3.140 \(\int \tan (c-b x) \tan (a+b x) \, dx\)
Optimal. Leaf size=34 \[ -\frac{\cot (a+c) \log (\cos (c-b x))}{b}+\frac{\cot (a+c) \log (\cos (a+b x))}{b}+x \]
[Out]
x - (Cot[a + c]*Log[Cos[c - b*x]])/b + (Cot[a + c]*Log[Cos[a + b*x]])/b
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Rubi [A] time = 0.0678364, antiderivative size = 34, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used =
{4612, 4610, 3475} \[ -\frac{\cot (a+c) \log (\cos (c-b x))}{b}+\frac{\cot (a+c) \log (\cos (a+b x))}{b}+x \]
Antiderivative was successfully verified.
[In]
Int[Tan[c - b*x]*Tan[a + b*x],x]
[Out]
x - (Cot[a + c]*Log[Cos[c - b*x]])/b + (Cot[a + c]*Log[Cos[a + b*x]])/b
Rule 4612
Int[Tan[(a_.) + (b_.)*(x_)]*Tan[(c_) + (d_.)*(x_)], x_Symbol] :> -Simp[(b*x)/d, x] + Dist[(b*Cos[(b*c - a*d)/d
])/d, Int[Sec[a + b*x]*Sec[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0
]
Rule 4610
Int[Sec[(a_.) + (b_.)*(x_)]*Sec[(c_) + (d_.)*(x_)], x_Symbol] :> -Dist[Csc[(b*c - a*d)/d], Int[Tan[a + b*x], x
], x] + Dist[Csc[(b*c - a*d)/b], Int[Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && Ne
Q[b*c - a*d, 0]
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \tan (c-b x) \tan (a+b x) \, dx &=x-\cos (a+c) \int \sec (c-b x) \sec (a+b x) \, dx\\ &=x-\cot (a+c) \int \tan (c-b x) \, dx-\cot (a+c) \int \tan (a+b x) \, dx\\ &=x-\frac{\cot (a+c) \log (\cos (c-b x))}{b}+\frac{\cot (a+c) \log (\cos (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.516919, size = 28, normalized size = 0.82 \[ \frac{\cot (a+c) (\log (\cos (a+b x))-\log (\cos (c-b x)))}{b}+x \]
Antiderivative was successfully verified.
[In]
Integrate[Tan[c - b*x]*Tan[a + b*x],x]
[Out]
x + (Cot[a + c]*(-Log[Cos[c - b*x]] + Log[Cos[a + b*x]]))/b
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Maple [C] time = 0.067, size = 145, normalized size = 4.3 \begin{align*} x+{\frac{i\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ){{\rm e}^{2\,i \left ( a+c \right ) }}}{b \left ({{\rm e}^{2\,i \left ( a+c \right ) }}-1 \right ) }}+{\frac{i\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) }{b \left ({{\rm e}^{2\,i \left ( a+c \right ) }}-1 \right ) }}-{\frac{i\ln \left ({{\rm e}^{2\,i \left ( a+c \right ) }}+{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ){{\rm e}^{2\,i \left ( a+c \right ) }}}{b \left ({{\rm e}^{2\,i \left ( a+c \right ) }}-1 \right ) }}-{\frac{i\ln \left ({{\rm e}^{2\,i \left ( a+c \right ) }}+{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{b \left ({{\rm e}^{2\,i \left ( a+c \right ) }}-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-tan(b*x-c)*tan(b*x+a),x)
[Out]
x+I/b/(exp(2*I*(a+c))-1)*ln(exp(2*I*(b*x+a))+1)*exp(2*I*(a+c))+I/b/(exp(2*I*(a+c))-1)*ln(exp(2*I*(b*x+a))+1)-I
/b/(exp(2*I*(a+c))-1)*ln(exp(2*I*(a+c))+exp(2*I*(b*x+a)))*exp(2*I*(a+c))-I/b/(exp(2*I*(a+c))-1)*ln(exp(2*I*(a+
c))+exp(2*I*(b*x+a)))
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Maxima [B] time = 1.08495, size = 392, normalized size = 11.53 \begin{align*} \frac{{\left (b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b\right )} x -{\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, b x\right ) - \sin \left (2 \, a\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, a\right )\right ) +{\left (\cos \left (2 \, a + 2 \, c\right )^{2} + \sin \left (2 \, a + 2 \, c\right )^{2} - 1\right )} \arctan \left (\sin \left (2 \, b x\right ) + \sin \left (2 \, c\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, c\right )\right ) + \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right ) \sin \left (2 \, a + 2 \, c\right ) - \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} + 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right ) \sin \left (2 \, a + 2 \, c\right )}{b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-tan(b*x-c)*tan(b*x+a),x, algorithm="maxima")
[Out]
((b*cos(2*a + 2*c)^2 + b*sin(2*a + 2*c)^2 - 2*b*cos(2*a + 2*c) + b)*x - (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 -
1)*arctan2(sin(2*b*x) - sin(2*a), cos(2*b*x) + cos(2*a)) + (cos(2*a + 2*c)^2 + sin(2*a + 2*c)^2 - 1)*arctan2(
sin(2*b*x) + sin(2*c), cos(2*b*x) + cos(2*c)) + log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*
b*x)^2 - 2*sin(2*b*x)*sin(2*a) + sin(2*a)^2)*sin(2*a + 2*c) - log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*c) + cos(2
*c)^2 + sin(2*b*x)^2 + 2*sin(2*b*x)*sin(2*c) + sin(2*c)^2)*sin(2*a + 2*c))/(b*cos(2*a + 2*c)^2 + b*sin(2*a + 2
*c)^2 - 2*b*cos(2*a + 2*c) + b)
