*c^4*d - 3*(a^3 - 3*a*b^2)*c^3*d^2 - 3*(3*a^2*b - b^3)*c^2*d^3 + (a^3 -
3*a*b^2)*c*d^4)*tan(f*x + e))*log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(tan(f*x + e)^2 + 1)) + 2*(3
*a*b^2*c^5 - 3*a^2*b*d^5 - 3*(2*a^2*b - b^3)*c^4*d + 3*(a^3 - 3*a*b^2)*c^3*d^2 + 3*(3*a^2*b - b^3)*c^2*d^3 - 3
*(a^3 - 2*a*b^2)*c*d^4 + 2*((a^3 - 3*a*b^2)*c^4*d + 3*(3*a^2*b - b^3)*c^3*d^2 - 3*(a^3 - 3*a*b^2)*c^2*d^3 - (3
*a^2*b - b^3)*c*d^4)*f*x)*tan(f*x + e))/((c^6*d^2 + 3*c^4*d^4 + 3*c^2*d^6 + d^8)*f*tan(f*x + e)^2 + 2*(c^7*d +
3*c^5*d^3 + 3*c^3*d^5 + c*d^7)*f*tan(f*x + e) + (c^8 + 3*c^6*d^2 + 3*c^4*d^4 + c^2*d^6)*f)

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giac [B]  time = 2.75, size = 830, normalized size = 3.46 \[ \frac {\frac {2 \, {\left (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}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left (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}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left (3 \, a^{2} b c^{3} d - b^{3} c^{3} d - 3 \, a^{3} c^{2} d^{2} + 9 \, a b^{2} c^{2} d^{2} - 9 \, a^{2} b c d^{3} + 3 \, b^{3} c d^{3} + a^{3} d^{4} - 3 \, a b^{2} d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}} + \frac {9 \, a^{2} b c^{3} d^{4} \tan \left (f x + e\right )^{2} - 3 \, b^{3} c^{3} d^{4} \tan \left (f x + e\right )^{2} - 9 \, a^{3} c^{2} d^{5} \tan \left (f x + e\right )^{2} + 27 \, a b^{2} c^{2} d^{5} \tan \left (f x + e\right )^{2} - 27 \, a^{2} b c d^{6} \tan \left (f x + e\right )^{2} + 9 \, b^{3} c d^{6} \tan \left (f x + e\right )^{2} + 3 \, a^{3} d^{7} \tan \left (f x + e\right )^{2} - 9 \, a b^{2} d^{7} \tan \left (f x + e\right )^{2} - 2 \, b^{3} c^{6} d \tan \left (f x + e\right ) + 24 \, a^{2} b c^{4} d^{3} \tan \left (f x + e\right ) - 14 \, b^{3} c^{4} d^{3} \tan \left (f x + e\right ) - 22 \, a^{3} c^{3} d^{4} \tan \left (f x + e\right ) + 66 \, a b^{2} c^{3} d^{4} \tan \left (f x + e\right ) - 54 \, a^{2} b c^{2} d^{5} \tan \left (f x + e\right ) + 12 \, b^{3} c^{2} d^{5} \tan \left (f x + e\right ) + 2 \, a^{3} c d^{6} \tan \left (f x + e\right ) - 6 \, a b^{2} c d^{6} \tan \left (f x + e\right ) - 6 \, a^{2} b d^{7} \tan \left (f x + e\right ) - b^{3} c^{7} - 3 \, a b^{2} c^{6} d + 18 \, a^{2} b c^{5} d^{2} - 9 \, b^{3} c^{5} d^{2} - 14 \, a^{3} c^{4} d^{3} + 33 \, a b^{2} c^{4} d^{3} - 21 \, a^{2} b c^{3} d^{4} + 4 \, b^{3} c^{3} d^{4} - 3 \, a^{3} c^{2} d^{5} - 3 \, a^{2} b c d^{6} - a^{3} d^{7}}{{\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.28, size = 1063, normalized size = 4.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^3,x)

[Out]

