Integrand size = 15, antiderivative size = 35 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\frac {\text {arctanh}(\sin (c+b x)) \sin ^2(a-c)}{b}+\frac {\sin (2 a-c+b x)}{b} \] Output:
arctanh(sin(b*x+c))*sin(a-c)^2/b+sin(b*x+2*a-c)/b
Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(35)=70\).
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.63 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\frac {(-1+\cos (2 a-2 c)) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )-\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{2 b}+\frac {(1-\cos (2 a-2 c)) \log \left (\cos \left (\frac {c}{2}+\frac {b x}{2}\right )+\sin \left (\frac {c}{2}+\frac {b x}{2}\right )\right )}{2 b}+\frac {\cos (b x) \sin (2 a-c)}{b}+\frac {\cos (2 a-c) \sin (b x)}{b} \] Input:
Integrate[Cos[a + b*x]^2*Sec[c + b*x],x]
Output:
((-1 + Cos[2*a - 2*c])*Log[Cos[c/2 + (b*x)/2] - Sin[c/2 + (b*x)/2]])/(2*b) + ((1 - Cos[2*a - 2*c])*Log[Cos[c/2 + (b*x)/2] + Sin[c/2 + (b*x)/2]])/(2* b) + (Cos[b*x]*Sin[2*a - c])/b + (Cos[2*a - c]*Sin[b*x])/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.
\(\displaystyle \int \cos ^2(a+b x) \sec (b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \cos ^2(a+b x) \sec (b x+c)dx\) |
Input:
Int[Cos[a + b*x]^2*Sec[c + b*x],x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.14
method | result | size |
risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{2 b}+\frac {\sin \left (b x +2 a -c \right )}{b}\) | \(145\) |
default | \(\frac {-\frac {2 \left (\left (-\cos \left (a \right ) \cos \left (c \right )-\sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )}+\frac {2 \left (-\cos \left (c \right )^{2} \sin \left (a \right )^{2}+2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \arctan \left (\frac {2 \left (\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-2 \sin \left (a \right ) \cos \left (c \right )+2 \cos \left (a \right ) \sin \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}}{b}\) | \(293\) |
Input:
int(cos(b*x+a)^2*sec(b*x+c),x,method=_RETURNVERBOSE)
Output:
-1/2/b*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))+1/2/b*ln(exp(I*(b*x+a))-I*exp(I*( a-c)))*cos(2*a-2*c)+1/2/b*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))-1/2/b*ln(exp(I *(b*x+a))+I*exp(I*(a-c)))*cos(2*a-2*c)+sin(b*x+2*a-c)/b
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (35) = 70\).
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.54 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=-\frac {4 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (-a + c\right ) + {\left (\cos \left (-a + c\right )^{2} - 1\right )} \log \left (\sin \left (b x + c\right ) + 1\right ) - {\left (\cos \left (-a + c\right )^{2} - 1\right )} \log \left (-\sin \left (b x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \sin \left (b x + c\right )}{2 \, b} \] Input:
integrate(cos(b*x+a)^2*sec(b*x+c),x, algorithm="fricas")
Output:
-1/2*(4*cos(b*x + c)*cos(-a + c)*sin(-a + c) + (cos(-a + c)^2 - 1)*log(sin (b*x + c) + 1) - (cos(-a + c)^2 - 1)*log(-sin(b*x + c) + 1) - 2*(2*cos(-a + c)^2 - 1)*sin(b*x + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (27) = 54\).
Time = 19.02 (sec) , antiderivative size = 3645, normalized size of antiderivative = 104.14 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)**2*sec(b*x+c),x)
Output:
-2*Piecewise((-sin(b*x)/b, Eq(c, pi/2)), (sin(b*x)/b, Eq(c, -pi/2)), (0, E q(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 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**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*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)* *2 + 2*b*tan(c/2)**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*t an(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2 )**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)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**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 + 2*b*tan(c/2) **2*tan(b*x/2)**2 + 2*b*tan(c/2)**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*tan(c/2)* *4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan (c/2)**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 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*...
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (35) = 70\).
