Integrand size = 16, antiderivative size = 34 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=-\frac {\text {arctanh}(\sin (c-b x)) \cos (a+c)}{b}-\frac {\sec (c-b x) \sin (a+c)}{b} \] Output:
arctanh(sin(b*x-c))*cos(a+c)/b-sec(b*x-c)*sin(a+c)/b
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.68 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=-\frac {2 i \arctan \left (\frac {\cos \left (\frac {b x}{2}\right )-\cos ^2(c) \cos \left (\frac {b x}{2}\right )-2 i \cos (c) \cos \left (\frac {b x}{2}\right ) \sin (c)+\cos \left (\frac {b x}{2}\right ) \sin ^2(c)+i \sin \left (\frac {b x}{2}\right )+i \cos ^2(c) \sin \left (\frac {b x}{2}\right )-2 \cos (c) \sin (c) \sin \left (\frac {b x}{2}\right )-i \sin ^2(c) \sin \left (\frac {b x}{2}\right )}{2 \cos (c) \cos \left (\frac {b x}{2}\right )+2 i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a+c)}{b}-\frac {\sec (c-b x) \sin (a+c)}{b} \] Input:
Integrate[Cos[a + b*x]*Sec[c - b*x]^2,x]
Output:
((-2*I)*ArcTan[(Cos[(b*x)/2] - Cos[c]^2*Cos[(b*x)/2] - (2*I)*Cos[c]*Cos[(b *x)/2]*Sin[c] + Cos[(b*x)/2]*Sin[c]^2 + I*Sin[(b*x)/2] + I*Cos[c]^2*Sin[(b *x)/2] - 2*Cos[c]*Sin[c]*Sin[(b*x)/2] - I*Sin[c]^2*Sin[(b*x)/2])/(2*Cos[c] *Cos[(b*x)/2] + (2*I)*Cos[(b*x)/2]*Sin[c])]*Cos[a + c])/b - (Sec[c - b*x]* Sin[a + c])/b
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos (a+b x) \sec ^2(c-b x) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \cos (a+b x) \sec ^2(c-b x)dx\) |
Input:
Int[Cos[a + b*x]*Sec[c - b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.29
| method | result | size |
| risch | \(\frac {i \left ({\mathrm e}^{i \left (b x +3 a +2 c \right )}-{\mathrm e}^{i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a +c \right )}\right ) \cos \left (a +c \right )}{b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a +c \right )}\right ) \cos \left (a +c \right )}{b}\) | \(112\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (\cos \left (c \right )^{2} \sin \left (a \right )^{2}+2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right )}-\frac {\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )}{\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}\right )}{\cos \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}-\sin \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}-2 \cos \left (c \right ) \sin \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-2 \sin \left (c \right ) \cos \left (a \right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )}-\frac {2 \left (\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right ) \arctan \left (\frac {2 \left (\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-2 \sin \left (a \right ) \cos \left (c \right )-2 \cos \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}}{b}\) | \(409\) |
Input:
int(cos(b*x+a)*sec(b*x-c)^2,x,method=_RETURNVERBOSE)
Output:
I/b/(exp(2*I*(a+c))+exp(2*I*(b*x+a)))*(exp(I*(b*x+3*a+2*c))-exp(I*(b*x+a)) )+ln(exp(I*(b*x+a))+I*exp(I*(a+c)))/b*cos(a+c)-ln(exp(I*(b*x+a))-I*exp(I*( a+c)))/b*cos(a+c)
Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (35) = 70\).
Time = 0.08 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.97 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=\frac {{\left (\cos \left (b x + a\right ) \cos \left (a + c\right )^{2} + \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right )\right )} \log \left (\frac {2 \, {\left (\cos \left (a + c\right ) \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \sin \left (a + c\right ) + 1\right )}}{\cos \left (a + c\right ) + 1}\right ) - {\left (\cos \left (b x + a\right ) \cos \left (a + c\right )^{2} + \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right )\right )} \log \left (-\frac {2 \, {\left (\cos \left (a + c\right ) \sin \left (b x + a\right ) - \cos \left (b x + a\right ) \sin \left (a + c\right ) - 1\right )}}{\cos \left (a + c\right ) + 1}\right ) - 2 \, \sin \left (a + c\right )}{2 \, {\left (b \cos \left (b x + a\right ) \cos \left (a + c\right ) + b \sin \left (b x + a\right ) \sin \left (a + c\right )\right )}} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^2,x, algorithm="fricas")
Output:
1/2*((cos(b*x + a)*cos(a + c)^2 + cos(a + c)*sin(b*x + a)*sin(a + c))*log( 2*(cos(a + c)*sin(b*x + a) - cos(b*x + a)*sin(a + c) + 1)/(cos(a + c) + 1) ) - (cos(b*x + a)*cos(a + c)^2 + cos(a + c)*sin(b*x + a)*sin(a + c))*log(- 2*(cos(a + c)*sin(b*x + a) - cos(b*x + a)*sin(a + c) - 1)/(cos(a + c) + 1) ) - 2*sin(a + c))/(b*cos(b*x + a)*cos(a + c) + b*sin(b*x + a)*sin(a + c))
Leaf count of result is larger than twice the leaf count of optimal. 4095 vs. \(2 (27) = 54\).
