\(\int \sec (c-b x) \sec (a+b x) \, dx\) [144]

Optimal result

Integrand size = 14, antiderivative size = 33 \[ \int \sec (c-b x) \sec (a+b x) \, dx=\frac {\csc (a+c) \log (\cos (c-b x))}{b}-\frac {\csc (a+c) \log (\cos (a+b x))}{b} \]

[Out]

csc(a+c)*ln(cos(b*x-c))/b-csc(a+c)*ln(cos(b*x+a))/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4706, 3556} \[ \int \sec (c-b x) \sec (a+b x) \, dx=\frac {\csc (a+c) \log (\cos (c-b x))}{b}-\frac {\csc (a+c) \log (\cos (a+b x))}{b} \]

[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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4706

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \sec (c-b x) \sec (a+b x) \, dx=\frac {\csc (a+c) (\log (\cos (c-b x))-\log (\cos (a+b x)))}{b} \]

[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

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61

method	result	size
default	\(\frac {\ln \left (\tan \left (x b +a \right ) \sin \left (a \right ) \cos \left (c \right )+\tan \left (x b +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )}{b \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}\)	\(53\)
risch	\(-\frac {2 i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) {\mathrm e}^{i \left (a +c \right )}}{\left ({\mathrm e}^{2 i \left (a +c \right )}-1\right ) b}+\frac {2 i \ln \left ({\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (x b +a \right )}\right ) {\mathrm e}^{i \left (a +c \right )}}{\left ({\mathrm e}^{2 i \left (a +c \right )}-1\right ) b}\)	\(80\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (34) = 68\).

Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.82 \[ \int \sec (c-b x) \sec (a+b x) \, dx=-\frac {\log \left (\cos \left (b x + a\right )^{2}\right ) - \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{2} + 1\right )}}{\cos \left (a + c\right )^{2} + 2 \, \cos \left (a + c\right ) + 1}\right )}{2 \, b \sin \left (a + c\right )} \]

[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))

Sympy [F]

\[ \int \sec (c-b x) \sec (a+b x) \, dx=\int \sec {\left (a + b x \right )} \sec {\left (b x - c \right )}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 322, normalized size of antiderivative = 9.76 \[ \int \sec (c-b x) \sec (a+b x) \, dx=\frac {2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) - \sin \left (2 \, a\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, a\right )\right ) - 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (2 \, b x\right ) + \sin \left (2 \, c\right ), \cos \left (2 \, b x\right ) + \cos \left (2 \, c\right )\right ) - {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right ) + {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, c\right ) + \cos \left (2 \, c\right )^{2} + \sin \left (2 \, b x\right )^{2} + 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, c\right ) + \sin \left (2 \, c\right )^{2}\right )}{b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (34) = 68\).

Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.12 \[ \int \sec (c-b x) \sec (a+b x) \, dx=-\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) + \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, c\right )^{2} - 1 \right |}\right )}{2 \, {\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )\right )} b} \]

[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)

Mupad [B] (verification not implemented)

Time = 33.69 (sec) , antiderivative size = 249, normalized size of antiderivative = 7.55 \[ \int \sec (c-b x) \sec (a+b x) \, dx=\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}\,\left (\ln \left (-\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}-1\right )}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )-\ln \left (-\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b-b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )} \]

[In]

int(1/(cos(a + b*x)*cos(c - b*x)),x)

[Out]

(2*(-exp(a*2i + c*2i))^(1/2)*(log(exp(a*1i)*exp(a*2i)*exp(c*1i)*exp(b*x*2i)*4i - (2*(-exp(a*2i)*exp(c*2i))^(1/
2)*(4*b*exp(a*2i)*exp(c*2i) + 2*b*exp(a*2i)*exp(b*x*2i) + 2*b*exp(a*4i)*exp(c*2i)*exp(b*x*2i)))/(b*(exp(a*2i)*
exp(c*2i) - 1))) - log(exp(a*1i)*exp(a*2i)*exp(c*1i)*exp(b*x*2i)*4i - (2*(-exp(a*2i)*exp(c*2i))^(1/2)*(4*b*exp
(a*2i)*exp(c*2i) + 2*b*exp(a*2i)*exp(b*x*2i) + 2*b*exp(a*4i)*exp(c*2i)*exp(b*x*2i)))/(b - b*exp(a*2i)*exp(c*2i
)))))/(b*(exp(a*2i + c*2i) - 1))