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

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Rubi [A]  time = 0.51, antiderivative size = 239, 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} \[ \frac {\left (a^2 \left (3 c^2-d^2\right )+8 a b c d-b^2 \left (c^2-3 d^2\right )\right ) (b c-a d) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac {x (a c+b d) \left (a^2 \left (c^2-3 d^2\right )+8 a b c d-b^2 \left (3 c^2-d^2\right )\right )}{\left (a^2+b^2\right )^3}-\frac {(b c-a d)^2 (c+d \tan (e+f x))}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {\left (a^2 d+4 a b c+5 b^2 d\right ) (b c-a d)^2}{2 b^2 f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))} \]

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 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 {(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*}

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Mathematica [C]  time = 5.61, 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)*(((-1/2*I)*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)*
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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fricas [B]  time = 1.05, size = 898, normalized size = 3.76 \[ -\frac {{\left (7 \, a^{2} b^{3} + b^{5}\right )} c^{3} - 3 \, {\left (5 \, a^{3} b^{2} - a b^{4}\right )} c^{2} d + 9 \, {\left (a^{4} b - a^{2} b^{3}\right )} c d^{2} - {\left (a^{5} - 5 \, a^{3} b^{2}\right )} d^{3} - 2 \, {\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} c^{3} + 3 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} c^{2} d - 3 \, {\left (a^{5} - 3 \, a^{3} b^{2}\right )} c d^{2} - {\left (3 \, a^{4} b - a^{2} b^{3}\right )} d^{3}\right )} f x - {\left ({\left (5 \, a^{2} b^{3} - b^{5}\right )} c^{3} - 9 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} d + 3 \, {\left (a^{4} b - 5 \, a^{2} b^{3}\right )} c d^{2} + {\left (a^{5} + 7 \, a^{3} b^{2}\right )} d^{3} + 2 \, {\left ({\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c^{3} + 3 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} c^{2} d - 3 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c d^{2} - {\left (3 \, a^{2} b^{3} - b^{5}\right )} d^{3}\right )} f x\right )} \tan \left (f x + e\right )^{2} - {\left ({\left (3 \, a^{4} b - a^{2} b^{3}\right )} c^{3} - 3 \, {\left (a^{5} - 3 \, a^{3} b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} c d^{2} + {\left (a^{5} - 3 \, a^{3} b^{2}\right )} d^{3} + {\left ({\left (3 \, a^{2} b^{3} - b^{5}\right )} c^{3} - 3 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c^{2} d - 3 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} c d^{2} + {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} c^{3} - 3 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c^{2} d - 3 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} c d^{2} + {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d^{3}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (3 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{3} - 3 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right )} c^{2} d + 3 \, {\left (a^{5} - 3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} c d^{2} + 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} d^{3} + 2 \, {\left ({\left (a^{4} b - 3 \, a^{2} b^{3}\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 (3 \, a^{3} b^{2} - a b^{4}\right )} d^{3}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} f \tan \left (f x + e\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} f\right )}} \]

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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giac [B]  time = 2.98, size = 830, normalized size = 3.47 \[ \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 )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{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 )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (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}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {9 \, a^{2} b^{5} c^{3} \tan \left (f x + e\right )^{2} - 3 \, b^{7} c^{3} \tan \left (f x + e\right )^{2} - 9 \, a^{3} b^{4} c^{2} d \tan \left (f x + e\right )^{2} + 27 \, a b^{6} c^{2} d \tan \left (f x + e\right )^{2} - 27 \, a^{2} b^{5} c d^{2} \tan \left (f x + e\right )^{2} + 9 \, b^{7} c d^{2} \tan \left (f x + e\right )^{2} + 3 \, a^{3} b^{4} d^{3} \tan \left (f x + e\right )^{2} - 9 \, a b^{6} d^{3} \tan \left (f x + e\right )^{2} + 22 \, a^{3} b^{4} c^{3} \tan \left (f x + e\right ) - 2 \, a b^{6} c^{3} \tan \left (f x + e\right ) - 24 \, a^{4} b^{3} c^{2} d \tan \left (f x + e\right ) + 54 \, a^{2} b^{5} c^{2} d \tan \left (f x + e\right ) + 6 \, b^{7} c^{2} d \tan \left (f x + e\right ) - 66 \, a^{3} b^{4} c d^{2} \tan \left (f x + e\right ) + 6 \, a b^{6} c d^{2} \tan \left (f x + e\right ) + 2 \, a^{6} b d^{3} \tan \left (f x + e\right ) + 14 \, a^{4} b^{3} d^{3} \tan \left (f x + e\right ) - 12 \, a^{2} b^{5} d^{3} \tan \left (f x + e\right ) + 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}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}}}{2 \, f} \]

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

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maple [B]  time = 0.26, size = 1063, normalized size = 4.45 \[ \text {result too large to display} \]

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]

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

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maxima [B]  time = 0.81, size = 533, normalized size = 2.23 \[ \frac {\frac {2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{3} + 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{2} - {\left (3 \, a^{2} b - b^{3}\right )} d^{3}\right )} {\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{3} - 3 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (5 \, a^{2} b^{3} + b^{5}\right )} c^{3} - 3 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} c^{2} d + 3 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c d^{2} + {\left (a^{5} + 5 \, a^{3} b^{2}\right )} d^{3} + 2 \, {\left (2 \, a b^{4} c^{3} - 6 \, a b^{4} c d^{2} - 3 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{2} d + {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} d^{3}\right )} \tan \left (f x + e\right )}{a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (f x + e\right )}}{2 \, f} \]

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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mupad [B]  time = 8.00, size = 466, normalized size = 1.95 \[ -\frac {\frac {a^5\,d^3+3\,a^4\,b\,c\,d^2-9\,a^3\,b^2\,c^2\,d+5\,a^3\,b^2\,d^3+5\,a^2\,b^3\,c^3-9\,a^2\,b^3\,c\,d^2+3\,a\,b^4\,c^2\,d+b^5\,c^3}{2\,b^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^4\,d^3-3\,a^2\,b^2\,c^2\,d+3\,a^2\,b^2\,d^3+2\,a\,b^3\,c^3-6\,a\,b^3\,c\,d^2+3\,b^4\,c^2\,d\right )}{b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{f\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (e+f\,x\right )+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-c^3\,1{}\mathrm {i}+3\,c^2\,d+c\,d^2\,3{}\mathrm {i}-d^3\right )}{2\,f\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\left (3\,c^2\,d-d^3\right )\,a^3+\left (9\,c\,d^2-3\,c^3\right )\,a^2\,b+\left (3\,d^3-9\,c^2\,d\right )\,a\,b^2+\left (c^3-3\,c\,d^2\right )\,b^3\right )}{f\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-c^3+c^2\,d\,3{}\mathrm {i}+3\,c\,d^2-d^3\,1{}\mathrm {i}\right )}{2\,f\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: AttributeError

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