Integrand size = 17, antiderivative size = 88 \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\frac {\text {arctanh}(\sin (c+b x)) \cos ^2(a-c)}{2 b}-\frac {\text {arctanh}(\sin (c+b x)) \cos (2 (a-c))}{b}+\frac {\sec (c+b x) \sin (2 (a-c))}{b}+\frac {\cos ^2(a-c) \sec (c+b x) \tan (c+b x)}{2 b} \] Output:
1/2*arctanh(sin(b*x+c))*cos(a-c)^2/b-arctanh(sin(b*x+c))*cos(2*a-2*c)/b+se c(b*x+c)*sin(2*a-2*c)/b+1/2*cos(a-c)^2*sec(b*x+c)*tan(b*x+c)/b
Time = 1.96 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.88 \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\frac {-\log \left (\cos \left (\frac {1}{2} (c+b x)\right )-\sin \left (\frac {1}{2} (c+b x)\right )\right )+3 \cos (2 (a-c)) \left (\log \left (\cos \left (\frac {1}{2} (c+b x)\right )-\sin \left (\frac {1}{2} (c+b x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+b x)\right )+\sin \left (\frac {1}{2} (c+b x)\right )\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+b x)\right )+\sin \left (\frac {1}{2} (c+b x)\right )\right )-4 \sec (c) \sin (2 (a-c))+4 \sec (c+b x) \sin (2 (a-c))+2 \cos ^2(a-c) \sec (c+b x) \tan (c+b x)}{4 b} \] Input:
Integrate[Sec[c + b*x]^3*Sin[a + b*x]^2,x]
Output:
(-Log[Cos[(c + b*x)/2] - Sin[(c + b*x)/2]] + 3*Cos[2*(a - c)]*(Log[Cos[(c + b*x)/2] - Sin[(c + b*x)/2]] - Log[Cos[(c + b*x)/2] + Sin[(c + b*x)/2]]) + Log[Cos[(c + b*x)/2] + Sin[(c + b*x)/2]] - 4*Sec[c]*Sin[2*(a - c)] + 4*S ec[c + b*x]*Sin[2*(a - c)] + 2*Cos[a - c]^2*Sec[c + b*x]*Tan[c + b*x])/(4* 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.
\(\displaystyle \int \sin ^2(a+b x) \sec ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \sec ^3(b x+c)dx\) |
Input:
Int[Sec[c + b*x]^3*Sin[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 7.97 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.84
method | result | size |
risch | \(-\frac {i \left (5 \,{\mathrm e}^{i \left (3 b x +6 a +c \right )}+2 \,{\mathrm e}^{i \left (3 b x +4 a +3 c \right )}-3 \,{\mathrm e}^{i \left (3 b x +2 a +5 c \right )}+3 \,{\mathrm e}^{i \left (b x +6 a -c \right )}-2 \,{\mathrm e}^{i \left (b x +4 a +c \right )}-5 \,{\mathrm e}^{i \left (b x +2 a +3 c \right )}\right )}{4 \left ({\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{2} b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right )}{4 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{4 b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right )}{4 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+i {\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (2 a -2 c \right )}{4 b}\) | \(250\) |
default | \(\text {Expression too large to display}\) | \(1166\) |
Input:
int(sec(b*x+c)^3*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*I/(exp(2*I*(b*x+a+c))+exp(2*I*a))^2/b*(5*exp(I*(3*b*x+6*a+c))+2*exp(I *(3*b*x+4*a+3*c))-3*exp(I*(3*b*x+2*a+5*c))+3*exp(I*(b*x+6*a-c))-2*exp(I*(b *x+4*a+c))-5*exp(I*(b*x+2*a+3*c)))-1/4/b*ln(exp(I*(b*x+a))-I*exp(I*(a-c))) +3/4/b*ln(exp(I*(b*x+a))-I*exp(I*(a-c)))*cos(2*a-2*c)+1/4/b*ln(exp(I*(b*x+ a))+I*exp(I*(a-c)))-3/4/b*ln(exp(I*(b*x+a))+I*exp(I*(a-c)))*cos(2*a-2*c)
Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (3 \, \cos \left (-a + c\right )^{2} - 2\right )} \cos \left (b x + c\right )^{2} \log \left (\sin \left (b x + c\right ) + 1\right ) - {\left (3 \, \cos \left (-a + c\right )^{2} - 2\right )} \cos \left (b x + c\right )^{2} \log \left (-\sin \left (b x + c\right ) + 1\right ) - 2 \, \cos \left (-a + c\right )^{2} \sin \left (b x + c\right ) + 8 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (-a + c\right )}{4 \, b \cos \left (b x + c\right )^{2}} \] Input:
integrate(sec(b*x+c)^3*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-1/4*((3*cos(-a + c)^2 - 2)*cos(b*x + c)^2*log(sin(b*x + c) + 1) - (3*cos( -a + c)^2 - 2)*cos(b*x + c)^2*log(-sin(b*x + c) + 1) - 2*cos(-a + c)^2*sin (b*x + c) + 8*cos(b*x + c)*cos(-a + c)*sin(-a + c))/(b*cos(b*x + c)^2)
