3.13.27 \(\int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\) [1227]

Optimal. Leaf size=286 \[ -\frac {\left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^3 f}+\frac {d^2 \left (8 a b c^3 d-a^2 d^2 \left (3 c^2-d^2\right )-b^2 \left (6 c^4+3 c^2 d^2+d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^3 f}+\frac {d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))} \]

[Out]

-(b*d*(3*c^2-d^2)-a*(c^3-3*c*d^2))*x/(a^2+b^2)/(c^2+d^2)^3+b^4*ln(a*cos(f*x+e)+b*sin(f*x+e))/(a^2+b^2)/(-a*d+b
*c)^3/f+d^2*(8*a*b*c^3*d-a^2*d^2*(3*c^2-d^2)-b^2*(6*c^4+3*c^2*d^2+d^4))*ln(c*cos(f*x+e)+d*sin(f*x+e))/(-a*d+b*
c)^3/(c^2+d^2)^3/f+1/2*d^2/(-a*d+b*c)/(c^2+d^2)/f/(c+d*tan(f*x+e))^2-d^2*(2*a*c*d-b*(3*c^2+d^2))/(-a*d+b*c)^2/
(c^2+d^2)^2/f/(c+d*tan(f*x+e))

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Rubi [A]
time = 0.67, antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3650, 3730, 3732, 3611} \begin {gather*} -\frac {x \left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right )}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {d^2 \left (-a^2 d^2 \left (3 c^2-d^2\right )+8 a b c^3 d-b^2 \left (6 c^4+3 c^2 d^2+d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3 (b c-a d)^3}+\frac {b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right ) (b c-a d)^3}-\frac {d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 (c+d \tan (e+f x))}+\frac {d^2}{2 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]

[Out]

-(((b*d*(3*c^2 - d^2) - a*(c^3 - 3*c*d^2))*x)/((a^2 + b^2)*(c^2 + d^2)^3)) + (b^4*Log[a*Cos[e + f*x] + b*Sin[e
+ f*x]])/((a^2 + b^2)*(b*c - a*d)^3*f) + (d^2*(8*a*b*c^3*d - a^2*d^2*(3*c^2 - d^2) - b^2*(6*c^4 + 3*c^2*d^2 +
d^4))*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)^3*(c^2 + d^2)^3*f) + d^2/(2*(b*c - a*d)*(c^2 + d^2)*
f*(c + d*Tan[e + f*x])^2) - (d^2*(2*a*c*d - b*(3*c^2 + d^2)))/((b*c - a*d)^2*(c^2 + d^2)^2*f*(c + d*Tan[e + f*
x]))

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3650

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

Rule 3730

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Ta
n[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3732

Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2)/(((a_) + (b_.)*tan[(e_.) + (f_.)
*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d)
)*(x/((a^2 + b^2)*(c^2 + d^2))), x] + (Dist[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d)*(a^2 + b^2)), Int[(b - a*Tan[
e + f*x])/(a + b*Tan[e + f*x]), x], x] - Dist[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)), Int[(d - c*Ta
n[e + f*x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ
[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx &=\frac {d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {-2 \left (a c d-b \left (c^2+d^2\right )\right )-2 d (b c-a d) \tan (e+f x)+2 b d^2 \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{2 (b c-a d) \left (c^2+d^2\right )}\\ &=\frac {d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {-2 \left (2 a b c^3 d-a^2 d^2 \left (c^2-d^2\right )-b^2 \left (c^2+d^2\right )^2\right )-4 c d (b c-a d)^2 \tan (e+f x)-2 b d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{2 (b c-a d)^2 \left (c^2+d^2\right )^2}\\ &=-\frac {\left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {b^4 \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right ) (b c-a d)^3}+\frac {\left (d^2 \left (8 a b c^3 d-a^2 d^2 \left (3 c^2-d^2\right )-b^2 \left (6 c^4+3 c^2 d^2+d^4\right )\right )\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^3 \left (c^2+d^2\right )^3}\\ &=-\frac {\left (b d \left (3 c^2-d^2\right )-a \left (c^3-3 c d^2\right )\right ) x}{\left (a^2+b^2\right ) \left (c^2+d^2\right )^3}+\frac {b^4 \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right ) (b c-a d)^3 f}+\frac {d^2 \left (8 a b c^3 d-a^2 d^2 \left (3 c^2-d^2\right )-b^2 \left (6 c^4+3 c^2 d^2+d^4\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^3 \left (c^2+d^2\right )^3 f}+\frac {d^2}{2 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 5.60, size = 409, normalized size = 1.43 \begin {gather*} \frac {\frac {(b c-a d)^3 \left (a \sqrt {-b^2} c \left (c^2-3 d^2\right )+b \left (-a+\sqrt {-b^2}\right ) d \left (-3 c^2+d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )-2 b^5 \left (c^2+d^2\right )^3 \log (a+b \tan (e+f x))-(b c-a d)^3 \left (a \sqrt {-b^2} c \left (c^2-3 d^2\right )+b \left (a+\sqrt {-b^2}\right ) d \left (-3 c^2+d^2\right )-b^2 \left (c^3-3 c d^2\right )\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )-2 b \left (a^2+b^2\right ) d^2 \left (8 a b c^3 d+a^2 d^2 \left (-3 c^2+d^2\right )-b^2 \left (6 c^4+3 c^2 d^2+d^4\right )\right ) \log (c+d \tan (e+f x))}{b \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )^2}-\frac {d^2}{(c+d \tan (e+f x))^2}+\frac {2 d^2 \left (-2 a c d+b \left (3 c^2+d^2\right )\right )}{(-b c+a d) \left (c^2+d^2\right ) (c+d \tan (e+f x))}}{2 (-b c+a d) \left (c^2+d^2\right ) f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])*(c + d*Tan[e + f*x])^3),x]

[Out]

(((b*c - a*d)^3*(a*Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*(-a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + b^2*(c^3 - 3*c*d^2))*Lo
g[Sqrt[-b^2] - b*Tan[e + f*x]] - 2*b^5*(c^2 + d^2)^3*Log[a + b*Tan[e + f*x]] - (b*c - a*d)^3*(a*Sqrt[-b^2]*c*(
c^2 - 3*d^2) + b*(a + Sqrt[-b^2])*d*(-3*c^2 + d^2) - b^2*(c^3 - 3*c*d^2))*Log[Sqrt[-b^2] + b*Tan[e + f*x]] - 2
*b*(a^2 + b^2)*d^2*(8*a*b*c^3*d + a^2*d^2*(-3*c^2 + d^2) - b^2*(6*c^4 + 3*c^2*d^2 + d^4))*Log[c + d*Tan[e + f*
x]])/(b*(a^2 + b^2)*(b*c - a*d)^2*(c^2 + d^2)^2) - d^2/(c + d*Tan[e + f*x])^2 + (2*d^2*(-2*a*c*d + b*(3*c^2 +
d^2)))/((-(b*c) + a*d)*(c^2 + d^2)*(c + d*Tan[e + f*x])))/(2*(-(b*c) + a*d)*(c^2 + d^2)*f)

