Integrand size = 17, antiderivative size = 1 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=0 \] Output:
0
Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.
Time = 0.13 (sec) , antiderivative size = 131, normalized size of antiderivative = 131.00 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\frac {\cos (3 a-2 c) \cos (b x)}{b}+\frac {\cos ^3(a-c) \sec (c+b x)}{b}-\frac {6 i \arctan \left (\frac {(i \cos (c)+\sin (c)) \left (\cos \left (\frac {b x}{2}\right ) \sin (c)+\cos (c) \sin \left (\frac {b x}{2}\right )\right )}{\cos (c) \cos \left (\frac {b x}{2}\right )-i \cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \cos ^2(a-c) \sin (a-c)}{b}-\frac {\sin (3 a-2 c) \sin (b x)}{b} \] Input:
Integrate[Sec[c + b*x]^2*Sin[a + b*x]^3,x]
Output:
(Cos[3*a - 2*c]*Cos[b*x])/b + (Cos[a - c]^3*Sec[c + b*x])/b - ((6*I)*ArcTa n[((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]^2*Sin[a - c])/b - (Sin [3*a - 2*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.
\(\displaystyle \int \sin ^3(a+b x) \sec ^2(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^3(a+b x) \sec ^2(b x+c)dx\) |
Input:
Int[Sec[c + b*x]^2*Sin[a + b*x]^3,x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 6.66 (sec) , antiderivative size = 235, normalized size of antiderivative = 235.00
method | result | size |
risch | \(\frac {{\mathrm e}^{i \left (b x +5 a -2 c \right )}+3 \,{\mathrm e}^{i \left (b x +3 a \right )}+3 \,{\mathrm e}^{i \left (b x +a +2 c \right )}+{\mathrm e}^{-i \left (-b x +a -4 c \right )}}{4 \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right ) b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (3 a -3 c \right )}{4 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (3 a -3 c \right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{4 b}+\frac {\cos \left (b x +3 a -2 c \right )}{b}\) | \(235\) |
default | \(\text {Expression too large to display}\) | \(652\) |
Input:
int(sec(b*x+c)^2*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/4/(exp(2*I*(b*x+a+c))+exp(2*I*a))/b*(exp(I*(b*x+5*a-2*c))+3*exp(I*(b*x+3 *a))+3*exp(I*(b*x+a+2*c))+exp(-I*(-b*x+a-4*c)))+3/4*ln(exp(I*(b*x+a))+I*ex p(I*(a-c)))/b*sin(3*a-3*c)+3/4*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))/b*sin(a-c )-3/4*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))/b*sin(3*a-3*c)-3/4*ln(exp(I*(b*x+a ))-I*exp(I*(a-c)))/b*sin(a-c)+cos(b*x+3*a-2*c)/b
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 149.00 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=-\frac {3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )^{2} \log \left (\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 3 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )^{2} \log \left (-\sin \left (b x + c\right ) + 1\right ) \sin \left (-a + c\right ) - 2 \, {\left (4 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - 2 \, {\left (4 \, \cos \left (-a + c\right )^{3} - 3 \, \cos \left (-a + c\right )\right )} \cos \left (b x + c\right )^{2} - 2 \, \cos \left (-a + c\right )^{3}}{2 \, b \cos \left (b x + c\right )} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^3,x, algorithm="fricas")
Output:
-1/2*(3*cos(b*x + c)*cos(-a + c)^2*log(sin(b*x + c) + 1)*sin(-a + c) - 3*c os(b*x + c)*cos(-a + c)^2*log(-sin(b*x + c) + 1)*sin(-a + c) - 2*(4*cos(-a + c)^2 - 1)*cos(b*x + c)*sin(b*x + c)*sin(-a + c) - 2*(4*cos(-a + c)^3 - 3*cos(-a + c))*cos(b*x + c)^2 - 2*cos(-a + c)^3)/(b*cos(b*x + c))
