Integrand size = 15, antiderivative size = 84 \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=-\frac {\csc ^3(a-c) \log (\cos (a+b x))}{b}+\frac {\csc ^3(a-c) \log (\cos (c+b x))}{b}-\frac {\csc (a-c) \sec ^2(c+b x)}{2 b}-\frac {\cot (a-c) \csc (a-c) \tan (c+b x)}{b} \] Output:
-csc(a-c)^3*ln(cos(b*x+a))/b+csc(a-c)^3*ln(cos(b*x+c))/b-1/2*csc(a-c)*sec( b*x+c)^2/b-cot(a-c)*csc(a-c)*tan(b*x+c)/b
Time = 0.68 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.82 \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\frac {\csc ^3(a-c) (2 \cos (2 a-3 c)-\cos (2 a-3 c-2 b x)+\cos (2 a-c+2 b x)-2 \cos (c+2 b x) \log (\cos (a+b x))-2 \cos (3 c+2 b x) \log (\cos (a+b x))+2 \cos (c+2 b x) \log (\cos (c+b x))+2 \cos (3 c+2 b x) \log (\cos (c+b x))+\cos (c) (-2-4 \log (\cos (a+b x))+4 \log (\cos (c+b x)))) \sec (c) \sec ^2(c+b x)}{8 b} \] Input:
Integrate[Sec[a + b*x]*Sec[c + b*x]^3,x]
Output:
(Csc[a - c]^3*(2*Cos[2*a - 3*c] - Cos[2*a - 3*c - 2*b*x] + Cos[2*a - c + 2 *b*x] - 2*Cos[c + 2*b*x]*Log[Cos[a + b*x]] - 2*Cos[3*c + 2*b*x]*Log[Cos[a + b*x]] + 2*Cos[c + 2*b*x]*Log[Cos[c + b*x]] + 2*Cos[3*c + 2*b*x]*Log[Cos[ c + b*x]] + Cos[c]*(-2 - 4*Log[Cos[a + b*x]] + 4*Log[Cos[c + b*x]]))*Sec[c ]*Sec[c + b*x]^2)/(8*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.
| \(\displaystyle \int \sec (a+b x) \sec ^3(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sec (a+b x) \sec ^3(b x+c)dx\) |
Input:
Int[Sec[a + b*x]*Sec[c + b*x]^3,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 3.36 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.51
| method | result | size |
| risch | \(\frac {4 i \left (2 \,{\mathrm e}^{i \left (2 b x +5 a +5 c \right )}+{\mathrm e}^{i \left (7 a +c \right )}+{\mathrm e}^{i \left (5 a +3 c \right )}\right )}{\left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} \left ({\mathrm e}^{2 i a}-{\mathrm e}^{2 i c}\right )^{2} b}+\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}-\frac {8 i \ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) {\mathrm e}^{3 i \left (a +c \right )}}{\left ({\mathrm e}^{6 i a}-3 \,{\mathrm e}^{2 i \left (2 a +c \right )}+3 \,{\mathrm e}^{2 i \left (a +2 c \right )}-{\mathrm e}^{6 i c}\right ) b}\) | \(211\) |
| default | \(\frac {-\frac {-2 \cos \left (a \right ) \cos \left (c \right )-2 \sin \left (a \right ) \sin \left (c \right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}+\frac {\ln \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )}{\left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{3}}+\frac {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}{2 \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )^{3} \left (\tan \left (b x +a \right ) \sin \left (a \right ) \cos \left (c \right )-\tan \left (b x +a \right ) \cos \left (a \right ) \sin \left (c \right )+\cos \left (a \right ) \cos \left (c \right )+\sin \left (a \right ) \sin \left (c \right )\right )^{2}}}{b}\) | \(214\) |
Input:
int(sec(b*x+a)*sec(b*x+c)^3,x,method=_RETURNVERBOSE)
Output:
4*I/(exp(2*I*(b*x+a+c))+exp(2*I*a))^2/(exp(2*I*a)-exp(2*I*c))^2/b*(2*exp(I *(2*b*x+5*a+5*c))+exp(I*(7*a+c))+exp(I*(5*a+3*c)))+8*I*ln(exp(2*I*(b*x+a)) +1)/(exp(6*I*a)-3*exp(2*I*(2*a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*exp(3* I*(a+c))-8*I*ln(exp(2*I*(b*x+a))+exp(2*I*(a-c)))/(exp(6*I*a)-3*exp(2*I*(2* a+c))+3*exp(2*I*(a+2*c))-exp(6*I*c))/b*exp(3*I*(a+c))
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (82) = 164\).
Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.14 \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + \cos \left (b x + c\right )^{2} \log \left (\cos \left (b x + c\right )^{2}\right ) - \cos \left (b x + c\right )^{2} \log \left (\frac {4 \, {\left (2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + {\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2} + 1\right )}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right ) + \cos \left (-a + c\right )^{2} - 1}{2 \, {\left (b \cos \left (-a + c\right )^{2} - b\right )} \cos \left (b x + c\right )^{2} \sin \left (-a + c\right )} \] Input:
integrate(sec(b*x+a)*sec(b*x+c)^3,x, algorithm="fricas")
Output:
1/2*(2*cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) + cos(b*x + c)^2* log(cos(b*x + c)^2) - cos(b*x + c)^2*log(4*(2*cos(b*x + c)*cos(-a + c)*sin (b*x + c)*sin(-a + c) + (2*cos(-a + c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c) ^2 + 1)/(cos(-a + c)^2 + 2*cos(-a + c) + 1)) + cos(-a + c)^2 - 1)/((b*cos( -a + c)^2 - b)*cos(b*x + c)^2*sin(-a + c))
\[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\int \sec {\left (a + b x \right )} \sec ^{3}{\left (b x + c \right )}\, dx \] Input:
integrate(sec(b*x+a)*sec(b*x+c)**3,x)
Output:
Integral(sec(a + b*x)*sec(b*x + c)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 69828 vs. \(2 (82) = 164\).
Time = 0.90 (sec) , antiderivative size = 69828, normalized size of antiderivative = 831.29 \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)*sec(b*x+c)^3,x, algorithm="maxima")
Output:
4*(9*((sin(4*a) + sin(4*c))*cos(3*a + c) - (cos(4*a) + cos(4*c))*sin(3*a + c) + 2*cos(2*a + 2*c)*sin(3*a + c) - 2*cos(3*a + c)*sin(2*a + 2*c))*cos(4 *a + 2*c)^2 + 9*((sin(4*a) + sin(4*c))*cos(3*a + c) + (sin(4*a) - 2*sin(2* a + 2*c) + sin(4*c))*cos(a + 3*c) - (cos(4*a) + cos(4*c))*sin(3*a + c) + 2 *cos(2*a + 2*c)*sin(3*a + c) - 2*cos(3*a + c)*sin(2*a + 2*c) - (cos(4*a) - 2*cos(2*a + 2*c) + cos(4*c))*sin(a + 3*c))*cos(2*a + 4*c)^2 + 9*((sin(4*a ) + sin(4*c))*cos(3*a + c) - (cos(4*a) + cos(4*c))*sin(3*a + c) + 2*cos(2* a + 2*c)*sin(3*a + c) - 2*cos(3*a + c)*sin(2*a + 2*c))*sin(4*a + 2*c)^2 + 2*(cos(6*a)^2 - 2*cos(6*a)*cos(6*c) + cos(6*c)^2 + sin(6*a)^2 - 2*sin(6*a) *sin(6*c) + sin(6*c)^2)*cos(2*a + 2*c)*sin(3*a + c) + 9*((sin(4*a) + sin(4 *c))*cos(3*a + c) + (sin(4*a) - 2*sin(2*a + 2*c) + sin(4*c))*cos(a + 3*c) - (cos(4*a) + cos(4*c))*sin(3*a + c) + 2*cos(2*a + 2*c)*sin(3*a + c) - 2*c os(3*a + c)*sin(2*a + 2*c) - (cos(4*a) - 2*cos(2*a + 2*c) + cos(4*c))*sin( a + 3*c))*sin(2*a + 4*c)^2 - 2*(cos(6*a)^2 - 2*cos(6*a)*cos(6*c) + cos(6*c )^2 + sin(6*a)^2 - 2*sin(6*a)*sin(6*c) + sin(6*c)^2)*cos(3*a + c)*sin(2*a + 2*c) - 2*(((cos(6*a) - 3*cos(4*a + 2*c) - cos(6*c))*cos(3*a + 3*c) + 3*c os(3*a + 3*c)*cos(2*a + 4*c) + (sin(6*a) - 3*sin(4*a + 2*c) - sin(6*c))*si n(3*a + 3*c) + 3*sin(3*a + 3*c)*sin(2*a + 4*c))*cos(4*b*x + 4*a + 4*c)^2 + 4*((cos(6*a) - 3*cos(4*a + 2*c) - cos(6*c))*cos(3*a + 3*c) + 3*cos(3*a + 3*c)*cos(2*a + 4*c) + (sin(6*a) - 3*sin(4*a + 2*c) - sin(6*c))*sin(3*a ...
Leaf count of result is larger than twice the leaf count of optimal. 2285 vs. \(2 (82) = 164\).