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Maple [A]
time = 0.72, size = 311, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-3 a \,c^{2} d +a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}-3 b \,c^{2} d +b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (c^{2}+d^{2}\right )^{3}}-\frac {d^{2}}{2 \left (a d -b c \right ) \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{2} \left (2 a c d -3 b \,c^{2}-b \,d^{2}\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{2} \left (3 a^{2} c^{2} d^{2}-a^{2} d^{4}-8 a b \,c^{3} d +6 b^{2} c^{4}+3 b^{2} c^{2} d^{2}+b^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{3}}-\frac {b^{4} \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (a d -b c \right )^{3}}}{f}\)	\(311\)
default	\(\frac {\frac {\frac {\left (-3 a \,c^{2} d +a \,d^{3}-b \,c^{3}+3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a \,c^{3}-3 a c \,d^{2}-3 b \,c^{2} d +b \,d^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (c^{2}+d^{2}\right )^{3}}-\frac {d^{2}}{2 \left (a d -b c \right ) \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}-\frac {d^{2} \left (2 a c d -3 b \,c^{2}-b \,d^{2}\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )^{2} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{2} \left (3 a^{2} c^{2} d^{2}-a^{2} d^{4}-8 a b \,c^{3} d +6 b^{2} c^{4}+3 b^{2} c^{2} d^{2}+b^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{3} \left (c^{2}+d^{2}\right )^{3}}-\frac {b^{4} \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right ) \left (a d -b c \right )^{3}}}{f}\)	\(311\)
norman	\(\frac {\frac {\left (a \,c^{3}-3 a c \,d^{2}-3 b \,c^{2} d +b \,d^{3}\right ) c^{2} x}{\left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {\left (-2 a \,d^{5} c +3 b \,c^{2} d^{4}+b \,d^{6}\right ) \tan \left (f x +e \right )}{f d \left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a \,c^{3}-3 a c \,d^{2}-3 b \,c^{2} d +b \,d^{3}\right ) d^{2} x \left (\tan ^{2}\left (f x +e \right )\right )}{\left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right ) \left (a^{2}+b^{2}\right )}+\frac {-5 a \,c^{2} d^{5}-a \,d^{7}+7 b \,c^{3} d^{4}+3 b c \,d^{6}}{2 f \,d^{2} \left (c^{4}+2 c^{2} d^{2}+d^{4}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 d \left (a \,c^{3}-3 a c \,d^{2}-3 b \,c^{2} d +b \,d^{3}\right ) c x \tan \left (f x +e \right )}{\left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {d^{2} \left (3 a^{2} c^{2} d^{2}-a^{2} d^{4}-8 a b \,c^{3} d +6 b^{2} c^{4}+3 b^{2} c^{2} d^{2}+b^{2} d^{4}\right ) \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (c^{6}+3 c^{4} d^{2}+3 c^{2} d^{4}+d^{6}\right )}-\frac {\left (3 a \,c^{2} d -a \,d^{3}+b \,c^{3}-3 b c \,d^{2}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{2} c^{6}+3 a^{2} c^{4} d^{2}+3 a^{2} c^{2} d^{4}+a^{2} d^{6}+b^{2} c^{6}+3 b^{2} c^{4} d^{2}+3 b^{2} c^{2} d^{4}+b^{2} d^{6}\right )}-\frac {b^{4} \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (a^{2}+b^{2}\right ) f}\)	\(695\)