Time = 95.50 (sec) , antiderivative size = 5545, normalized size of antiderivative = 163.09 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)**2,x)
Output:
-Piecewise((log(tan(b*x/2))/b, Eq(c, -pi/2) | Eq(c, pi/2)), (0, Eq(b, 0)), (-2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) - 1) + 1/(tan(c/2) - 1))*tan(c/2) **3*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b*tan(c /2)**3*tan(b*x/2) + 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log (tan(b*x/2) + tan(c/2)/(tan(c/2) - 1) + 1/(tan(c/2) - 1))*tan(c/2)**3/(b*t an(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b*tan(c/2)**3*tan(b*x/2) + 4* b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 8*log(tan(b*x/2) + tan(c/2) /(tan(c/2) - 1) + 1/(tan(c/2) - 1))*tan(c/2)**2*tan(b*x/2)/(b*tan(c/2)**4* tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b*tan(c/2)**3*tan(b*x/2) + 4*b*tan(c/2)* tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) - 1) + 1/(tan(c/2) - 1))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)* *2 - b*tan(c/2)**4 + 4*b*tan(c/2)**3*tan(b*x/2) + 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + tan(c/2)/(tan(c/2) - 1) + 1/(t an(c/2) - 1))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b* tan(c/2)**3*tan(b*x/2) + 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - tan(c/2)/(tan(c/2) + 1) + 1/(tan(c/2) + 1))*tan(c/2)**3 *tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b*tan(c/2) **3*tan(b*x/2) + 4*b*tan(c/2)*tan(b*x/2) - b*tan(b*x/2)**2 + b) - 2*log(ta n(b*x/2) - tan(c/2)/(tan(c/2) + 1) + 1/(tan(c/2) + 1))*tan(c/2)**3/(b*tan( c/2)**4*tan(b*x/2)**2 - b*tan(c/2)**4 + 4*b*tan(c/2)**3*tan(b*x/2) + 4*...
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (35) = 70\).
Time = 0.17 (sec) , antiderivative size = 368, normalized size of antiderivative = 10.82 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=-\frac {2 \, {\left (\sin \left (b x + 2 \, a + 2 \, c\right ) - \sin \left (b x\right )\right )} \cos \left (2 \, b x + a\right ) + {\left (\cos \left (2 \, b x + a\right )^{2} \cos \left (a + c\right ) + 2 \, \cos \left (2 \, b x + a\right ) \cos \left (a + 2 \, c\right ) \cos \left (a + c\right ) + \cos \left (a + 2 \, c\right )^{2} \cos \left (a + c\right ) + \cos \left (a + c\right ) \sin \left (2 \, b x + a\right )^{2} + 2 \, \cos \left (a + c\right ) \sin \left (2 \, b x + a\right ) \sin \left (a + 2 \, c\right ) + \cos \left (a + c\right ) \sin \left (a + 2 \, c\right )^{2}\right )} \log \left (\frac {\cos \left (b x\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x\right ) + \sin \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x\right ) + \sin \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) - 2 \, {\left (\cos \left (b x + 2 \, a + 2 \, c\right ) - \cos \left (b x\right )\right )} \sin \left (2 \, b x + a\right ) + 2 \, \cos \left (a + 2 \, c\right ) \sin \left (b x + 2 \, a + 2 \, c\right ) - 2 \, \cos \left (a + 2 \, c\right ) \sin \left (b x\right ) - 2 \, \cos \left (b x + 2 \, a + 2 \, c\right ) \sin \left (a + 2 \, c\right ) + 2 \, \cos \left (b x\right ) \sin \left (a + 2 \, c\right )}{2 \, {\left (b \cos \left (2 \, b x + a\right )^{2} + 2 \, b \cos \left (2 \, b x + a\right ) \cos \left (a + 2 \, c\right ) + b \cos \left (a + 2 \, c\right )^{2} + b \sin \left (2 \, b x + a\right )^{2} + 2 \, b \sin \left (2 \, b x + a\right ) \sin \left (a + 2 \, c\right ) + b \sin \left (a + 2 \, c\right )^{2}\right )}} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^2,x, algorithm="maxima")
Output:
-1/2*(2*(sin(b*x + 2*a + 2*c) - sin(b*x))*cos(2*b*x + a) + (cos(2*b*x + a) ^2*cos(a + c) + 2*cos(2*b*x + a)*cos(a + 2*c)*cos(a + c) + cos(a + 2*c)^2* cos(a + c) + cos(a + c)*sin(2*b*x + a)^2 + 2*cos(a + c)*sin(2*b*x + a)*sin (a + 2*c) + cos(a + c)*sin(a + 2*c)^2)*log((cos(b*x)^2 + cos(c)^2 - 2*cos( c)*sin(b*x) + sin(b*x)^2 + 2*cos(b*x)*sin(c) + sin(c)^2)/(cos(b*x)^2 + cos (c)^2 + 2*cos(c)*sin(b*x) + sin(b*x)^2 - 2*cos(b*x)*sin(c) + sin(c)^2)) - 2*(cos(b*x + 2*a + 2*c) - cos(b*x))*sin(2*b*x + a) + 2*cos(a + 2*c)*sin(b* x + 2*a + 2*c) - 2*cos(a + 2*c)*sin(b*x) - 2*cos(b*x + 2*a + 2*c)*sin(a + 2*c) + 2*cos(b*x)*sin(a + 2*c))/(b*cos(2*b*x + a)^2 + 2*b*cos(2*b*x + a)*c os(a + 2*c) + b*cos(a + 2*c)^2 + b*sin(2*b*x + a)^2 + 2*b*sin(2*b*x + a)*s in(a + 2*c) + b*sin(a + 2*c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 1342 vs. \(2 (35) = 70\).