Time = 0.24 (sec) , antiderivative size = 2285, normalized size of antiderivative = 27.20 \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+a)*sec(b*x+c)^3,x, algorithm="giac")
Output:
1/8*((tan(1/2*a)^6*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 3*tan(1/2* a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan(1/2*a)^4*tan(1/2*c )^4 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)^6 + 9*tan(1/2*a)^4*tan(1/2* c)^2 + 9*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*c)^6 + 3*tan(1/2*a)^4 + 9*tan (1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*c)^4 + 3*tan(1/2*a)^2 + 3*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) - ta n(1/2*c)^2 + 1))/(tan(1/2*a)^6*tan(1/2*c)^3 - 3*tan(1/2*a)^5*tan(1/2*c)^4 + 3*tan(1/2*a)^4*tan(1/2*c)^5 - tan(1/2*a)^3*tan(1/2*c)^6 + 3*tan(1/2*a)^5 *tan(1/2*c)^2 - 9*tan(1/2*a)^4*tan(1/2*c)^3 + 9*tan(1/2*a)^3*tan(1/2*c)^4 - 3*tan(1/2*a)^2*tan(1/2*c)^5 + 3*tan(1/2*a)^4*tan(1/2*c) - 9*tan(1/2*a)^3 *tan(1/2*c)^2 + 9*tan(1/2*a)^2*tan(1/2*c)^3 - 3*tan(1/2*a)*tan(1/2*c)^4 + 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) - 2*(3*tan(b*x + a)^2*tan(1/2*a)^8*tan(1/2*c)^7 - 3*tan(b*x + a )^2*tan(1/2*a)^7*tan(1/2*c)^8 + tan(b*x + a)*tan(1/2*a)^8*tan(1/2*c)^8 + 9 *tan(b*x + a)^2*tan(1/2*a)^8*tan(1/2*c)^5 - 6*tan(b*x + a)^2*tan(1/2*a)^7* tan(1/2*c)^6 + 2*tan(b*x + a)*tan(1/2*a)^8*tan(1/2*c)^6 + 6*tan(b*x + a)^2 *tan(1/2*a)^6*tan(1/2*c)^7 + 4*tan(b*x + a)*tan(1/2*a)^7*tan(1/2*c)^7 + ta n(1/2*a)^8*tan(1/2*c)^7 - 9*tan(b*x + a)^2*tan(1/2*a)^5*tan(1/2*c)^8 + ...
Timed out. \[ \int \sec (a+b x) \sec ^3(c+b x) \, dx=\text {Hanged} \] Input:
int(1/(cos(a + b*x)*cos(c + b*x)^3),x)
Output:
\text{Hanged}
\[ \int \sec (a+b x) \sec ^3(c+b x) \, dx =\text {Too large to display} \] Input:
int(sec(b*x+a)*sec(b*x+c)^3,x)
Output:
(2*int(sin(b*x + c)**2/(sin(b*x + c)**2 - 1),x)*sin(b*x + c)**2*b - 2*int( sin(b*x + c)**2/(sin(b*x + c)**2 - 1),x)*b + 2*int(sin(b*x + c)**2/(cos(a + b*x)*sin(b*x + c)**2 - cos(a + b*x)),x)*sin(b*x + c)**2*b - 2*int(sin(b* x + c)**2/(cos(a + b*x)*sin(b*x + c)**2 - cos(a + b*x)),x)*b - 2*int(1/(si n(b*x + c)**2 - 1),x)*sin(b*x + c)**2*b + 2*int(1/(sin(b*x + c)**2 - 1),x) *b - 2*int(1/(cos(b*x + c)*cos(a + b*x)*sin(b*x + c)**2 - cos(b*x + c)*cos (a + b*x)),x)*sin(b*x + c)**2*b + 2*int(1/(cos(b*x + c)*cos(a + b*x)*sin(b *x + c)**2 - cos(b*x + c)*cos(a + b*x)),x)*b - 2*int(1/(cos(b*x + c)*sin(b *x + c)**2 - cos(b*x + c)),x)*sin(b*x + c)**2*b + 2*int(1/(cos(b*x + c)*si n(b*x + c)**2 - cos(b*x + c)),x)*b - 2*int(1/(cos(a + b*x)*sin(b*x + c)**2 - cos(a + b*x)),x)*sin(b*x + c)**2*b + 2*int(1/(cos(a + b*x)*sin(b*x + c) **2 - cos(a + b*x)),x)*b + log(tan((b*x + c)/2) - 1)*sin(b*x + c)**2 - log (tan((b*x + c)/2) - 1) - log(tan((b*x + c)/2) + 1)*sin(b*x + c)**2 + log(t an((b*x + c)/2) + 1) + 2*log(tan((a + b*x)/2) - 1)*sin(b*x + c)**2 - 2*log (tan((a + b*x)/2) - 1) - 2*log(tan((a + b*x)/2) + 1)*sin(b*x + c)**2 + 2*l og(tan((a + b*x)/2) + 1) - 2*sin(b*x + c)**2*b*x + sin(b*x + c) + 2*b*x)/( 2*b*(sin(b*x + c)**2 - 1))