risch	\(\text {Expression too large to display}\)	\(4160\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(1/(a^2+b^2)/(c^2+d^2)^3*(1/2*(-3*a*c^2*d+a*d^3-b*c^3+3*b*c*d^2)*ln(1+tan(f*x+e)^2)+(a*c^3-3*a*c*d^2-3*b*c
^2*d+b*d^3)*arctan(tan(f*x+e)))-1/2*d^2/(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^2-d^2*(2*a*c*d-3*b*c^2-b*d^2)/(a*
d-b*c)^2/(c^2+d^2)^2/(c+d*tan(f*x+e))+d^2*(3*a^2*c^2*d^2-a^2*d^4-8*a*b*c^3*d+6*b^2*c^4+3*b^2*c^2*d^2+b^2*d^4)/
(a*d-b*c)^3/(c^2+d^2)^3*ln(c+d*tan(f*x+e))-b^4/(a^2+b^2)/(a*d-b*c)^3*ln(a+b*tan(f*x+e)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (290) = 580\).
time = 0.60, size = 806, normalized size = 2.82 \begin {gather*} \frac {\frac {2 \, b^{4} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} b^{3} + b^{5}\right )} c^{3} - 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} c^{2} d + 3 \, {\left (a^{4} b + a^{2} b^{3}\right )} c d^{2} - {\left (a^{5} + a^{3} b^{2}\right )} d^{3}} + \frac {2 \, {\left (a c^{3} - 3 \, b c^{2} d - 3 \, a c d^{2} + b d^{3}\right )} {\left (f x + e\right )}}{{\left (a^{2} + b^{2}\right )} c^{6} + 3 \, {\left (a^{2} + b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4} + {\left (a^{2} + b^{2}\right )} d^{6}} - \frac {2 \, {\left (6 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + 3 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4} - {\left (a^{2} - b^{2}\right )} d^{6}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{3} c^{9} - 3 \, a b^{2} c^{8} d + 3 \, a^{2} b c d^{8} - a^{3} d^{9} + 3 \, {\left (a^{2} b + b^{3}\right )} c^{7} d^{2} - {\left (a^{3} + 9 \, a b^{2}\right )} c^{6} d^{3} + 3 \, {\left (3 \, a^{2} b + b^{3}\right )} c^{5} d^{4} - 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} c^{4} d^{5} + {\left (9 \, a^{2} b + b^{3}\right )} c^{3} d^{6} - 3 \, {\left (a^{3} + a b^{2}\right )} c^{2} d^{7}} - \frac {{\left (b c^{3} + 3 \, a c^{2} d - 3 \, b c d^{2} - a d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} c^{6} + 3 \, {\left (a^{2} + b^{2}\right )} c^{4} d^{2} + 3 \, {\left (a^{2} + b^{2}\right )} c^{2} d^{4} + {\left (a^{2} + b^{2}\right )} d^{6}} + \frac {7 \, b c^{3} d^{2} - 5 \, a c^{2} d^{3} + 3 \, b c d^{4} - a d^{5} + 2 \, {\left (3 \, b c^{2} d^{3} - 2 \, a c d^{4} + b d^{5}\right )} \tan \left (f x + e\right )}{b^{2} c^{8} - 2 \, a b c^{7} d - 4 \, a b c^{5} d^{3} - 2 \, a b c^{3} d^{5} + a^{2} c^{2} d^{6} + {\left (a^{2} + 2 \, b^{2}\right )} c^{6} d^{2} + {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{4} + {\left (b^{2} c^{6} d^{2} - 2 \, a b c^{5} d^{3} - 4 \, a b c^{3} d^{5} - 2 \, a b c d^{7} + a^{2} d^{8} + {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{4} + {\left (2 \, a^{2} + b^{2}\right )} c^{2} d^{6}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (b^{2} c^{7} d - 2 \, a b c^{6} d^{2} - 4 \, a b c^{4} d^{4} - 2 \, a b c^{2} d^{6} + a^{2} c d^{7} + {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{3} + {\left (2 \, a^{2} + b^{2}\right )} c^{3} d^{5}\right )} \tan \left (f x + e\right )}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1855 vs. \(2 (290) = 580\).