Time = 0.22 (sec) , antiderivative size = 1342, normalized size of antiderivative = 39.47 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)*sec(b*x-c)^2,x, algorithm="giac")
Output:
((tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan (1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c) - 5*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/ 2*a)*tan(1/2*c)^3 + tan(1/2*a)^3 + 5*tan(1/2*a)^2*tan(1/2*c) + 5*tan(1/2*a )*tan(1/2*c)^2 + tan(1/2*c)^3 + tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2*c) + t an(1/2*c)^2 - tan(1/2*a) - tan(1/2*c) - 1)*log(abs(-tan(1/2*b*x + 1/2*a)*t an(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a)*tan(1/2*a) + tan(1/2*b*x + 1/2 *a)*tan(1/2*c) - tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a) - tan(1/2*a) - tan(1/2*c) + 1))/(tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c) - tan(1/2*a)^2*tan( 1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 - tan(1/2*a)^2*tan(1/2*c ) - tan(1/2*a)*tan(1/2*c)^2 - tan(1/2*c)^3 - tan(1/2*a)^2 + tan(1/2*a)*tan (1/2*c) - tan(1/2*c)^2 - tan(1/2*a) - tan(1/2*c) - 1) - (tan(1/2*a)^3*tan( 1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2 *a)^3*tan(1/2*c) - 5*tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 - 5*tan(1/2*a)^2*tan(1/2*c) - 5*tan(1/2*a)*tan(1/2*c)^2 - ta n(1/2*c)^3 + tan(1/2*a)^2 + 5*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 + tan(1 /2*a) + tan(1/2*c) - 1)*log(abs(-tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c ) - tan(1/2*b*x + 1/2*a)*tan(1/2*a) - tan(1/2*b*x + 1/2*a)*tan(1/2*c) + ta n(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a) - tan(1/2*a) - tan(1/2*c) - 1)) /(tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^2 + tan(1/2*a)^2*...
Time = 26.59 (sec) , antiderivative size = 246, normalized size of antiderivative = 7.24 \[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \] Input:
int(cos(a + b*x)/cos(c - b*x)^2,x)
Output:
(log(- exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(c*2i) + 1) - (exp(a*2i)*exp(c* 2i)*(exp(a*2i)*exp(c*2i) + 1)*1i)/(exp(a*2i)*exp(c*2i))^(1/2))*(exp(a*2i + c*2i) + 1))/(2*b*exp(a*2i + c*2i)^(1/2)) - (log((exp(a*2i)*exp(c*2i)*(exp (a*2i)*exp(c*2i) + 1)*1i)/(exp(a*2i)*exp(c*2i))^(1/2) - exp(a*1i)*exp(b*x* 1i)*(exp(a*2i)*exp(c*2i) + 1))*(exp(a*2i + c*2i) + 1))/(2*b*exp(a*2i + c*2 i)^(1/2)) + (exp(a*1i + b*x*1i)*(exp(a*2i + c*2i) - 1)*1i)/(b*(exp(a*2i + c*2i) + exp(a*2i + b*x*2i)))
\[ \int \cos (a+b x) \sec ^2(c-b x) \, dx=\frac {-\cos \left (b x -c \right ) \left (\int \frac {\sin \left (b x -c \right )^{2}}{\sin \left (b x -c \right )^{2}-1}d x \right ) b -\cos \left (b x -c \right ) \left (\int \frac {\cos \left (b x +a \right ) \sin \left (b x -c \right )^{2}}{\sin \left (b x -c \right )^{2}-1}d x \right ) b +\cos \left (b x -c \right ) \sin \left (b x +a \right )+2 \cos \left (b x -c \right ) a +\cos \left (b x -c \right ) b x -\sin \left (b x -c \right )}{\cos \left (b x -c \right ) b} \] Input:
int(cos(b*x+a)*sec(b*x-c)^2,x)
Output:
( - cos(b*x - c)*int(sin(b*x - c)**2/(sin(b*x - c)**2 - 1),x)*b - cos(b*x - c)*int((cos(a + b*x)*sin(b*x - c)**2)/(sin(b*x - c)**2 - 1),x)*b + cos(b *x - c)*sin(a + b*x) + 2*cos(b*x - c)*a + cos(b*x - c)*b*x - sin(b*x - c)) /(cos(b*x - c)*b)