3.13.14 \(\int \frac {1}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx\) [1214]

Optimal. Leaf size=183 \[ \frac {\left (a^2 c-b^2 c-2 a b d\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}+\frac {b^2 \left (2 a b c-3 a^2 d-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^2 f}+\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^2 \left (c^2+d^2\right ) f}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3650, 3732, 3611} \begin {gather*} \frac {x \left (a^2 c-2 a b d-b^2 c\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}-\frac {b^2}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}+\frac {b^2 \left (-3 a^2 d+2 a b c-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^2 (b c-a d)^2}+\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right ) (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])),x]

[Out]

((a^2*c - b^2*c - 2*a*b*d)*x)/((a^2 + b^2)^2*(c^2 + d^2)) + (b^2*(2*a*b*c - 3*a^2*d - b^2*d)*Log[a*Cos[e + f*x
] + b*Sin[e + f*x]])/((a^2 + b^2)^2*(b*c - a*d)^2*f) + (d^3*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/((b*c - a*d)
^2*(c^2 + d^2)*f) - b^2/((a^2 + b^2)*(b*c - a*d)*f*(a + b*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 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))^2 (c+d \tan (e+f x))} \, dx &=-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac {\int \frac {-a b c+a^2 d+b^2 d+b (b c-a d) \tan (e+f x)+b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=\frac {\left (a^2 c-b^2 c-2 a b d\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}+\frac {\left (b^2 \left (2 a b c-3 a^2 d-b^2 d\right )\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^2}+\frac {d^3 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(b c-a d)^2 \left (c^2+d^2\right )}\\ &=\frac {\left (a^2 c-b^2 c-2 a b d\right ) x}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}+\frac {b^2 \left (2 a b c-3 a^2 d-b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^2 f}+\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{(b c-a d)^2 \left (c^2+d^2\right ) f}-\frac {b^2}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 3.35, size = 302, normalized size = 1.65 \begin {gather*} -\frac {\frac {\left (2 a b c+a^2 d-b^2 d+\frac {\sqrt {-b^2} \left (a^2 c-b^2 c-2 a b d\right )}{b}\right ) \log \left (\sqrt {-b^2}-b \tan (e+f x)\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}+\frac {2 b^2 \left (-2 a b c+3 a^2 d+b^2 d\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right )^2 (b c-a d)^2}+\frac {\left (2 a b c+a^2 d-b^2 d+\frac {\sqrt {-b^2} \left (-a^2 c+b^2 c+2 a b d\right )}{b}\right ) \log \left (\sqrt {-b^2}+b \tan (e+f x)\right )}{\left (a^2+b^2\right )^2 \left (c^2+d^2\right )}-\frac {2 d^3 \log (c+d \tan (e+f x))}{(b c-a d)^2 \left (c^2+d^2\right )}+\frac {2 b^2}{\left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Tan[e + f*x])^2*(c + d*Tan[e + f*x])),x]

[Out]

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

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Maple [A]
time = 0.46, size = 202, normalized size = 1.10

