Integrand size = 17, antiderivative size = 50 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=-x \cos (2 (a-c))-\frac {\log (\cos (c+b x)) \sin (2 (a-c))}{b}+\frac {\cos ^2(a-c) \tan (c+b x)}{b} \] Output:
-x*cos(2*a-2*c)-ln(cos(b*x+c))*sin(2*a-2*c)/b+cos(a-c)^2*tan(b*x+c)/b
Leaf count is larger than twice the leaf count of optimal. \(177\) vs. \(2(50)=100\).
Time = 0.35 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.54 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=-\frac {\sec (c) \sec (c+b x) (b x \cos (2 a-4 c-b x)+b x \cos (2 a-2 c-b x)+b x \cos (2 a+b x)+b x \cos (2 a-2 c+b x)-2 \sin (b x)+\log (\cos (c+b x)) \sin (2 a-4 c-b x)+\sin (2 a-2 c-b x)+\log (\cos (c+b x)) \sin (2 a-2 c-b x)+\log (\cos (c+b x)) \sin (2 a+b x)-\sin (2 a-2 c+b x)+\log (\cos (c+b x)) \sin (2 a-2 c+b x))}{4 b} \] Input:
Integrate[Sec[c + b*x]^2*Sin[a + b*x]^2,x]
Output:
-1/4*(Sec[c]*Sec[c + b*x]*(b*x*Cos[2*a - 4*c - b*x] + b*x*Cos[2*a - 2*c - b*x] + b*x*Cos[2*a + b*x] + b*x*Cos[2*a - 2*c + b*x] - 2*Sin[b*x] + Log[Co s[c + b*x]]*Sin[2*a - 4*c - b*x] + Sin[2*a - 2*c - b*x] + Log[Cos[c + b*x] ]*Sin[2*a - 2*c - b*x] + Log[Cos[c + b*x]]*Sin[2*a + b*x] - Sin[2*a - 2*c + b*x] + Log[Cos[c + b*x]]*Sin[2*a - 2*c + 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.
| \(\displaystyle \int \sin ^2(a+b x) \sec ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sin ^2(a+b x) \sec ^2(b x+c)dx\) |
Input:
Int[Sec[c + b*x]^2*Sin[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.34
| method | result | size |
| risch | \(-x \,{\mathrm e}^{2 i \left (a -c \right )}+2 i \sin \left (2 a -2 c \right ) x +\frac {2 i \sin \left (2 a -2 c \right ) a}{b}+\frac {i {\mathrm e}^{2 i \left (2 a -c \right )}}{2 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {i {\mathrm e}^{2 i a}}{b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}+\frac {i {\mathrm e}^{2 i c}}{2 b \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (a -c \right )}\right ) \sin \left (2 a -2 c \right )}{b}\) | \(167\) |
| default | \(\frac {-\frac {\left (2 \cos \left (c \right )^{3} \sin \left (a \right )^{2} \cos \left (a \right )+2 \cos \left (c \right )^{2} \sin \left (c \right ) \sin \left (a \right )^{3}-4 \cos \left (c \right )^{2} \sin \left (c \right ) \cos \left (a \right )^{2} \sin \left (a \right )-4 \cos \left (c \right ) \sin \left (c \right )^{2} \cos \left (a \right ) \sin \left (a \right )^{2}+2 \cos \left (c \right ) \sin \left (c \right )^{2} \cos \left (a \right )^{3}+2 \sin \left (c \right )^{3} \sin \left (a \right ) \cos \left (a \right )^{2}\right ) \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 (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right )^{2} \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )^{2} \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )}-\frac {\cos \left (a \right )^{2} \cos \left (c \right )^{2}+2 \cos \left (a \right ) \cos \left (c \right ) \sin \left (a \right ) \sin \left (c \right )+\sin \left (a \right )^{2} \sin \left (c \right )^{2}}{\left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right ) \left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right ) \left (\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right ) \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 {\frac {\left (-2 \cos \left (a \right )^{2} \cos \left (c \right ) \sin \left (c \right )+2 \cos \left (c \right )^{2} \cos \left (a \right ) \sin \left (a \right )-2 \cos \left (a \right ) \sin \left (a \right ) \sin \left (c \right )^{2}+2 \sin \left (a \right )^{2} \cos \left (c \right ) \sin \left (c \right )\right ) \ln \left (\tan \left (b x +a \right )^{2}+1\right )}{2}+\left (-\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}-4 \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}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \arctan \left (\tan \left (b x +a \right )\right )}{\left (\cos \left (c \right )^{2}+\sin \left (c \right )^{2}\right )^{2} \left (\cos \left (a \right )^{2}+\sin \left (a \right )^{2}\right )^{2}}}{b}\) | \(394\) |
Input:
int(sec(b*x+c)^2*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-x*exp(2*I*(a-c))+2*I*sin(2*a-2*c)*x+2*I/b*sin(2*a-2*c)*a+1/2*I/b/(exp(2*I *(b*x+a+c))+exp(2*I*a))*exp(2*I*(2*a-c))+I/b/(exp(2*I*(b*x+a+c))+exp(2*I*a ))*exp(2*I*a)+1/2*I/b/(exp(2*I*(b*x+a+c))+exp(2*I*a))*exp(2*I*c)-ln(exp(2* I*(b*x+a))+exp(2*I*(a-c)))/b*sin(2*a-2*c)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.64 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=\frac {2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \log \left (-\cos \left (b x + c\right )\right ) \sin \left (-a + c\right ) + \cos \left (-a + c\right )^{2} \sin \left (b x + c\right ) - {\left (2 \, b x \cos \left (-a + c\right )^{2} - b x\right )} \cos \left (b x + c\right )}{b \cos \left (b x + c\right )} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^2,x, algorithm="fricas")
Output:
(2*cos(b*x + c)*cos(-a + c)*log(-cos(b*x + c))*sin(-a + c) + cos(-a + c)^2 *sin(b*x + c) - (2*b*x*cos(-a + c)^2 - b*x)*cos(b*x + c))/(b*cos(b*x + c))
Exception generated. \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(sec(b*x+c)**2*sin(b*x+a)**2,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (50) = 100\).