Timed out. \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Timed out} \] Input:
integrate(sec(b*x+c)**3*sin(b*x+a)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1263 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 1263, normalized size of antiderivative = 14.35 \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+c)^3*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/8*(2*(5*sin(3*b*x + 4*a + 2*c) + 2*sin(3*b*x + 2*a + 4*c) - 3*sin(3*b*x + 6*c) + 3*sin(b*x + 4*a) - 2*sin(b*x + 2*a + 2*c) - 5*sin(b*x + 4*c))*cos (4*b*x + 2*a + 5*c) - 10*(2*sin(2*b*x + 2*a + 3*c) + sin(2*a + c))*cos(3*b *x + 4*a + 2*c) - 4*(2*sin(2*b*x + 2*a + 3*c) + sin(2*a + c))*cos(3*b*x + 2*a + 4*c) + 6*(2*sin(2*b*x + 2*a + 3*c) + sin(2*a + c))*cos(3*b*x + 6*c) + 4*(3*sin(b*x + 4*a) - 2*sin(b*x + 2*a + 2*c) - 5*sin(b*x + 4*c))*cos(2*b *x + 2*a + 3*c) + ((3*cos(-2*a + 2*c) - 1)*cos(4*b*x + 2*a + 5*c)^2 + 4*(3 *cos(-2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c)^2 + (3*cos(-2*a + 2*c) - 1)*s in(4*b*x + 2*a + 5*c)^2 + 4*(3*cos(-2*a + 2*c) - 1)*sin(2*b*x + 2*a + 3*c) ^2 + 2*(2*(3*cos(-2*a + 2*c) - 1)*cos(2*b*x + 2*a + 3*c) + 3*cos(2*a + c)* cos(-2*a + 2*c) - cos(2*a + c))*cos(4*b*x + 2*a + 5*c) + 4*(3*cos(2*a + c) *cos(-2*a + 2*c) - cos(2*a + c))*cos(2*b*x + 2*a + 3*c) - cos(2*a + c)^2 + 3*(cos(2*a + c)^2 + sin(2*a + c)^2)*cos(-2*a + 2*c) + 2*(2*(3*cos(-2*a + 2*c) - 1)*sin(2*b*x + 2*a + 3*c) + 3*cos(-2*a + 2*c)*sin(2*a + c) - sin(2* a + c))*sin(4*b*x + 2*a + 5*c) + 4*(3*cos(-2*a + 2*c)*sin(2*a + c) - sin(2 *a + c))*sin(2*b*x + 2*a + 3*c) - sin(2*a + 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)*s in(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*cos(b*x + 2*c)*sin(c) + sin(c)^2)) - 2*(5*cos(3*b*x + 4*a + 2*c) + 2*cos(3*b*x + 2*a + 4*c) - 3*cos(3*b*x + 6*c) + 3*cos(b*...
Leaf count of result is larger than twice the leaf count of optimal. 1502 vs. \(2 (84) = 168\).
Time = 0.17 (sec) , antiderivative size = 1502, normalized size of antiderivative = 17.07 \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sec(b*x+c)^3*sin(b*x+a)^2,x, algorithm="giac")
Output:
-1/2*((tan(1/2*a)^4*tan(1/2*c)^4 - 10*tan(1/2*a)^4*tan(1/2*c)^2 + 24*tan(1 /2*a)^3*tan(1/2*c)^3 - 10*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 - 24*ta n(1/2*a)^3*tan(1/2*c) + 52*tan(1/2*a)^2*tan(1/2*c)^2 - 24*tan(1/2*a)*tan(1 /2*c)^3 + tan(1/2*c)^4 - 10*tan(1/2*a)^2 + 24*tan(1/2*a)*tan(1/2*c) - 10*t an(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x + 1/2*c) + 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*tan(1/2*a)^2 + 2*ta n(1/2*c)^2 + 1) - (tan(1/2*a)^4*tan(1/2*c)^4 - 10*tan(1/2*a)^4*tan(1/2*c)^ 2 + 24*tan(1/2*a)^3*tan(1/2*c)^3 - 10*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2* a)^4 - 24*tan(1/2*a)^3*tan(1/2*c) + 52*tan(1/2*a)^2*tan(1/2*c)^2 - 24*tan( 1/2*a)*tan(1/2*c)^3 + tan(1/2*c)^4 - 10*tan(1/2*a)^2 + 24*tan(1/2*a)*tan(1 /2*c) - 10*tan(1/2*c)^2 + 1)*log(abs(tan(1/2*b*x + 1/2*c) - 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*tan(1/2 *a)^2 + 2*tan(1/2*c)^2 + 1) - 2*(tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1 /2*c)^4 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c)^2 + 8*tan(1/2*b *x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^3 - 8*tan(1/2*b*x + 1/2*c)^2*tan(1/2 *a)^4*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^4 + 8*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*b*x + 1/2*c)* tan(1/2*a)^4*tan(1/2*c)^4 + tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4 - 8*tan...
Timed out. \[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\text {Hanged} \] Input:
int(sin(a + b*x)^2/cos(c + b*x)^3,x)
Output:
\text{Hanged}
\[ \int \sec ^3(c+b x) \sin ^2(a+b x) \, dx=\int \sec \left (b x +c \right )^{3} \sin \left (b x +a \right )^{2}d x \] Input:
int(sec(b*x+c)^3*sin(b*x+a)^2,x)
Output:
int(sec(b*x + c)**3*sin(a + b*x)**2,x)