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Fricas [B] time = 2.57033, size = 374, normalized size = 11. \begin{align*} \frac{2 \, b x \sin \left (2 \, a + 2 \, c\right ) -{\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (-\frac{{\left (\cos \left (2 \, a + 2 \, c\right ) - 1\right )} \tan \left (b x + a\right )^{2} - 2 \, \sin \left (2 \, a + 2 \, c\right ) \tan \left (b x + a\right ) - \cos \left (2 \, a + 2 \, c\right ) - 1}{{\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \tan \left (b x + a\right )^{2} + \cos \left (2 \, a + 2 \, c\right ) + 1}\right ) +{\left (\cos \left (2 \, a + 2 \, c\right ) + 1\right )} \log \left (\frac{1}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b \sin \left (2 \, a + 2 \, c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-tan(b*x-c)*tan(b*x+a),x, algorithm="fricas")
[Out]
1/2*(2*b*x*sin(2*a + 2*c) - (cos(2*a + 2*c) + 1)*log(-((cos(2*a + 2*c) - 1)*tan(b*x + a)^2 - 2*sin(2*a + 2*c)*
tan(b*x + a) - cos(2*a + 2*c) - 1)/((cos(2*a + 2*c) + 1)*tan(b*x + a)^2 + cos(2*a + 2*c) + 1)) + (cos(2*a + 2*
c) + 1)*log(1/(tan(b*x + a)^2 + 1)))/(b*sin(2*a + 2*c))
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Sympy [B] time = 10.1553, size = 7720, normalized size = 227.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-tan(b*x-c)*tan(b*x+a),x)
[Out]
Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (-2*b*x*tan(c)/(2*b*tan(c)**2 + 2*b) + 2*log(tan(b*x) + 1/tan(c
))/(2*b*tan(c)**2 + 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(c)**2 + 2*b), Eq(a, 0)), (-4*b*x*tan(c)**2*tan(b*x)/(
2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) -
4*b*x*tan(c)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*ta
n(b*x) + 2*b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)**3*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*ta
n(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)**2/(2*b*tan(
c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + 2*log(t
an(b*x) + 1/tan(c))*tan(c)*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan
(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + 2*log(tan(b*x) + 1/tan(c))/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b
*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + log(tan(b*x)**2 + 1)*tan(c)**3*tan(b*x)/(2*
b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + l
og(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2
+ 2*b*tan(c)*tan(b*x) + 2*b) - log(tan(b*x)**2 + 1)*tan(c)*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 +
4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(c)**5*tan
(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2*tan(c)**2/(2*b
*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2/
(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b),
Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (0, Eq(b, 0)), (2*b*x*tan(a)/(2*b*tan
(a)**2 + 2*b) + 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(a)**2 + 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(a)**2 + 2*b),
Eq(c, 0)), (2*b*x*tan(a)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**
2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*b*x*tan(c)**2/(2*b*tan(a)**3*ta
n(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) +
2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(
a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2
*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*tan(a)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)
**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x)
+ 1/tan(c))*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**
2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) + 1/tan(c))*tan(c)
/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)
**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**
2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*t
an(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*
tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) -
log(tan(b*x)**2 + 1)*tan(a)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**
2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(c)/(2*b*
tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 +
2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True))*tan(a) - Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (-2*b
*x*tan(c)/(2*b*tan(c)**2 + 2*b) + 2*log(tan(b*x) + 1/tan(c))/(2*b*tan(c)**2 + 2*b) - log(tan(b*x)**2 + 1)/(2*b
*tan(c)**2 + 2*b), Eq(a, 0)), (-4*b*x*tan(c)**2*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)*
*3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 4*b*x*tan(c)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**
4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)**3
*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*
x) + 2*b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)**2/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b