time = 2.90, size = 1855, normalized size = 6.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/2*(9*(a^2*b^2 + b^4)*c^4*d^4 - 16*(a^3*b + a*b^3)*c^3*d^5 + (7*a^4 + 10*a^2*b^2 + 3*b^4)*c^2*d^6 - 4*(a^3*b
+ a*b^3)*c*d^7 + (a^4 + a^2*b^2)*d^8 + 2*(a*b^3*c^8 - a^3*b*c^2*d^6 - 3*(a^2*b^2 + b^4)*c^7*d + 3*(a^3*b + 2*a
*b^3)*c^6*d^2 - (a^4 - b^4)*c^5*d^3 - 3*(2*a^3*b + a*b^3)*c^4*d^4 + 3*(a^4 + a^2*b^2)*c^3*d^5)*f*x - (7*(a^2*b
^2 + b^4)*c^4*d^4 - 12*(a^3*b + a*b^3)*c^3*d^5 + (5*a^4 + 6*a^2*b^2 + b^4)*c^2*d^6 - (a^4 + a^2*b^2)*d^8 - 2*(
a*b^3*c^6*d^2 - a^3*b*d^8 - 3*(a^2*b^2 + b^4)*c^5*d^3 + 3*(a^3*b + 2*a*b^3)*c^4*d^4 - (a^4 - b^4)*c^3*d^5 - 3*
(2*a^3*b + a*b^3)*c^2*d^6 + 3*(a^4 + a^2*b^2)*c*d^7)*f*x)*tan(f*x + e)^2 + (b^4*c^8 + 3*b^4*c^6*d^2 + 3*b^4*c^
4*d^4 + b^4*c^2*d^6 + (b^4*c^6*d^2 + 3*b^4*c^4*d^4 + 3*b^4*c^2*d^6 + b^4*d^8)*tan(f*x + e)^2 + 2*(b^4*c^7*d +
3*b^4*c^5*d^3 + 3*b^4*c^3*d^5 + b^4*c*d^7)*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)) - (6*(a^2*b^2 + b^4)*c^6*d^2 - 8*(a^3*b + a*b^3)*c^5*d^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*c^4*
d^4 - (a^4 - b^4)*c^2*d^6 + (6*(a^2*b^2 + b^4)*c^4*d^4 - 8*(a^3*b + a*b^3)*c^3*d^5 + 3*(a^4 + 2*a^2*b^2 + b^4)
*c^2*d^6 - (a^4 - b^4)*d^8)*tan(f*x + e)^2 + 2*(6*(a^2*b^2 + b^4)*c^5*d^3 - 8*(a^3*b + a*b^3)*c^4*d^4 + 3*(a^4
+ 2*a^2*b^2 + b^4)*c^3*d^5 - (a^4 - b^4)*c*d^7)*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*(4*(a^2*b^2 + b^4)*c^5*d^3 - 7*(a^3*b + a*b^3)*c^4*d^4 + 3*(a^4 - b^4)*c^3*d^5
+ 6*(a^3*b + a*b^3)*c^2*d^6 - (3*a^4 + 4*a^2*b^2 + b^4)*c*d^7 + (a^3*b + a*b^3)*d^8 - 2*(a*b^3*c^7*d - a^3*b*c
*d^7 - 3*(a^2*b^2 + b^4)*c^6*d^2 + 3*(a^3*b + 2*a*b^3)*c^5*d^3 - (a^4 - b^4)*c^4*d^4 - 3*(2*a^3*b + a*b^3)*c^3
*d^5 + 3*(a^4 + a^2*b^2)*c^2*d^6)*f*x)*tan(f*x + e))/(((a^2*b^3 + b^5)*c^9*d^2 - 3*(a^3*b^2 + a*b^4)*c^8*d^3 +
3*(a^4*b + 2*a^2*b^3 + b^5)*c^7*d^4 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^6*d^5 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^
5*d^6 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^4*d^7 + (9*a^4*b + 10*a^2*b^3 + b^5)*c^3*d^8 - 3*(a^5 + 2*a^3*b^2 + a*
b^4)*c^2*d^9 + 3*(a^4*b + a^2*b^3)*c*d^10 - (a^5 + a^3*b^2)*d^11)*f*tan(f*x + e)^2 + 2*((a^2*b^3 + b^5)*c^10*d
- 3*(a^3*b^2 + a*b^4)*c^9*d^2 + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^8*d^3 - (a^5 + 10*a^3*b^2 + 9*a*b^4)*c^7*d^4 +
3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^6*d^5 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^5*d^6 + (9*a^4*b + 10*a^2*b^3 + b^5)*c
^4*d^7 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^3*d^8 + 3*(a^4*b + a^2*b^3)*c^2*d^9 - (a^5 + a^3*b^2)*c*d^10)*f*tan(f*x
+ e) + ((a^2*b^3 + b^5)*c^11 - 3*(a^3*b^2 + a*b^4)*c^10*d + 3*(a^4*b + 2*a^2*b^3 + b^5)*c^9*d^2 - (a^5 + 10*a
^3*b^2 + 9*a*b^4)*c^8*d^3 + 3*(3*a^4*b + 4*a^2*b^3 + b^5)*c^7*d^4 - 3*(a^5 + 4*a^3*b^2 + 3*a*b^4)*c^6*d^5 + (9
*a^4*b + 10*a^2*b^3 + b^5)*c^5*d^6 - 3*(a^5 + 2*a^3*b^2 + a*b^4)*c^4*d^7 + 3*(a^4*b + a^2*b^3)*c^3*d^8 - (a^5
+ a^3*b^2)*c^2*d^9)*f)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))**3,x)

