Integrand size = 11, antiderivative size = 1 \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=0 \] Output:
0
Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(1)=2\).
Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 41.00 \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=\frac {a}{b}+x+\frac {16-40 (a+b x)^2+45 (a+b x)^4}{5 b (a+b x)^5} \] Input:
Integrate[Sin[3*ArcSec[a + b*x]]^2,x]
Output:
a/b + x + (16 - 40*(a + b*x)^2 + 45*(a + b*x)^4)/(5*b*(a + b*x)^5)
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.
\(\displaystyle \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int \sin ^2\left (3 \sec ^{-1}(a+b x)\right )d(a+b x)}{b}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {\int \sin ^2\left (3 \sec ^{-1}(a+b x)\right )d(a+b x)}{b}\) |
Input:
Int[Sin[3*ArcSec[a + b*x]]^2,x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(1)=2\).
Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 39.00
method | result | size |
default | \(x +\frac {16}{5 b \left (b x +a \right )^{5}}+\frac {9}{\left (b x +a \right ) b}-\frac {8}{b \left (b x +a \right )^{3}}\) | \(39\) |
orering | \(-\frac {\left (-5 b^{6} x^{6}+75 a^{2} b^{4} x^{4}+200 a^{3} b^{3} x^{3}+225 a^{4} b^{2} x^{2}-45 x^{4} b^{4}+120 a^{5} b x -180 a \,b^{3} x^{3}+25 a^{6}-270 a^{2} b^{2} x^{2}-180 a^{3} b x -45 a^{4}+40 x^{2} b^{2}+80 a b x +40 a^{2}-16\right ) \left (b x +a \right ) \sin \left (3 \,\operatorname {arcsec}\left (b x +a \right )\right )^{2}}{5 b \left (b x +a -2\right )^{2} \left (b x +a +1\right ) \left (b x +a -1\right ) \left (b x +a +2\right )^{2}}\) | \(167\) |
Input:
int(sin(3*arcsec(b*x+a))^2,x,method=_RETURNVERBOSE)
Output:
x+16/5/b/(b*x+a)^5+9/(b*x+a)/b-8/b/(b*x+a)^3
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 154.00 \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=\frac {5 \, b^{6} x^{6} + 25 \, a b^{5} x^{5} + 5 \, {\left (10 \, a^{2} + 9\right )} b^{4} x^{4} + 10 \, {\left (5 \, a^{3} + 18 \, a\right )} b^{3} x^{3} + 5 \, {\left (5 \, a^{4} + 54 \, a^{2} - 8\right )} b^{2} x^{2} + 45 \, a^{4} + 5 \, {\left (a^{5} + 36 \, a^{3} - 16 \, a\right )} b x - 40 \, a^{2} + 16}{5 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \] Input:
integrate(sin(3*arcsec(b*x+a))^2,x, algorithm="fricas")
Output:
1/5*(5*b^6*x^6 + 25*a*b^5*x^5 + 5*(10*a^2 + 9)*b^4*x^4 + 10*(5*a^3 + 18*a) *b^3*x^3 + 5*(5*a^4 + 54*a^2 - 8)*b^2*x^2 + 45*a^4 + 5*(a^5 + 36*a^3 - 16* a)*b*x - 40*a^2 + 16)/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3 *x^2 + 5*a^4*b^2*x + a^5*b)
\[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=\int \sin ^{2}{\left (3 \operatorname {asec}{\left (a + b x \right )} \right )}\, dx \] Input:
integrate(sin(3*asec(b*x+a))**2,x)
Output:
Integral(sin(3*asec(a + b*x))**2, x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 94.23 (sec) , antiderivative size = 502, normalized size of antiderivative = 502.00 \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=-\frac {48 \, b^{10} x^{10} + 480 \, a b^{9} x^{9} + 40 \, {\left (54 \, a^{2} - 5\right )} b^{8} x^{8} + 320 \, {\left (18 \, a^{3} - 5 \, a\right )} b^{7} x^{7} + 10 \, {\left (1008 \, a^{4} - 560 \, a^{2} + 29\right )} b^{6} x^{6} + 28 \, {\left (432 \, a^{5} - 400 \, a^{3} + 75 \, a\right )} b^{5} x^{5} + 48 \, a^{10} + 5 \, {\left (2016 \, a^{6} - 2800 \, a^{4} + 1230 \, a^{2} - 273\right )} b^{4} x^{4} - 200 \, a^{8} + 20 \, {\left (288 \, a^{7} - 560 \, a^{5} + 470 \, a^{3} - 273 \, a\right )} b^{3} x^{3} + 650 \, a^{6} + 10 \, {\left (216 \, a^{8} - 560 \, a^{6} + 795 \, a^{4} - 819 \, a^{2} + 128\right )} b^{2} x^{2} - 1365 \, a^{4} + 20 \, {\left (24 \, a^{9} - 80 \, a^{7} + 177 \, a^{5} - 273 \, a^{3} + 128 \, a\right )} b x - {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )} {\left (48 \, b^{4} x^{5} + 240 \, a b^{3} x^{4} + 40 \, {\left (12 \, a^{2} - 5\right )} b^{2} x^{3} + 120 \, {\left (4 \, a^{3} - 5 \, a\right )} b x^{2} + 30 \, {\left (8 \, a^{4} - 20 \, a^{2} + 15\right )} x + \frac {3 \, {\left (25 \, b^{4} x^{4} + 100 \, a b^{3} x^{3} + 150 \, a^{2} b^{2} x^{2} + 100 \, a^{3} b x + 25 \, a^{4} + 1\right )}}{b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b}\right )} + 1280 \, a^{2} - 509}{160 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \] Input:
integrate(sin(3*arcsec(b*x+a))^2,x, algorithm="maxima")
Output:
-1/160*(48*b^10*x^10 + 480*a*b^9*x^9 + 40*(54*a^2 - 5)*b^8*x^8 + 320*(18*a ^3 - 5*a)*b^7*x^7 + 10*(1008*a^4 - 560*a^2 + 29)*b^6*x^6 + 28*(432*a^5 - 4 00*a^3 + 75*a)*b^5*x^5 + 48*a^10 + 5*(2016*a^6 - 2800*a^4 + 1230*a^2 - 273 )*b^4*x^4 - 200*a^8 + 20*(288*a^7 - 560*a^5 + 470*a^3 - 273*a)*b^3*x^3 + 6 50*a^6 + 10*(216*a^8 - 560*a^6 + 795*a^4 - 819*a^2 + 128)*b^2*x^2 - 1365*a ^4 + 20*(24*a^9 - 80*a^7 + 177*a^5 - 273*a^3 + 128*a)*b*x - (b^6*x^5 + 5*a *b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)*(48*b^4* x^5 + 240*a*b^3*x^4 + 40*(12*a^2 - 5)*b^2*x^3 + 120*(4*a^3 - 5*a)*b*x^2 + 30*(8*a^4 - 20*a^2 + 15)*x + 3*(25*b^4*x^4 + 100*a*b^3*x^3 + 150*a^2*b^2*x ^2 + 100*a^3*b*x + 25*a^4 + 1)/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 1 0*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)) + 1280*a^2 - 509)/(b^6*x^5 + 5*a*b^5 *x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2*x + a^5*b)
Exception generated. \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sin(3*arcsec(b*x+a))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:int(sage0,sageVARx) Error: Bad Arg ument Value
Time = 39.66 (sec) , antiderivative size = 38, normalized size of antiderivative = 38.00 \[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=x+\frac {9}{b\,\left (a+b\,x\right )}-\frac {8}{b\,{\left (a+b\,x\right )}^3}+\frac {16}{5\,b\,{\left (a+b\,x\right )}^5} \] Input:
int(sin(3*acos(1/(a + b*x)))^2,x)
Output:
x + 9/(b*(a + b*x)) - 8/(b*(a + b*x)^3) + 16/(5*b*(a + b*x)^5)
\[ \int \sin ^2\left (3 \sec ^{-1}(a+b x)\right ) \, dx=\int \sin \left (3 \mathit {asec} \left (b x +a \right )\right )^{2}d x \] Input:
int(sin(3*asec(b*x+a))^2,x)
Output:
int(sin(3*asec(a + b*x))**2,x)