Time = 0.06 (sec) , antiderivative size = 534, normalized size of antiderivative = 10.68 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^2,x, algorithm="maxima")
Output:
-1/2*(2*(b*cos(2*a + 2*c)*cos(4*c) + b*sin(2*a + 2*c)*sin(4*c))*x + (2*b*x *cos(4*c) + sin(4*a) + 2*sin(2*a + 2*c) + sin(4*c))*cos(2*b*x + 2*a + 4*c) + 2*(b*x*cos(2*b*x + 2*a + 4*c) + b*x*cos(2*a + 2*c))*cos(2*b*x + 6*c) + (sin(4*a) + sin(4*c))*cos(2*a + 2*c) - (cos(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c) + 2*cos(2*b*x + 2*a + 4*c)*cos(2*a + 2*c)*sin(-2*a + 2*c) + cos(2*a + 2*c)^2*sin(-2*a + 2*c) + sin(2*b*x + 2*a + 4*c)^2*sin(-2*a + 2*c) + 2*s in(2*b*x + 2*a + 4*c)*sin(2*a + 2*c)*sin(-2*a + 2*c) + sin(2*a + 2*c)^2*si n(-2*a + 2*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) + (2*b*x*sin(4*c) - cos(4* a) - 2*cos(2*a + 2*c) - cos(4*c))*sin(2*b*x + 2*a + 4*c) + 2*(b*x*sin(2*b* x + 2*a + 4*c) + b*x*sin(2*a + 2*c))*sin(2*b*x + 6*c) - (cos(4*a) + cos(4* c))*sin(2*a + 2*c))/(b*cos(2*b*x + 2*a + 4*c)^2 + 2*b*cos(2*b*x + 2*a + 4* c)*cos(2*a + 2*c) + b*cos(2*a + 2*c)^2 + b*sin(2*b*x + 2*a + 4*c)^2 + 2*b* sin(2*b*x + 2*a + 4*c)*sin(2*a + 2*c) + b*sin(2*a + 2*c)^2)
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (50) = 100\).
Time = 0.14 (sec) , antiderivative size = 760, normalized size of antiderivative = 15.20 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^2,x, algorithm="giac")
Output:
-((tan(1/2*a)^4*tan(1/2*c)^4 - 6*tan(1/2*a)^4*tan(1/2*c)^2 + 16*tan(1/2*a) ^3*tan(1/2*c)^3 - 6*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 - 16*tan(1/2* a)^3*tan(1/2*c) + 36*tan(1/2*a)^2*tan(1/2*c)^2 - 16*tan(1/2*a)*tan(1/2*c)^ 3 + tan(1/2*c)^4 - 6*tan(1/2*a)^2 + 16*tan(1/2*a)*tan(1/2*c) - 6*tan(1/2*c )^2 + 1)*(b*x + c)/(tan(1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^ 2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c) ^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) - 2*(tan(1/2*a)^4 *tan(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c) + 6*ta n(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)*tan(1/2 *c)^4 - tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) - 6*tan(1/2*a)*tan(1/2*c) ^2 + tan(1/2*c)^3 + tan(1/2*a) - tan(1/2*c))*log(tan(b*x + c)^2 + 1)/(tan( 1/2*a)^4*tan(1/2*c)^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1 /2*c)^4 + tan(1/2*a)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*ta n(1/2*a)^2 + 2*tan(1/2*c)^2 + 1) - (tan(b*x + c)*tan(1/2*a)^4*tan(1/2*c)^4 - 2*tan(b*x + c)*tan(1/2*a)^4*tan(1/2*c)^2 + 8*tan(b*x + c)*tan(1/2*a)^3* tan(1/2*c)^3 - 2*tan(b*x + c)*tan(1/2*a)^2*tan(1/2*c)^4 + tan(b*x + c)*tan (1/2*a)^4 - 8*tan(b*x + c)*tan(1/2*a)^3*tan(1/2*c) + 20*tan(b*x + c)*tan(1 /2*a)^2*tan(1/2*c)^2 - 8*tan(b*x + c)*tan(1/2*a)*tan(1/2*c)^3 + tan(b*x + c)*tan(1/2*c)^4 - 2*tan(b*x + c)*tan(1/2*a)^2 + 8*tan(b*x + c)*tan(1/2*a)* tan(1/2*c) - 2*tan(b*x + c)*tan(1/2*c)^2 + tan(b*x + c))/(tan(1/2*a)^4*...
