3.144 \(\int \sec (c-b x) \sec (a+b x) \, dx\)

Optimal. Leaf size=33 \[ \frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b} \]

[Out]

(Csc[a + c]*Log[Cos[c - b*x]])/b - (Csc[a + c]*Log[Cos[a + b*x]])/b

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Rubi [A]  time = 0.018378, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4610, 3475} \[ \frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c - b*x]*Sec[a + b*x],x]

[Out]

(Csc[a + c]*Log[Cos[c - b*x]])/b - (Csc[a + c]*Log[Cos[a + b*x]])/b

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 \sec (c-b x) \sec (a+b x) \, dx &=\csc (a+c) \int \tan (c-b x) \, dx+\csc (a+c) \int \tan (a+b x) \, dx\\ &=\frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.228994, size = 26, normalized size = 0.79 \[ \frac{\csc (a+c) (\log (\cos (c-b x))-\log (\cos (a+b x)))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c - b*x]*Sec[a + b*x],x]

[Out]

(Csc[a + c]*(Log[Cos[c - b*x]] - Log[Cos[a + b*x]]))/b

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Maple [A]  time = 0.221, size = 53, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \sin \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \cos \left ( c \right ) -\sin \left ( a \right ) \sin \left ( c \right ) \right ) }{b \left ( \sin \left ( a \right ) \cos \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(b*x-c)*sec(b*x+a),x)

[Out]

1/b/(sin(a)*cos(c)+cos(a)*sin(c))*ln(tan(b*x+a)*cos(a)*sin(c)+tan(b*x+a)*sin(a)*cos(c)+cos(a)*cos(c)-sin(a)*si
n(c))

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Maxima [B]  time = 1.09119, size = 435, normalized size = 13.18 \begin{align*} \frac{2 \,{\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\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 ) - 2 \,{\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\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 ) -{\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + 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 ) +{\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\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 )}{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(sec(b*x-c)*sec(b*x+a),x, algorithm="maxima")

[Out]

(2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(2*b*x) - sin(2*a), cos(2*b
*x) + cos(2*a)) - 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(2*b*x) +
sin(2*c), cos(2*b*x) + cos(2*c)) - (cos(a + c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + c))*log(co
s(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) + (cos(a
+ c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + 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))/(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 = 2.52903, size = 263, normalized size = 7.97 \begin{align*} -\frac{\log \left (\cos \left (b x + a\right )^{2}\right ) - \log \left (\frac{4 \,{\left (2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) +{\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{2} + 1\right )}}{\cos \left (a + c\right )^{2} + 2 \, \cos \left (a + c\right ) + 1}\right )}{2 \, b \sin \left (a + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x-c)*sec(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(log(cos(b*x + a)^2) - log(4*(2*cos(b*x + a)*cos(a + c)*sin(b*x + a)*sin(a + c) + (2*cos(a + c)^2 - 1)*co
s(b*x + a)^2 - cos(a + c)^2 + 1)/(cos(a + c)^2 + 2*cos(a + c) + 1)))/(b*sin(a + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec{\left (a + b x \right )} \sec{\left (b x - c \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x-c)*sec(b*x+a),x)

[Out]

Integral(sec(a + b*x)*sec(b*x - c), x)

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Giac [B]  time = 1.24222, size = 228, normalized size = 6.91 \begin{align*} -\frac{{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} + \tan \left (\frac{1}{2} \, a\right )^{2} + \tan \left (\frac{1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, a\right ) + \tan \left (\frac{1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac{1}{2} \, c\right ) + 4 \, \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, c\right )^{2} - 1 \right |}\right )}{2 \,{\left (\tan \left (\frac{1}{2} \, a\right )^{2} \tan \left (\frac{1}{2} \, c\right ) + \tan \left (\frac{1}{2} \, a\right ) \tan \left (\frac{1}{2} \, c\right )^{2} - \tan \left (\frac{1}{2} \, a\right ) - \tan \left (\frac{1}{2} \, c\right )\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(b*x-c)*sec(b*x+a),x, algorithm="giac")

[Out]

-1/2*(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*log(abs(2*tan(b*x + a)*tan(1/2*a)^2*tan(1/2
*c) + 2*tan(b*x + a)*tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^2 - 2*tan(b*x + a)*tan(1/2*a) + tan(1/2
*a)^2 - 2*tan(b*x + a)*tan(1/2*c) + 4*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 - 1))/((tan(1/2*a)^2*tan(1/2*c) + t
an(1/2*a)*tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c))*b)