Integrand size = 15, antiderivative size = 35 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=\frac {\text {arctanh}(\sin (c+b x)) \cos (a-c)}{b}-\frac {\sec (c+b x) \sin (a-c)}{b} \] 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 = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.54 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=-\frac {2 i \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos (a-c)}{b}-\frac {\sec (c+b x) \sin (a-c)}{b} \] Input:
Integrate[Cos[a + b*x]*Sec[c + b*x]^2,x]
Output:
((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x )/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Sec [c + b*x]*Sin[a - c])/b
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5094, 3042, 3086, 24, 4257}
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.
| \(\displaystyle \int \cos (a+b x) \sec ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 5094 |
| \(\displaystyle \cos (a-c) \int \sec (c+b x)dx-\sin (a-c) \int \sec (c+b x) \tan (c+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )dx-\sin (a-c) \int \sec (c+b x) \tan (c+b x)dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )dx-\frac {\sin (a-c) \int 1d\sec (c+b x)}{b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \cos (a-c) \int \csc \left (c+b x+\frac {\pi }{2}\right )dx-\frac {\sin (a-c) \sec (b x+c)}{b}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\cos (a-c) \text {arctanh}(\sin (b x+c))}{b}-\frac {\sin (a-c) \sec (b x+c)}{b}\) |
Input:
Int[Cos[a + b*x]*Sec[c + b*x]^2,x]
Output:
(ArcTanh[Sin[c + b*x]]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cos[v_]*Sec[w_]^(n_.), x_Symbol] :> Simp[-Sin[v - w] Int[Tan[w]*Sec[w ]^(n - 1), x], x] + Simp[Cos[v - w] Int[Sec[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.40
| method | result | size |
| risch | \(\frac {i \left ({\mathrm e}^{i \left (b x +3 a \right )}-{\mathrm e}^{i \left (b x +a +2 c \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{b}\) | \(119\) |
| 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}\) | \(406\) |
Input:
int(cos(b*x+a)*sec(b*x+c)^2,x,method=_RETURNVERBOSE)
Output:
I/b/(exp(2*I*(b*x+a+c))+exp(2*I*a))*(exp(I*(b*x+3*a))-exp(I*(b*x+a+2*c)))- 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)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=\frac {\cos \left (b x + c\right ) \cos \left (-a + c\right ) \log \left (\sin \left (b x + c\right ) + 1\right ) - \cos \left (b x + c\right ) \cos \left (-a + c\right ) \log \left (-\sin \left (b x + c\right ) + 1\right ) + 2 \, \sin \left (-a + c\right )}{2 \, b \cos \left (b x + c\right )} \] Input:
integrate(cos(b*x+a)*sec(b*x+c)^2,x, algorithm="fricas")
Output:
1/2*(cos(b*x + c)*cos(-a + c)*log(sin(b*x + c) + 1) - cos(b*x + c)*cos(-a + c)*log(-sin(b*x + c) + 1) + 2*sin(-a + c))/(b*cos(b*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 4095 vs. \(2 (27) = 54\).
Time = 97.14 (sec) , antiderivative size = 5545, normalized size of antiderivative = 158.43 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=\text {Too large to display} \] 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. 391 vs. \(2 (35) = 70\).
Time = 0.17 (sec) , antiderivative size = 391, normalized size of antiderivative = 11.17 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=-\frac {2 \, {\left (\sin \left (b x + 2 \, a\right ) - \sin \left (b x + 2 \, c\right )\right )} \cos \left (2 \, b x + a + 2 \, c\right ) + {\left (\cos \left (2 \, b x + a + 2 \, c\right )^{2} \cos \left (-a + c\right ) + 2 \, \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) \cos \left (-a + c\right ) + \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, \cos \left (-a + c\right ) \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} \cos \left (-a + c\right )\right )} \log \left (\frac {\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) - 2 \, {\left (\cos \left (b x + 2 \, a\right ) - \cos \left (b x + 2 \, c\right )\right )} \sin \left (2 \, b x + a + 2 \, c\right ) + 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, a\right ) - 2 \, \cos \left (a\right ) \sin \left (b x + 2 \, c\right ) - 2 \, \cos \left (b x + 2 \, a\right ) \sin \left (a\right ) + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (a\right )}{2 \, {\left (b \cos \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, b x + a + 2 \, c\right ) \cos \left (a\right ) + b \sin \left (2 \, b x + a + 2 \, c\right )^{2} + 2 \, b \sin \left (2 \, b x + a + 2 \, c\right ) \sin \left (a\right ) + {\left (\cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right )} b\right )}} \] Input:
integrate(cos(b*x+a)*sec(b*x+c)^2,x, algorithm="maxima")
Output:
-1/2*(2*(sin(b*x + 2*a) - sin(b*x + 2*c))*cos(2*b*x + a + 2*c) + (cos(2*b* x + a + 2*c)^2*cos(-a + c) + 2*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + c os(-a + c)*sin(2*b*x + a + 2*c)^2 + 2*cos(-a + c)*sin(2*b*x + a + 2*c)*sin (a) + (cos(a)^2 + sin(a)^2)*cos(-a + c))*log((cos(b*x + 2*c)^2 + cos(c)^2 - 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 + 2*cos(b*x + 2*c)*sin(c) + s in(c)^2)/(cos(b*x + 2*c)^2 + cos(c)^2 + 2*cos(c)*sin(b*x + 2*c) + sin(b*x + 2*c)^2 - 2*cos(b*x + 2*c)*sin(c) + sin(c)^2)) - 2*(cos(b*x + 2*a) - cos( b*x + 2*c))*sin(2*b*x + a + 2*c) + 2*cos(a)*sin(b*x + 2*a) - 2*cos(a)*sin( b*x + 2*c) - 2*cos(b*x + 2*a)*sin(a) + 2*cos(b*x + 2*c)*sin(a))/(b*cos(2*b *x + a + 2*c)^2 + 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*x + a + 2*c) ^2 + 2*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b)
Leaf count of result is larger than twice the leaf count of optimal. 1341 vs. \(2 (35) = 70\).
Time = 0.21 (sec) , antiderivative size = 1341, normalized size of antiderivative = 38.31 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=\text {Too large to display} \] 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*ta n(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) - 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) + 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)*ta n(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 + t an(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) - t an(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 = 0.00 (sec) , antiderivative size = 246, normalized size of antiderivative = 7.03 \[ \int \cos (a+b x) \sec ^2(c+b x) \, dx=\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {\ln \left (-{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{2\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )} \] 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)*e xp(b*x*1i)*(exp(a*2i)*exp(-c*2i) + 1))*(exp(a*2i - c*2i) + 1))/(2*b*exp(a* 2i - c*2i)^(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)