[Out]

Exception raised: NotImplementedError >> no valid subset found

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (290) = 580\).
time = 0.68, size = 1112, normalized size = 3.89 \begin {gather*} \frac {\frac {2 \, b^{5} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{2} b^{4} c^{3} + b^{6} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 3 \, a b^{5} c^{2} d + 3 \, a^{4} b^{2} c d^{2} + 3 \, a^{2} b^{4} c d^{2} - a^{5} b d^{3} - a^{3} b^{3} d^{3}} + \frac {2 \, {\left (a c^{3} - 3 \, b c^{2} d - 3 \, a c d^{2} + b d^{3}\right )} {\left (f x + e\right )}}{a^{2} c^{6} + b^{2} c^{6} + 3 \, a^{2} c^{4} d^{2} + 3 \, b^{2} c^{4} d^{2} + 3 \, a^{2} c^{2} d^{4} + 3 \, b^{2} c^{2} d^{4} + a^{2} d^{6} + b^{2} d^{6}} - \frac {{\left (b c^{3} + 3 \, a c^{2} d - 3 \, b c d^{2} - a d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} c^{6} + b^{2} c^{6} + 3 \, a^{2} c^{4} d^{2} + 3 \, b^{2} c^{4} d^{2} + 3 \, a^{2} c^{2} d^{4} + 3 \, b^{2} c^{2} d^{4} + a^{2} d^{6} + b^{2} d^{6}} - \frac {2 \, {\left (6 \, b^{2} c^{4} d^{3} - 8 \, a b c^{3} d^{4} + 3 \, a^{2} c^{2} d^{5} + 3 \, b^{2} c^{2} d^{5} - a^{2} d^{7} + b^{2} d^{7}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{3} c^{9} d - 3 \, a b^{2} c^{8} d^{2} + 3 \, a^{2} b c^{7} d^{3} + 3 \, b^{3} c^{7} d^{3} - a^{3} c^{6} d^{4} - 9 \, a b^{2} c^{6} d^{4} + 9 \, a^{2} b c^{5} d^{5} + 3 \, b^{3} c^{5} d^{5} - 3 \, a^{3} c^{4} d^{6} - 9 \, a b^{2} c^{4} d^{6} + 9 \, a^{2} b c^{3} d^{7} + b^{3} c^{3} d^{7} - 3 \, a^{3} c^{2} d^{8} - 3 \, a b^{2} c^{2} d^{8} + 3 \, a^{2} b c d^{9} - a^{3} d^{10}} + \frac {18 \, b^{2} c^{4} d^{4} \tan \left (f x + e\right )^{2} - 24 \, a b c^{3} d^{5} \tan \left (f x + e\right )^{2} + 9 \, a^{2} c^{2} d^{6} \tan \left (f x + e\right )^{2} + 9 \, b^{2} c^{2} d^{6} \tan \left (f x + e\right )^{2} - 3 \, a^{2} d^{8} \tan \left (f x + e\right )^{2} + 3 \, b^{2} d^{8} \tan \left (f x + e\right )^{2} + 42 \, b^{2} c^{5} d^{3} \tan \left (f x + e\right ) - 58 \, a b c^{4} d^{4} \tan \left (f x + e\right ) + 22 \, a^{2} c^{3} d^{5} \tan \left (f x + e\right ) + 26 \, b^{2} c^{3} d^{5} \tan \left (f x + e\right ) - 12 \, a b c^{2} d^{6} \tan \left (f x + e\right ) - 2 \, a^{2} c d^{7} \tan \left (f x + e\right ) + 8 \, b^{2} c d^{7} \tan \left (f x + e\right ) - 2 \, a b d^{8} \tan \left (f x + e\right ) + 25 \, b^{2} c^{6} d^{2} - 36 \, a b c^{5} d^{3} + 14 \, a^{2} c^{4} d^{4} + 19 \, b^{2} c^{4} d^{4} - 16 \, a b c^{3} d^{5} + 3 \, a^{2} c^{2} d^{6} + 6 \, b^{2} c^{2} d^{6} - 4 \, a b c d^{7} + a^{2} d^{8}}{{\left (b^{3} c^{9} - 3 \, a b^{2} c^{8} d + 3 \, a^{2} b c^{7} d^{2} + 3 \, b^{3} c^{7} d^{2} - a^{3} c^{6} d^{3} - 9 \, a b^{2} c^{6} d^{3} + 9 \, a^{2} b c^{5} d^{4} + 3 \, b^{3} c^{5} d^{4} - 3 \, a^{3} c^{4} d^{5} - 9 \, a b^{2} c^{4} d^{5} + 9 \, a^{2} b c^{3} d^{6} + b^{3} c^{3} d^{6} - 3 \, a^{3} c^{2} d^{7} - 3 \, a b^{2} c^{2} d^{7} + 3 \, a^{2} b c d^{8} - a^{3} d^{9}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*b^5*log(abs(b*tan(f*x + e) + a))/(a^2*b^4*c^3 + b^6*c^3 - 3*a^3*b^3*c^2*d - 3*a*b^5*c^2*d + 3*a^4*b^2*c
*d^2 + 3*a^2*b^4*c*d^2 - a^5*b*d^3 - a^3*b^3*d^3) + 2*(a*c^3 - 3*b*c^2*d - 3*a*c*d^2 + b*d^3)*(f*x + e)/(a^2*c
^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) - (b*c^3 + 3
*a*c^2*d - 3*b*c*d^2 - a*d^3)*log(tan(f*x + e)^2 + 1)/(a^2*c^6 + b^2*c^6 + 3*a^2*c^4*d^2 + 3*b^2*c^4*d^2 + 3*a
^2*c^2*d^4 + 3*b^2*c^2*d^4 + a^2*d^6 + b^2*d^6) - 2*(6*b^2*c^4*d^3 - 8*a*b*c^3*d^4 + 3*a^2*c^2*d^5 + 3*b^2*c^2
*d^5 - a^2*d^7 + b^2*d^7)*log(abs(d*tan(f*x + e) + c))/(b^3*c^9*d - 3*a*b^2*c^8*d^2 + 3*a^2*b*c^7*d^3 + 3*b^3*
c^7*d^3 - a^3*c^6*d^4 - 9*a*b^2*c^6*d^4 + 9*a^2*b*c^5*d^5 + 3*b^3*c^5*d^5 - 3*a^3*c^4*d^6 - 9*a*b^2*c^4*d^6 +
9*a^2*b*c^3*d^7 + b^3*c^3*d^7 - 3*a^3*c^2*d^8 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10) + (18*b^2*c^4*d^4*
tan(f*x + e)^2 - 24*a*b*c^3*d^5*tan(f*x + e)^2 + 9*a^2*c^2*d^6*tan(f*x + e)^2 + 9*b^2*c^2*d^6*tan(f*x + e)^2 -
3*a^2*d^8*tan(f*x + e)^2 + 3*b^2*d^8*tan(f*x + e)^2 + 42*b^2*c^5*d^3*tan(f*x + e) - 58*a*b*c^4*d^4*tan(f*x +
e) + 22*a^2*c^3*d^5*tan(f*x + e) + 26*b^2*c^3*d^5*tan(f*x + e) - 12*a*b*c^2*d^6*tan(f*x + e) - 2*a^2*c*d^7*tan
(f*x + e) + 8*b^2*c*d^7*tan(f*x + e) - 2*a*b*d^8*tan(f*x + e) + 25*b^2*c^6*d^2 - 36*a*b*c^5*d^3 + 14*a^2*c^4*d
^4 + 19*b^2*c^4*d^4 - 16*a*b*c^3*d^5 + 3*a^2*c^2*d^6 + 6*b^2*c^2*d^6 - 4*a*b*c*d^7 + a^2*d^8)/((b^3*c^9 - 3*a*
b^2*c^8*d + 3*a^2*b*c^7*d^2 + 3*b^3*c^7*d^2 - a^3*c^6*d^3 - 9*a*b^2*c^6*d^3 + 9*a^2*b*c^5*d^4 + 3*b^3*c^5*d^4
- 3*a^3*c^4*d^5 - 9*a*b^2*c^4*d^5 + 9*a^2*b*c^3*d^6 + b^3*c^3*d^6 - 3*a^3*c^2*d^7 - 3*a*b^2*c^2*d^7 + 3*a^2*b*
c*d^8 - a^3*d^9)*(d*tan(f*x + e) + c)^2))/f

