\(\int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [278]

Optimal result

Integrand size = 23, antiderivative size = 111 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a A b-a^2 B+b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {A b-a B}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

(A*a^2-A*b^2+2*B*a*b)*x/(a^2+b^2)^2+(2*A*a*b-B*a^2+B*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^2/d+(-A*b+B*
a)/(a^2+b^2)/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3610, 3612, 3611} \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {A b-a B}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (a^2 (-B)+2 a A b+b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2} \]

[In]

Int[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2,x]

[Out]

((a^2*A - A*b^2 + 2*a*b*B)*x)/(a^2 + b^2)^2 + ((2*a*A*b - a^2*B + b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])
/((a^2 + b^2)^2*d) - (A*b - a*B)/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))

Rule 3610

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

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 3612

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {A b-a B}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a A+b B-(A b-a B) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2} \\ & = \frac {\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}-\frac {A b-a B}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (2 a A b-a^2 B+b^2 B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = \frac {\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a A b-a^2 B+b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {A b-a B}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.33 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.71 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {B ((-i a-b) \log (i-\tan (c+d x))+i (a+i b) \log (i+\tan (c+d x))+2 b \log (a+b \tan (c+d x)))}{a^2+b^2}-(A b-a B) \left (\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}-\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 b \left (-2 a \log (a+b \tan (c+d x))+\frac {a^2+b^2}{a+b \tan (c+d x)}\right )}{\left (a^2+b^2\right )^2}\right )}{2 b d} \]

[In]

Integrate[(A + B*Tan[c + d*x])/(a + b*Tan[c + d*x])^2,x]

[Out]

((B*(((-I)*a - b)*Log[I - Tan[c + d*x]] + I*(a + I*b)*Log[I + Tan[c + d*x]] + 2*b*Log[a + b*Tan[c + d*x]]))/(a
^2 + b^2) - (A*b - a*B)*((I*Log[I - Tan[c + d*x]])/(a + I*b)^2 - (I*Log[I + Tan[c + d*x]])/(a - I*b)^2 + (2*b*
(-2*a*Log[a + b*Tan[c + d*x]] + (a^2 + b^2)/(a + b*Tan[c + d*x])))/(a^2 + b^2)^2))/(2*b*d)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A b -B a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(141\)
default	\(\frac {\frac {\frac {\left (-2 A a b +B \,a^{2}-B \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (A \,a^{2}-A \,b^{2}+2 B a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {A b -B a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(141\)
norman	\(\frac {\frac {a \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b \left (A \,a^{2}-A \,b^{2}+2 B a b \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\left (A b -B a \right ) b \tan \left (d x +c \right )}{a d \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}+\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (2 A a b -B \,a^{2}+B \,b^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(226\)
parallelrisch	\(-\frac {-2 A \,b^{4} \tan \left (d x +c \right )-B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b +2 B \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{3} b -2 A \,a^{2} b^{2} \tan \left (d x +c \right )+2 B \,a^{3} b \tan \left (d x +c \right )+2 B a \,b^{3} \tan \left (d x +c \right )-B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}-4 A \ln \left (a +b \tan \left (d x +c \right )\right ) a^{3} b +2 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{2}-4 A \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a^{2} b^{2}+B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-2 B \ln \left (a +b \tan \left (d x +c \right )\right ) \tan \left (d x +c \right ) a \,b^{3}-2 B \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2} b^{2}+2 B \ln \left (a +b \tan \left (d x +c \right )\right ) a^{4}+2 A \,a^{2} b^{2} d x -4 B \,a^{3} b d x -2 A x \,a^{4} d -2 A \tan \left (d x +c \right ) a^{3} b d x +2 A x \tan \left (d x +c \right ) a \,b^{3} d -4 B x \tan \left (d x +c \right ) a^{2} b^{2} d +2 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b +B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}}{2 \left (a +b \tan \left (d x +c \right )\right ) d a \left (a^{2}+b^{2}\right )^{2}}\)	\(415\)
risch	\(\frac {i x B}{2 i b a -a^{2}+b^{2}}-\frac {x A}{2 i b a -a^{2}+b^{2}}-\frac {4 i a b A x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i a^{2} B x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i B \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 i a b A c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 i a^{2} B c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 i B \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2} A}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 i b B a}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 a b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(482\)

[In]

int((A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^2*(1/2*(-2*A*a*b+B*a^2-B*b^2)*ln(1+tan(d*x+c)^2)+(A*a^2-A*b^2+2*B*a*b)*arctan(tan(d*x+c)))-(A
*b-B*a)/(a^2+b^2)/(a+b*tan(d*x+c))+(2*A*a*b-B*a^2+B*b^2)/(a^2+b^2)^2*ln(a+b*tan(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.00 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {2 \, B a b^{2} - 2 \, A b^{3} + 2 \, {\left (A a^{3} + 2 \, B a^{2} b - A a b^{2}\right )} d x - {\left (B a^{3} - 2 \, A a^{2} b - B a b^{2} + {\left (B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (B a^{2} b - A a b^{2} - {\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]

[In]

integrate((A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/2*(2*B*a*b^2 - 2*A*b^3 + 2*(A*a^3 + 2*B*a^2*b - A*a*b^2)*d*x - (B*a^3 - 2*A*a^2*b - B*a*b^2 + (B*a^2*b - 2*A
*a*b^2 - B*b^3)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - 2*(B
*a^2*b - A*a*b^2 - (A*a^2*b + 2*B*a*b^2 - A*b^3)*d*x)*tan(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d*tan(d*x + c)
+ (a^5 + 2*a^3*b^2 + a*b^4)*d)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.83 (sec) , antiderivative size = 2878, normalized size of antiderivative = 25.93 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*tan(d*x+c))/(a+b*tan(d*x+c))**2,x)