Exception generated. \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate(sec(b*x+c)**2*sin(b*x+a)**3,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.22 (sec) , antiderivative size = 871, normalized size of antiderivative = 871.00 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^3,x, algorithm="maxima")
Output:
1/8*(4*(cos(3*b*x + 3*a + 4*c) + cos(b*x + 3*a + 2*c))*cos(4*b*x + 6*a + 2 *c) + 2*(3*cos(2*b*x + 6*a) + 3*cos(2*b*x + 4*a + 2*c) + 3*cos(2*b*x + 2*a + 4*c) + 3*cos(2*b*x + 6*c) + 2*cos(4*c))*cos(3*b*x + 3*a + 4*c) + 6*cos( 2*b*x + 6*a)*cos(b*x + 3*a + 2*c) + 6*cos(2*b*x + 4*a + 2*c)*cos(b*x + 3*a + 2*c) + 6*cos(2*b*x + 2*a + 4*c)*cos(b*x + 3*a + 2*c) + 6*cos(2*b*x + 6* c)*cos(b*x + 3*a + 2*c) + 4*cos(b*x + 3*a + 2*c)*cos(4*c) + 3*((sin(-a + c ) + sin(-3*a + 3*c))*cos(3*b*x + 3*a + 4*c)^2 + 2*(sin(-a + c) + sin(-3*a + 3*c))*cos(3*b*x + 3*a + 4*c)*cos(b*x + 3*a + 2*c) + (sin(-a + c) + sin(- 3*a + 3*c))*cos(b*x + 3*a + 2*c)^2 + (sin(-a + c) + sin(-3*a + 3*c))*sin(3 *b*x + 3*a + 4*c)^2 + 2*(sin(-a + c) + sin(-3*a + 3*c))*sin(3*b*x + 3*a + 4*c)*sin(b*x + 3*a + 2*c) + (sin(-a + c) + sin(-3*a + 3*c))*sin(b*x + 3*a + 2*c)^2)*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 + co s(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)) + 4*(sin(3*b*x + 3*a + 4*c) + sin(b*x + 3*a + 2*c))*sin(4 *b*x + 6*a + 2*c) + 2*(3*sin(2*b*x + 6*a) + 3*sin(2*b*x + 4*a + 2*c) + 3*s in(2*b*x + 2*a + 4*c) + 3*sin(2*b*x + 6*c) + 2*sin(4*c))*sin(3*b*x + 3*a + 4*c) + 6*sin(2*b*x + 6*a)*sin(b*x + 3*a + 2*c) + 6*sin(2*b*x + 4*a + 2*c) *sin(b*x + 3*a + 2*c) + 6*sin(2*b*x + 2*a + 4*c)*sin(b*x + 3*a + 2*c) + 6* sin(2*b*x + 6*c)*sin(b*x + 3*a + 2*c) + 4*sin(b*x + 3*a + 2*c)*sin(4*c)...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.18 (sec) , antiderivative size = 2956, normalized size of antiderivative = 2956.00 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+c)^2*sin(b*x+a)^3,x, algorithm="giac")
Output:
2*(3*(tan(1/2*a)^6*tan(1/2*c)^5 - tan(1/2*a)^5*tan(1/2*c)^6 - 2*tan(1/2*a) ^6*tan(1/2*c)^3 + 11*tan(1/2*a)^5*tan(1/2*c)^4 - 11*tan(1/2*a)^4*tan(1/2*c )^5 + 2*tan(1/2*a)^3*tan(1/2*c)^6 + tan(1/2*a)^6*tan(1/2*c) - 11*tan(1/2*a )^5*tan(1/2*c)^2 + 38*tan(1/2*a)^4*tan(1/2*c)^3 - 38*tan(1/2*a)^3*tan(1/2* c)^4 + 11*tan(1/2*a)^2*tan(1/2*c)^5 - tan(1/2*a)*tan(1/2*c)^6 + tan(1/2*a) ^5 - 11*tan(1/2*a)^4*tan(1/2*c) + 38*tan(1/2*a)^3*tan(1/2*c)^2 - 38*tan(1/ 2*a)^2*tan(1/2*c)^3 + 11*tan(1/2*a)*tan(1/2*c)^4 - tan(1/2*c)^5 - 2*tan(1/ 2*a)^3 + 11*tan(1/2*a)^2*tan(1/2*c) - 11*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1 /2*c)^3 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*c) + 1))/(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*t an(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) - 3* (tan(1/2*a)^6*tan(1/2*c)^5 - tan(1/2*a)^5*tan(1/2*c)^6 - 2*tan(1/2*a)^6*ta n(1/2*c)^3 + 11*tan(1/2*a)^5*tan(1/2*c)^4 - 11*tan(1/2*a)^4*tan(1/2*c)^5 + 2*tan(1/2*a)^3*tan(1/2*c)^6 + tan(1/2*a)^6*tan(1/2*c) - 11*tan(1/2*a)^5*t an(1/2*c)^2 + 38*tan(1/2*a)^4*tan(1/2*c)^3 - 38*tan(1/2*a)^3*tan(1/2*c)^4 + 11*tan(1/2*a)^2*tan(1/2*c)^5 - tan(1/2*a)*tan(1/2*c)^6 + tan(1/2*a)^5 - 11*tan(1/2*a)^4*tan(1/2*c) + 38*tan(1/2*a)^3*tan(1/2*c)^2 - 38*tan(1/2*...