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

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maxima [B]  time = 0.95, size = 528, normalized size = 2.20 \[ \frac {\frac {2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{3} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{2} - {\left (3 \, a^{2} b - b^{3}\right )} d^{3}\right )} {\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {2 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac {{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac {b^{3} c^{5} + 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c d^{4} + a^{3} d^{5} - {\left (9 \, a^{2} b - 5 \, b^{3}\right )} c^{3} d^{2} + {\left (5 \, a^{3} - 9 \, a b^{2}\right )} c^{2} d^{3} + 2 \, {\left (b^{3} c^{4} d + 3 \, a^{2} b d^{5} - 3 \, {\left (a^{2} b - b^{3}\right )} c^{2} d^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )} \tan \left (f x + e\right )}{c^{6} d^{2} + 2 \, c^{4} d^{4} + c^{2} d^{6} + {\left (c^{4} d^{4} + 2 \, c^{2} d^{6} + d^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (c^{5} d^{3} + 2 \, c^{3} d^{5} + c d^{7}\right )} \tan \left (f x + e\right )}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))^3/(c+d*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)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^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(d*tan(f*x + e) + c)/(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^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)/(c^6 + 3*
c^4*d^2 + 3*c^2*d^4 + d^6) - (b^3*c^5 + 3*a*b^2*c^4*d + 3*a^2*b*c*d^4 + a^3*d^5 - (9*a^2*b - 5*b^3)*c^3*d^2 +
(5*a^3 - 9*a*b^2)*c^2*d^3 + 2*(b^3*c^4*d + 3*a^2*b*d^5 - 3*(a^2*b - b^3)*c^2*d^3 + 2*(a^3 - 3*a*b^2)*c*d^4)*ta
n(f*x + e))/(c^6*d^2 + 2*c^4*d^4 + c^2*d^6 + (c^4*d^4 + 2*c^2*d^6 + d^8)*tan(f*x + e)^2 + 2*(c^5*d^3 + 2*c^3*d
^5 + c*d^7)*tan(f*x + e)))/f

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mupad [B]  time = 8.29, size = 466, normalized size = 1.94 \[ -\frac {\frac {5\,a^3\,c^2\,d^3+a^3\,d^5-9\,a^2\,b\,c^3\,d^2+3\,a^2\,b\,c\,d^4+3\,a\,b^2\,c^4\,d-9\,a\,b^2\,c^2\,d^3+b^3\,c^5+5\,b^3\,c^3\,d^2}{2\,d^2\,\left (c^4+2\,c^2\,d^2+d^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (2\,a^3\,c\,d^3-3\,a^2\,b\,c^2\,d^2+3\,a^2\,b\,d^4-6\,a\,b^2\,c\,d^3+b^3\,c^4+3\,b^3\,c^2\,d^2\right )}{d\,\left (c^4+2\,c^2\,d^2+d^4\right )}}{f\,\left (c^2+2\,c\,d\,\mathrm {tan}\left (e+f\,x\right )+d^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}{2\,f\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\left (3\,a^2\,b-b^3\right )\,c^3+\left (9\,a\,b^2-3\,a^3\right )\,c^2\,d+\left (3\,b^3-9\,a^2\,b\right )\,c\,d^2+\left (a^3-3\,a\,b^2\right )\,d^3\right )}{f\,\left (c^6+3\,c^4\,d^2+3\,c^2\,d^4+d^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}{2\,f\,\left (-c^3\,1{}\mathrm {i}-3\,c^2\,d+c\,d^2\,3{}\mathrm {i}+d^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))^3/(c + d*tan(e + f*x))^3,x)

[Out]

- ((a^3*d^5 + b^3*c^5 + 5*a^3*c^2*d^3 + 5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 - 9*a^2*b*c^3*d^2 + 3*a*b^2*c^4*d + 3*
a^2*b*c*d^4)/(2*d^2*(c^4 + d^4 + 2*c^2*d^2)) + (tan(e + f*x)*(b^3*c^4 + 3*a^2*b*d^4 + 2*a^3*c*d^3 + 3*b^3*c^2*
d^2 - 3*a^2*b*c^2*d^2 - 6*a*b^2*c*d^3))/(d*(c^4 + d^4 + 2*c^2*d^2)))/(f*(c^2 + d^2*tan(e + f*x)^2 + 2*c*d*tan(
e + f*x))) - (log(tan(e + f*x) - 1i)*(a*b^2*3i + 3*a^2*b - a^3*1i - b^3))/(2*f*(3*c*d^2 - c^2*d*3i - c^3 + d^3
*1i)) - (log(c + d*tan(e + f*x))*(c^3*(3*a^2*b - b^3) - d^3*(3*a*b^2 - a^3) + c^2*d*(9*a*b^2 - 3*a^3) - c*d^2*
(9*a^2*b - 3*b^3)))/(f*(c^6 + d^6 + 3*c^2*d^4 + 3*c^4*d^2)) - (log(tan(e + f*x) + 1i)*(3*a*b^2 + a^2*b*3i - a^
3 - b^3*1i))/(2*f*(c*d^2*3i - 3*c^2*d - c^3*1i + d^3))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**3,x)

[Out]

Exception raised: AttributeError

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