*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + 2*log(tan(b*x) + 1/tan(c))*tan(c)*tan(b*x)/(2*b*tan(c)**5*t
an(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + 2*log(tan(b*x)
+ 1/tan(c))/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan
(b*x) + 2*b) + log(tan(b*x)**2 + 1)*tan(c)**3*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3
*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) + log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(c)**5*tan(b*x
) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - log(tan(b*x)**2 + 1)
*tan(c)*tan(b*x)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)
*tan(b*x) + 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*tan(b*x) + 4*b
*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2*tan(c)**2/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)**3*
tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b) - 2/(2*b*tan(c)**5*tan(b*x) + 2*b*tan(c)**4 + 4*b*tan(c)
**3*tan(b*x) + 4*b*tan(c)**2 + 2*b*tan(c)*tan(b*x) + 2*b), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*
floor(c/pi - 1/2))), (0, Eq(b, 0)), (2*b*x*tan(a)/(2*b*tan(a)**2 + 2*b) + 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(
a)**2 + 2*b) - log(tan(b*x)**2 + 1)/(2*b*tan(a)**2 + 2*b), Eq(c, 0)), (2*b*x*tan(a)**2/(2*b*tan(a)**3*tan(c)**
2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*t
an(c)**3 + 2*b*tan(c)) - 2*b*x*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 +
2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/ta
n(a))*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(
c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*tan(a)/(2*b*
tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 +
2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) + 1/tan(c))*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**
2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*t
an(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) + 1/tan(c))*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(
a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log
(tan(b*x)**2 + 1)*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*ta
n(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)
*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan
(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(a)/(2*b*tan(a)**3*tan(c)**
2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*t
an(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*
tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True))*tan
(c) + Piecewise((0, Eq(a, 0) & Eq(b, 0) & Eq(c, 0)), (2*b*x*tan(c)/(2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*
x) + 1/tan(c))/(2*b*tan(c)**3 + 2*b*tan(c)) - log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(c)**3 + 2*b*tan(c)), Eq(
a, 0)), (-b*x*tan(c)**4*tan(b*x)/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3
+ b*tan(c)**2*tan(b*x) + b*tan(c)) - b*x*tan(c)**3/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x
) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) + b*x*tan(c)**2*tan(b*x)/(b*tan(c)**6*tan(b*x) + b*tan(c)
**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) + b*x*tan(c)/(b*tan(c)**6*tan(
b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) + 2*log(tan(b*x
) + 1/tan(c))*tan(c)**3*tan(b*x)/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3
+ b*tan(c)**2*tan(b*x) + b*tan(c)) + 2*log(tan(b*x) + 1/tan(c))*tan(c)**2/(b*tan(c)**6*tan(b*x) + b*tan(c)**5
+ 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) - log(tan(b*x)**2 + 1)*tan(c)**3*t
an(b*x)/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) +
b*tan(c)) - log(tan(b*x)**2 + 1)*tan(c)**2/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*tan(b*x) + 2*b*
tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) + tan(c)**2/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c)**4*t
an(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)) + 1/(b*tan(c)**6*tan(b*x) + b*tan(c)**5 + 2*b*tan(c
)**4*tan(b*x) + 2*b*tan(c)**3 + b*tan(c)**2*tan(b*x) + b*tan(c)), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi
) - pi*floor(c/pi - 1/2))), (0, Eq(b, 0)), (2*b*x*tan(a)/(2*b*tan(a)**3 + 2*b*tan(a)) + 2*log(tan(b*x) - 1/tan
(a))/(2*b*tan(a)**3 + 2*b*tan(a)) + log(tan(b*x)**2 + 1)*tan(a)**2/(2*b*tan(a)**3 + 2*b*tan(a)), Eq(c, 0)), (2
*b*x*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c
) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(a)*tan(c)**2/(2*b*tan(a)**3*ta
n(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) +
2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(a)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3
+ 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*b*x*tan(c)/(2*b*
tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 +