method	result	size
derivativedivides	\(\frac {\frac {b^{2}}{\left (a d -b c \right ) \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}-\frac {b^{2} \left (3 a^{2} d -2 a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-a^{2} d -2 a b c +b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c -2 a b d -b^{2} c \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )}+\frac {d^{3} \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )}}{f}\)	\(202\)
default	\(\frac {\frac {b^{2}}{\left (a d -b c \right ) \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (f x +e \right )\right )}-\frac {b^{2} \left (3 a^{2} d -2 a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (-a^{2} d -2 a b c +b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (a^{2} c -2 a b d -b^{2} c \right ) \arctan \left (\tan \left (f x +e \right )\right )}{\left (a^{2}+b^{2}\right )^{2} \left (c^{2}+d^{2}\right )}+\frac {d^{3} \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (a d -b c \right )^{2} \left (c^{2}+d^{2}\right )}}{f}\)	\(202\)
norman	\(\frac {\frac {a \left (a^{2} c -2 a b d -b^{2} c \right ) x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (c^{2}+d^{2}\right )}+\frac {b^{2}}{\left (a d -b c \right ) f \left (a^{2}+b^{2}\right )}+\frac {b \left (a^{2} c -2 a b d -b^{2} c \right ) x \tan \left (f x +e \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (c^{2}+d^{2}\right )}}{a +b \tan \left (f x +e \right )}+\frac {d^{3} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (a^{2} c^{2} d^{2}+a^{2} d^{4}-2 a b \,c^{3} d -2 a b c \,d^{3}+b^{2} c^{4}+b^{2} c^{2} d^{2}\right )}-\frac {\left (a^{2} d +2 a b c -b^{2} d \right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (c^{2}+d^{2}\right )}-\frac {b^{2} \left (3 a^{2} d -2 a b c +b^{2} d \right ) \ln \left (a +b \tan \left (f x +e \right )\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) f \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(346\)
risch	\(-\frac {x}{i a^{2} d +2 i a b c -i b^{2} d -a^{2} c +2 a b d +b^{2} c}-\frac {2 i d^{3} x}{a^{2} c^{2} d^{2}+a^{2} d^{4}-2 a b \,c^{3} d -2 a b c \,d^{3}+b^{2} c^{4}+b^{2} c^{2} d^{2}}-\frac {2 i d^{3} e}{f \left (a^{2} c^{2} d^{2}+a^{2} d^{4}-2 a b \,c^{3} d -2 a b c \,d^{3}+b^{2} c^{4}+b^{2} c^{2} d^{2}\right )}+\frac {6 i b^{2} a^{2} d x}{a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}}+\frac {6 i b^{2} a^{2} d e}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}-\frac {4 i b^{3} a c x}{a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}}-\frac {4 i b^{3} a c e}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}+\frac {2 i b^{4} d x}{a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}}+\frac {2 i b^{4} d e}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}-\frac {2 i b^{3}}{\left (-i a +b \right ) f \left (-a d +b c \right ) \left (i a +b \right )^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )} b +i a \,{\mathrm e}^{2 i \left (f x +e \right )}-b +i a \right )}+\frac {d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{f \left (a^{2} c^{2} d^{2}+a^{2} d^{4}-2 a b \,c^{3} d -2 a b c \,d^{3}+b^{2} c^{4}+b^{2} c^{2} d^{2}\right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) a^{2} d}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) a c}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}-\frac {b^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i b +a}{i b -a}\right ) d}{f \left (a^{6} d^{2}-2 a^{5} b c d +a^{4} b^{2} c^{2}+2 a^{4} b^{2} d^{2}-4 a^{3} b^{3} c d +2 a^{2} b^{4} c^{2}+a^{2} b^{4} d^{2}-2 a \,b^{5} c d +b^{6} c^{2}\right )}\)	\(1273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (188) = 376\).
time = 0.58, size = 389, normalized size = 2.13 \begin {gather*} \frac {\frac {2 \, d^{3} \log \left (d \tan \left (f x + e\right ) + c\right )}{b^{2} c^{4} - 2 \, a b c^{3} d - 2 \, a b c d^{3} + a^{2} d^{4} + {\left (a^{2} + b^{2}\right )} c^{2} d^{2}} - \frac {2 \, {\left (2 \, a b d - {\left (a^{2} - b^{2}\right )} c\right )} {\left (f x + e\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} - \frac {2 \, b^{2}}{{\left (a^{3} b + a b^{3}\right )} c - {\left (a^{4} + a^{2} b^{2}\right )} d + {\left ({\left (a^{2} b^{2} + b^{4}\right )} c - {\left (a^{3} b + a b^{3}\right )} d\right )} \tan \left (f x + e\right )} + \frac {2 \, {\left (2 \, a b^{3} c - {\left (3 \, a^{2} b^{2} + b^{4}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} c^{2} - 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} c d + {\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d^{2}} - \frac {{\left (2 \, a b c + {\left (a^{2} - b^{2}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} c^{2} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (188) = 376\).
time = 1.51, size = 766, normalized size = 4.19 \begin {gather*} -\frac {2 \, b^{5} c^{3} - 2 \, a b^{4} c^{2} d + 2 \, b^{5} c d^{2} - 2 \, a b^{4} d^{3} + 2 \, {\left (2 \, a^{4} b c^{2} d + 2 \, a^{4} b d^{3} - {\left (a^{3} b^{2} - a b^{4}\right )} c^{3} - {\left (a^{5} + 3 \, a^{3} b^{2}\right )} c d^{2}\right )} f x - {\left (2 \, a^{2} b^{3} c^{3} + 2 \, a^{2} b^{3} c d^{2} - {\left (3 \, a^{3} b^{2} + a b^{4}\right )} c^{2} d - {\left (3 \, a^{3} b^{2} + a b^{4}\right )} d^{3} + {\left (2 \, a b^{4} c^{3} + 2 \, a b^{4} c d^{2} - {\left (3 \, a^{2} b^{3} + b^{5}\right )} c^{2} d - {\left (3 \, a^{2} b^{3} + b^{5}\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 ) - {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d^{3} \tan \left (f x + e\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d^{3}\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 (a b^{4} c^{3} - a^{2} b^{3} c^{2} d + a b^{4} c d^{2} - a^{2} b^{3} d^{3} - {\left (2 \, a^{3} b^{2} c^{2} d + 2 \, a^{3} b^{2} d^{3} - {\left (a^{2} b^{3} - b^{5}\right )} c^{3} - {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} c d^{2}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left ({\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} c^{4} - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} c^{3} d + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} c^{2} d^{2} - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} c d^{3} + {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d^{4}\right )} f \tan \left (f x + e\right ) + {\left ({\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} c^{4} - 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c^{3} d + {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} c^{2} d^{2} - 2 \, {\left (a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} c d^{3} + {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d^{4}\right )} f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e)),x, algorithm="fricas")

[Out]