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Mupad [B]
time = 12.64, size = 719, normalized size = 2.51 \begin {gather*} \frac {\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d^4\,\left (3\,a^2\,c^2+3\,b^2\,c^2\right )-d^6\,\left (a^2-b^2\right )+6\,b^2\,c^4\,d^2-8\,a\,b\,c^3\,d^3\right )}{f\,\left (a^3\,c^6\,d^3+3\,a^3\,c^4\,d^5+3\,a^3\,c^2\,d^7+a^3\,d^9-3\,a^2\,b\,c^7\,d^2-9\,a^2\,b\,c^5\,d^4-9\,a^2\,b\,c^3\,d^6-3\,a^2\,b\,c\,d^8+3\,a\,b^2\,c^8\,d+9\,a\,b^2\,c^6\,d^3+9\,a\,b^2\,c^4\,d^5+3\,a\,b^2\,c^2\,d^7-b^3\,c^9-3\,b^3\,c^7\,d^2-3\,b^3\,c^5\,d^4-b^3\,c^3\,d^6\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (\frac {-b\,c^3-3\,a\,c^2\,d+3\,b\,c\,d^2+a\,d^3}{\left (a^2+b^2\right )\,{\left (c^2+d^2\right )}^3}+\frac {d^2\,\left (3\,c^2-d^2\right )}{\left (a\,d-b\,c\right )\,{\left (c^2+d^2\right )}^3}+\frac {b^2\,d^2}{{\left (a\,d-b\,c\right )}^3\,\left (c^2+d^2\right )}-\frac {2\,b\,c\,d^2}{{\left (a\,d-b\,c\right )}^2\,{\left (c^2+d^2\right )}^2}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}{2\,f\,\left (a\,c^3\,1{}\mathrm {i}+a\,d^3-b\,c^3+b\,d^3\,1{}\mathrm {i}-a\,c\,d^2\,3{}\mathrm {i}-3\,a\,c^2\,d+3\,b\,c\,d^2-b\,c^2\,d\,3{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{2\,f\,\left (a\,c^3\,1{}\mathrm {i}-a\,d^3+b\,c^3+b\,d^3\,1{}\mathrm {i}-a\,c\,d^2\,3{}\mathrm {i}+3\,a\,c^2\,d-3\,b\,c\,d^2-b\,c^2\,d\,3{}\mathrm {i}\right )}-\frac {\frac {-7\,b\,c^3\,d^2+5\,a\,c^2\,d^3-3\,b\,c\,d^4+a\,d^5}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (c^4+2\,c^2\,d^2+d^4\right )}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (3\,b\,c^2\,d^3-2\,a\,c\,d^4+b\,d^5\right )}{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^3),x)

[Out]

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

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