2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b
*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**
3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**
3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*x) +
1/tan(c))*tan(a)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c)
+ 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*x) + 1/tan(c))/(2*b*tan(a)**3*
tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a)
+ 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(a)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b
*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c))
- log(tan(b*x)**2 + 1)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(
a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True)) - Piecewise((-x, Eq(a,
0) & Eq(b, 0) & Eq(c, 0)), (-2*b*x/(2*b*tan(c)**2 + 2*b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)/(2*b*tan(c)**2 +
2*b) + log(tan(b*x)**2 + 1)*tan(c)/(2*b*tan(c)**2 + 2*b), Eq(a, 0)), (b*x*tan(c)**3*tan(b*x)/(b*tan(c)**5*tan(
b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) + b*x*tan(c)**2/(b*tan(c)
**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) - b*x*tan(c)*tan(
b*x)/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) - b
*x/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) - 2*l
og(tan(b*x) + 1/tan(c))*tan(c)**2*tan(b*x)/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*
tan(c)**2 + b*tan(c)*tan(b*x) + b) - 2*log(tan(b*x) + 1/tan(c))*tan(c)/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2
*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) + log(tan(b*x)**2 + 1)*tan(c)**2*tan(b*x)/(b*ta
n(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b*x) + b) + log(tan(b*x
)**2 + 1)*tan(c)/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*tan(b
*x) + b) + tan(c)**3/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*t
an(b*x) + b) + tan(c)/(b*tan(c)**5*tan(b*x) + b*tan(c)**4 + 2*b*tan(c)**3*tan(b*x) + 2*b*tan(c)**2 + b*tan(c)*
tan(b*x) + b), Eq(a, -atan(tan(c)) - pi*floor((c - pi/2)/pi) - pi*floor(c/pi - 1/2))), (-x, Eq(b, 0)), (-2*b*x
/(2*b*tan(a)**2 + 2*b) + 2*log(tan(b*x) - 1/tan(a))*tan(a)/(2*b*tan(a)**2 + 2*b) - log(tan(b*x)**2 + 1)*tan(a)
/(2*b*tan(a)**2 + 2*b), Eq(c, 0)), (-2*b*x*tan(a)**2*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan
(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*
b*x*tan(a)*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c)
+ 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*b*x*tan(a)/(2*b*tan(a)**3*tan(c)**2 + 2
*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)
**3 + 2*b*tan(c)) - 2*b*x*tan(c)/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(
a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*ta
n(a)**2*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) +
2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + 2*log(tan(b*x) - 1/tan(a))*tan(a)**2/(2*b*ta
n(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*
b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*x) + 1/tan(c))*tan(a)**2*tan(c)**2/(2*b*tan(a)**3*tan(c)*
*2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*
tan(c)**3 + 2*b*tan(c)) - 2*log(tan(b*x) + 1/tan(c))*tan(c)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*
tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) -
log(tan(b*x)**2 + 1)*tan(a)**2/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a
)**2*tan(c) + 2*b*tan(a)*tan(c)**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)) + log(tan(b*x)**2 + 1)*tan(c)**2
/(2*b*tan(a)**3*tan(c)**2 + 2*b*tan(a)**3 + 2*b*tan(a)**2*tan(c)**3 + 2*b*tan(a)**2*tan(c) + 2*b*tan(a)*tan(c)
**2 + 2*b*tan(a) + 2*b*tan(c)**3 + 2*b*tan(c)), True))*tan(a)*tan(c)
________________________________________________________________________________________
Giac [B] time = 1.16567, size = 109, normalized size = 3.21 \begin{align*} x - \frac{{\left (\tan \left (a\right )^{2} \tan \left (c\right ) - \tan \left (a\right )\right )} \log \left ({\left | \tan \left (b x\right ) \tan \left (a\right ) - 1 \right |}\right )}{b \tan \left (a\right )^{2} + b \tan \left (a\right ) \tan \left (c\right )} + \frac{{\left (\tan \left (a\right ) \tan \left (c\right )^{2} - \tan \left (c\right )\right )} \log \left ({\left | \tan \left (b x\right ) \tan \left (c\right ) + 1 \right |}\right )}{b \tan \left (a\right ) \tan \left (c\right ) + b \tan \left (c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(-tan(b*x-c)*tan(b*x+a),x, algorithm="giac")
[Out]
x - (tan(a)^2*tan(c) - tan(a))*log(abs(tan(b*x)*tan(a) - 1))/(b*tan(a)^2 + b*tan(a)*tan(c)) + (tan(a)*tan(c)^2
- tan(c))*log(abs(tan(b*x)*tan(c) + 1))/(b*tan(a)*tan(c) + b*tan(c)^2)