-1/2*(2*b^5*c^3 - 2*a*b^4*c^2*d + 2*b^5*c*d^2 - 2*a*b^4*d^3 + 2*(2*a^4*b*c^2*d + 2*a^4*b*d^3 - (a^3*b^2 - a*b^
4)*c^3 - (a^5 + 3*a^3*b^2)*c*d^2)*f*x - (2*a^2*b^3*c^3 + 2*a^2*b^3*c*d^2 - (3*a^3*b^2 + a*b^4)*c^2*d - (3*a^3*
b^2 + a*b^4)*d^3 + (2*a*b^4*c^3 + 2*a*b^4*c*d^2 - (3*a^2*b^3 + b^5)*c^2*d - (3*a^2*b^3 + b^5)*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)) - ((a^4*b + 2*a^2*b^3 + b^5)*d^3*
tan(f*x + e) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3)*log((d^2*tan(f*x + e)^2 + 2*c*d*tan(f*x + e) + c^2)/(tan(f*x + e
)^2 + 1)) - 2*(a*b^4*c^3 - a^2*b^3*c^2*d + a*b^4*c*d^2 - a^2*b^3*d^3 - (2*a^3*b^2*c^2*d + 2*a^3*b^2*d^3 - (a^2
*b^3 - b^5)*c^3 - (a^4*b + 3*a^2*b^3)*c*d^2)*f*x)*tan(f*x + e))/(((a^4*b^3 + 2*a^2*b^5 + b^7)*c^4 - 2*(a^5*b^2
+ 2*a^3*b^4 + a*b^6)*c^3*d + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c^2*d^2 - 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*
c*d^3 + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d^4)*f*tan(f*x + e) + ((a^5*b^2 + 2*a^3*b^4 + a*b^6)*c^4 - 2*(a^6*b + 2*
a^4*b^3 + a^2*b^5)*c^3*d + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c^2*d^2 - 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*c*d
^3 + (a^7 + 2*a^5*b^2 + a^3*b^4)*d^4)*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))**2/(c+d*tan(f*x+e)),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. 542 vs. \(2 (188) = 376\).
time = 0.57, size = 542, normalized size = 2.96 \begin {gather*} \frac {\frac {2 \, d^{4} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} + b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}} + \frac {2 \, {\left (a^{2} c - b^{2} c - 2 \, a b d\right )} {\left (f x + e\right )}}{a^{4} c^{2} + 2 \, a^{2} b^{2} c^{2} + b^{4} c^{2} + a^{4} d^{2} + 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} - \frac {{\left (2 \, a b c + a^{2} d - b^{2} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{4} c^{2} + 2 \, a^{2} b^{2} c^{2} + b^{4} c^{2} + a^{4} d^{2} + 2 \, a^{2} b^{2} d^{2} + b^{4} d^{2}} + \frac {2 \, {\left (2 \, a b^{4} c - 3 \, a^{2} b^{3} d - b^{5} d\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{4} b^{3} c^{2} + 2 \, a^{2} b^{5} c^{2} + b^{7} c^{2} - 2 \, a^{5} b^{2} c d - 4 \, a^{3} b^{4} c d - 2 \, a b^{6} c d + a^{6} b d^{2} + 2 \, a^{4} b^{3} d^{2} + a^{2} b^{5} d^{2}} - \frac {2 \, {\left (2 \, a b^{4} c \tan \left (f x + e\right ) - 3 \, a^{2} b^{3} d \tan \left (f x + e\right ) - b^{5} d \tan \left (f x + e\right ) + 3 \, a^{2} b^{3} c + b^{5} c - 4 \, a^{3} b^{2} d - 2 \, a b^{4} d\right )}}{{\left (a^{4} b^{2} c^{2} + 2 \, a^{2} b^{4} c^{2} + b^{6} c^{2} - 2 \, a^{5} b c d - 4 \, a^{3} b^{3} c d - 2 \, a b^{5} c d + a^{6} d^{2} + 2 \, a^{4} b^{2} d^{2} + a^{2} b^{4} d^{2}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}}}{2 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(f*x+e))^2/(c+d*tan(f*x+e)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 6.97, size = 309, normalized size = 1.69 \begin {gather*} \frac {b^2}{f\,\left (a\,d-b\,c\right )\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (d\,\left (3\,a^2\,b^2+b^4\right )-2\,a\,b^3\,c\right )}{f\,\left (a^6\,d^2-2\,a^5\,b\,c\,d+a^4\,b^2\,c^2+2\,a^4\,b^2\,d^2-4\,a^3\,b^3\,c\,d+2\,a^2\,b^4\,c^2+a^2\,b^4\,d^2-2\,a\,b^5\,c\,d+b^6\,c^2\right )}+\frac {d^3\,\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}{f\,{\left (a\,d-b\,c\right )}^2\,\left (c^2+d^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,f\,\left (a^2\,c-b^2\,c-2\,a\,b\,d+a^2\,d\,1{}\mathrm {i}-b^2\,d\,1{}\mathrm {i}+a\,b\,c\,2{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,f\,\left (b^2\,c-a^2\,c+2\,a\,b\,d+a^2\,d\,1{}\mathrm {i}-b^2\,d\,1{}\mathrm {i}+a\,b\,c\,2{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*tan(e + f*x))^2*(c + d*tan(e + f*x))),x)

[Out]

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

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