[Out]

Piecewise((zoo*x*(A + B*tan(c))/tan(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((A*x + B*log(tan(c + d*x)**2 + 1)
/(2*d))/a**2, Eq(b, 0)), (-A*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*
d) + 2*I*A*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + A*d*x/(4*b**2*d*
tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - A*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*
tan(c + d*x) - 4*b**2*d) + 2*I*A/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + I*B*d*x*tan
(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*B*d*x*tan(c + d*x)/(4*b**2*d*
tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - I*B*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c +
d*x) - 4*b**2*d) + I*B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I
*b)), (-A*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 2*I*A*d*x*tan(
c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + A*d*x/(4*b**2*d*tan(c + d*x)**2 + 8
*I*b**2*d*tan(c + d*x) - 4*b**2*d) - A*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b*
*2*d) - 2*I*A/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - I*B*d*x*tan(c + d*x)**2/(4*b**
2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*B*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8
*I*b**2*d*tan(c + d*x) - 4*b**2*d) + I*B*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) -
I*B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, I*b)), (x*(A + B*tan(
c))/(a + b*tan(c))**2, Eq(d, 0)), (2*A*a**3*d*x/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b
**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 2*A*a**2*b*d*x*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*
tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 4*A*a**2*b*l
og(a/b + tan(c + d*x))/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*
b**4*d + 2*b**5*d*tan(c + d*x)) - 2*A*a**2*b*log(tan(c + d*x)**2 + 1)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*
a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*A*a**2*b/(2*a**5*d + 2*a**4
*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*A*a*b
**2*d*x/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5
*d*tan(c + d*x)) + 4*A*a*b**2*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**
3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*A*a*b**2*log(tan(c + d*x)**2 +
1)*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d
+ 2*b**5*d*tan(c + d*x)) - 2*A*b**3*d*x*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*
a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*A*b**3/(2*a**5*d + 2*a**4*b*d*tan(c + d*x)
+ 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*B*a**3*log(a/b + tan(c
+ d*x))/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5
*d*tan(c + d*x)) + B*a**3*log(tan(c + d*x)**2 + 1)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**
2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 2*B*a**3/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4
*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 4*B*a**2*b*d*x/(2*a**5*d + 2
*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - 2*
B*a**2*b*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**
3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + B*a**2*b*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**
5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x
)) + 4*B*a*b**2*d*x*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d
*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 2*B*a*b**2*log(a/b + tan(c + d*x))/(2*a**5*d + 2*a**4*b*d*tan(c +
d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - B*a*b**2*log(tan(c +
d*x)**2 + 1)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d +
2*b**5*d*tan(c + d*x)) + 2*B*a*b**2/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c
+ d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) + 2*B*b**3*log(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*d + 2*a**
4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)) - B*b**3
*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**5*d + 2*a**4*b*d*tan(c + d*x) + 4*a**3*b**2*d + 4*a**2*b**3*d*tan
(c + d*x) + 2*a*b**4*d + 2*b**5*d*tan(c + d*x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.59 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (A a^{2} + 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (B a - A b\right )}}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{2 \, d} \]

[In]

integrate((A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(2*(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - 2*(B*a^2 - 2*A*a*b - B*b^2)*log(b*tan(d*x
+ c) + a)/(a^4 + 2*a^2*b^2 + b^4) + (B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4)
+ 2*(B*a - A*b)/(a^3 + a*b^2 + (a^2*b + b^3)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (111) = 222\).

Time = 0.40 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.11 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (A a^{2} + 2 \, B a b - A b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {2 \, {\left (B a^{2} b \tan \left (d x + c\right ) - 2 \, A a b^{2} \tan \left (d x + c\right ) - B b^{3} \tan \left (d x + c\right ) + 2 \, B a^{3} - 3 \, A a^{2} b - A b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \]

[In]

integrate((A+B*tan(d*x+c))/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(2*(A*a^2 + 2*B*a*b - A*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) + (B*a^2 - 2*A*a*b - B*b^2)*log(tan(d*x + c
)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2*(B*a^2*b - 2*A*a*b^2 - B*b^3)*log(abs(b*tan(d*x + c) + a))/(a^4*b + 2*a^2
*b^3 + b^5) + 2*(B*a^2*b*tan(d*x + c) - 2*A*a*b^2*tan(d*x + c) - B*b^3*tan(d*x + c) + 2*B*a^3 - 3*A*a^2*b - A*
b^3)/((a^4 + 2*a^2*b^2 + b^4)*(b*tan(d*x + c) + a)))/d

Mupad [B] (verification not implemented)

Time = 8.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.38 \[ \int \frac {A+B \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-B\,a^2+2\,A\,a\,b+B\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {A\,b-B\,a}{d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (A+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )} \]

[In]

int((A + B*tan(c + d*x))/(a + b*tan(c + d*x))^2,x)

[Out]

(log(a + b*tan(c + d*x))*(B*b^2 - B*a^2 + 2*A*a*b))/(d*(a^2 + b^2)^2) - (A*b - B*a)/(d*(a^2 + b^2)*(a + b*tan(
c + d*x))) - (log(tan(c + d*x) + 1i)*(A*1i + B))/(2*d*(a*b*2i - a^2 + b^2)) - (log(tan(c + d*x) - 1i)*(A + B*1
i))/(2*d*(2*a*b - a^2*1i + b^2*1i))