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.00 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\frac {{\left (\cos \left (-2 \, a + 2 \, c\right ) - 1\right )} \log \left (\frac {\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) + 4 \, \sin \left (b x + 2 \, a - c\right )}{4 \, b} \] Input:
integrate(cos(b*x+a)^2*sec(b*x+c),x, algorithm="maxima")
Output:
1/4*((cos(-2*a + 2*c) - 1)*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) + sin(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*c os(b*x + 2*c)*sin(c) + sin(c)^2)) + 4*sin(b*x + 2*a - c))/b
Leaf count of result is larger than twice the leaf count of optimal. 1708 vs. \(2 (35) = 70\).
Time = 0.22 (sec) , antiderivative size = 1708, normalized size of antiderivative = 48.80 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cos(b*x+a)^2*sec(b*x+c),x, algorithm="giac")
Output:
-2*(2*(tan(1/2*a)^5*tan(1/2*c)^3 - 2*tan(1/2*a)^4*tan(1/2*c)^4 + tan(1/2*a )^3*tan(1/2*c)^5 - tan(1/2*a)^5*tan(1/2*c)^2 + 3*tan(1/2*a)^4*tan(1/2*c)^3 - 3*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*a)^2*tan(1/2*c)^5 + 3*tan(1/2*a)^ 4*tan(1/2*c)^2 - 6*tan(1/2*a)^3*tan(1/2*c)^3 + 3*tan(1/2*a)^2*tan(1/2*c)^4 - 2*tan(1/2*a)^4*tan(1/2*c) + 6*tan(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^ 2*tan(1/2*c)^3 + 2*tan(1/2*a)*tan(1/2*c)^4 + 3*tan(1/2*a)^3*tan(1/2*c) - 6 *tan(1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*a)^3 + 3* tan(1/2*a)^2*tan(1/2*c) - 3*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*c)^3 + tan(1 /2*a)^2 - 2*tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2)*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)^5*tan(1/2*c)^5 - tan(1/2*a)^5*tan(1 /2*c)^4 + tan(1/2*a)^4*tan(1/2*c)^5 + 2*tan(1/2*a)^5*tan(1/2*c)^3 + tan(1/ 2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^3*tan(1/2*c)^5 - 2*tan(1/2*a)^5*tan(1/2 *c)^2 + 2*tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(1/2*a)^3*tan(1/2*c)^4 + 2*tan( 1/2*a)^2*tan(1/2*c)^5 + tan(1/2*a)^5*tan(1/2*c) + 2*tan(1/2*a)^4*tan(1/2*c )^2 + 4*tan(1/2*a)^3*tan(1/2*c)^3 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2* a)*tan(1/2*c)^5 - tan(1/2*a)^5 + tan(1/2*a)^4*tan(1/2*c) - 4*tan(1/2*a)^3* tan(1/2*c)^2 + 4*tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)*tan(1/2*c)^4 + tan (1/2*c)^5 + tan(1/2*a)^4 + 2*tan(1/2*a)^3*tan(1/2*c) + 4*tan(1/2*a)^2*t...
Time = 19.14 (sec) , antiderivative size = 217, normalized size of antiderivative = 6.20 \[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (-\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}^2}{4\,b}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}\,\ln \left (\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{-c\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}^2}{4\,b} \] Input:
int(cos(a + b*x)^2/cos(c + b*x),x)
Output:
(exp(c*1i - a*2i - b*x*1i)*1i)/(2*b) - (exp(a*2i - c*1i + b*x*1i)*1i)/(2*b ) + (exp(c*2i - a*2i)*log((exp(c*1i)*exp(b*x*1i)*(exp(a*4i)*exp(-c*4i) - 2 *exp(a*2i)*exp(-c*2i) + 1))/2 - ((exp(a*2i)*exp(-c*2i) - 1)^2*1i)/2)*(exp( a*2i - c*2i) - 1)^2)/(4*b) - (exp(c*2i - a*2i)*log(((exp(a*2i)*exp(-c*2i) - 1)^2*1i)/2 + (exp(c*1i)*exp(b*x*1i)*(exp(a*4i)*exp(-c*4i) - 2*exp(a*2i)* exp(-c*2i) + 1))/2)*(exp(a*2i - c*2i) - 1)^2)/(4*b)
\[ \int \cos ^2(a+b x) \sec (c+b x) \, dx=\int \cos \left (b x +a \right )^{2} \sec \left (b x +c \right )d x \] Input:
int(cos(b*x+a)^2*sec(b*x+c),x)
Output:
int(cos(a + b*x)**2*sec(b*x + c),x)