Time = 24.93 (sec) , antiderivative size = 382, normalized size of antiderivative = 382.00 \[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\frac {{\mathrm {e}}^{-a\,3{}\mathrm {i}+c\,2{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,2{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}+c\,2{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}-c\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}-c\,6{}\mathrm {i}}+1\right )\,1{}\mathrm {i}}{4\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}+{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}+\frac {3\,\sin \left (2\,a-2\,c\right )\,\ln \left (-\frac {3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (\sin \left (2\,a-2\,c\right )+\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )}{2}-\frac {\sin \left (2\,a-2\,c\right )\,{\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 )\,3{}\mathrm {i}}{2\,\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 )}{4\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}-\frac {3\,\sin \left (2\,a-2\,c\right )\,\ln \left (-\frac {3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (\sin \left (2\,a-2\,c\right )+\sin \left (2\,a-2\,c\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\right )}{2}+\frac {\sin \left (2\,a-2\,c\right )\,{\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 )\,3{}\mathrm {i}}{2\,\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 )}{4\,b\,\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \] Input:
int(sin(a + b*x)^3/cos(c + b*x)^2,x)
Output:
exp(c*2i - a*3i - b*x*1i)/(2*b) + exp(a*3i - c*2i + b*x*1i)/(2*b) + (exp(c *2i - a*1i + b*x*1i)*(3*exp(a*2i - c*2i) + 3*exp(a*4i - c*4i) + exp(a*6i - c*6i) + 1)*1i)/(4*b*(exp(a*2i - c*2i)*1i + exp(a*2i + b*x*2i)*1i)) + (3*s in(2*a - 2*c)*log(- (3*exp(a*1i)*exp(b*x*1i)*(sin(2*a - 2*c) + sin(2*a - 2 *c)*exp(a*2i)*exp(-c*2i)))/2 - (sin(2*a - 2*c)*exp(a*2i)*exp(-c*2i)*(exp(a *2i)*exp(-c*2i) + 1)*3i)/(2*(exp(a*2i)*exp(-c*2i))^(1/2)))*(exp(a*2i - c*2 i) + 1))/(4*b*exp(a*2i - c*2i)^(1/2)) - (3*sin(2*a - 2*c)*log((sin(2*a - 2 *c)*exp(a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) + 1)*3i)/(2*(exp(a*2i)*exp( -c*2i))^(1/2)) - (3*exp(a*1i)*exp(b*x*1i)*(sin(2*a - 2*c) + sin(2*a - 2*c) *exp(a*2i)*exp(-c*2i)))/2)*(exp(a*2i - c*2i) + 1))/(4*b*exp(a*2i - c*2i)^( 1/2))
\[ \int \sec ^2(c+b x) \sin ^3(a+b x) \, dx=\int \sec \left (b x +c \right )^{2} \sin \left (b x +a \right )^{3}d x \] Input:
int(sec(b*x+c)^2*sin(b*x+a)^3,x)
Output:
int(sec(b*x + c)**2*sin(a + b*x)**3,x)