Time = 0.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.92 \[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx=-x\,\left (\cos \left (2\,a-2\,c\right )-\sin \left (2\,a-2\,c\right )\,1{}\mathrm {i}\right )+\frac {\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}-c\,4{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{2\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\right )}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}+c\,4{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )\,\left (2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-2\,b\,{\mathrm {e}}^{a\,6{}\mathrm {i}-c\,6{}\mathrm {i}}\right )\,1{}\mathrm {i}}{4\,b^2} \] Input:
int(sin(a + b*x)^2/cos(c + b*x)^2,x)
Output:
((2*exp(a*2i - c*2i) + exp(a*4i - c*4i) + 1)*1i)/(2*b*(exp(a*2i - c*2i) + exp(a*2i + b*x*2i))) - x*(cos(2*a - 2*c) - sin(2*a - 2*c)*1i) - (exp(c*4i - a*4i)*log(exp(a*2i)*exp(b*x*2i) + exp(a*2i)*exp(-c*2i))*(2*b*exp(a*2i - c*2i) - 2*b*exp(a*6i - c*6i))*1i)/(4*b^2)
\[ \int \sec ^2(c+b x) \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:
int(sec(b*x+c)^2*sin(b*x+a)^2,x)
Output:
(7*cos(b*x + c)*cos(a + b*x)*sin(a + b*x) + 96*cos(b*x + c)*int(tan((b*x + c)/2)**2/(tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 + 2*tan((b*x + c)/2)**4 *tan((a + b*x)/2)**2 + tan((b*x + c)/2)**4 - 2*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 - 4*tan((b*x + c)/2)**2*tan((a + b*x)/2)**2 - 2*tan((b*x + c) /2)**2 + tan((a + b*x)/2)**4 + 2*tan((a + b*x)/2)**2 + 1),x)*b + 96*cos(b* x + c)*int(tan((a + b*x)/2)**2/(tan((b*x + c)/2)**4*tan((a + b*x)/2)**4 + 2*tan((b*x + c)/2)**4*tan((a + b*x)/2)**2 + tan((b*x + c)/2)**4 - 2*tan((b *x + c)/2)**2*tan((a + b*x)/2)**4 - 4*tan((b*x + c)/2)**2*tan((a + b*x)/2) **2 - 2*tan((b*x + c)/2)**2 + tan((a + b*x)/2)**4 + 2*tan((a + b*x)/2)**2 + 1),x)*b - 128*cos(b*x + c)*int((tan((b*x + c)/2)*tan((a + b*x)/2))/(tan( (b*x + c)/2)**4*tan((a + b*x)/2)**4 + 2*tan((b*x + c)/2)**4*tan((a + b*x)/ 2)**2 + tan((b*x + c)/2)**4 - 2*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 - 4*tan((b*x + c)/2)**2*tan((a + b*x)/2)**2 - 2*tan((b*x + c)/2)**2 + tan((a + b*x)/2)**4 + 2*tan((a + b*x)/2)**2 + 1),x)*b - 32*cos(b*x + c)*int(1/(t an((b*x + c)/2)**4*tan((a + b*x)/2)**4 + 2*tan((b*x + c)/2)**4*tan((a + b* x)/2)**2 + tan((b*x + c)/2)**4 - 2*tan((b*x + c)/2)**2*tan((a + b*x)/2)**4 - 4*tan((b*x + c)/2)**2*tan((a + b*x)/2)**2 - 2*tan((b*x + c)/2)**2 + tan ((a + b*x)/2)**4 + 2*tan((a + b*x)/2)**2 + 1),x)*b + 16*cos(b*x + c)*sin(a + b*x) + 9*cos(b*x + c)*a + 9*cos(b*x + c)*b*x - 8*cos(a + b*x)*sin(b*x + c) + 8*cos(a + b*x)*sin(a + b*x) + 4*sin(b*x + c)*sin(a + b*x)**2 - 8*...