Integrand size = 31, antiderivative size = 73 \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {5-4 \sec ^2(x)}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {5-4 \sec ^2(x)}}{\sqrt {5}}\right )}{5 \sqrt {5}}-\frac {2}{15 \sqrt {5-4 \sec ^2(x)}} \] Output:
-1/18*arctanh(1/3*(5-4*sec(x)^2)^(1/2)*3^(1/2))*3^(1/2)-1/25*arctanh(1/5*( 5-4*sec(x)^2)^(1/2)*5^(1/2))*5^(1/2)-2/15/(5-4*sec(x)^2)^(1/2)
Time = 3.70 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.77 \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=-\frac {\sqrt {-\cos ^2(x)} \left (60 \sqrt {-\cos ^2(x)}+9 \text {arcsinh}\left (\frac {1}{2} \sqrt {5} \sqrt {-\cos ^2(x)}\right ) \sqrt {30-50 \cos (2 x)}+25 \text {arctanh}\left (\frac {\sqrt {3} \sqrt {-\cos ^2(x)}}{\sqrt {-1+5 \sin ^2(x)}}\right ) \sqrt {-3+15 \sin ^2(x)}\right ) \sqrt {\sec ^2(x)-5 \tan ^2(x)}}{450 \left (-1+5 \sin ^2(x)\right )} \] Input:
Integrate[((3 + Sin[x]^2)*Tan[x]^3)/((-2 + Cos[x]^2)*(5 - 4*Sec[x]^2)^(3/2 )),x]
Output:
-1/450*(Sqrt[-Cos[x]^2]*(60*Sqrt[-Cos[x]^2] + 9*ArcSinh[(Sqrt[5]*Sqrt[-Cos [x]^2])/2]*Sqrt[30 - 50*Cos[2*x]] + 25*ArcTanh[(Sqrt[3]*Sqrt[-Cos[x]^2])/S qrt[-1 + 5*Sin[x]^2]]*Sqrt[-3 + 15*Sin[x]^2])*Sqrt[Sec[x]^2 - 5*Tan[x]^2]) /(-1 + 5*Sin[x]^2)
Time = 1.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 4873, 25, 7276, 2009}
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.
| \(\displaystyle \int \frac {\left (\sin ^2(x)+3\right ) \tan ^3(x)}{\left (\cos ^2(x)-2\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (\sin (x)^2+3\right ) \tan (x)^3}{\left (\cos (x)^2-2\right ) \left (5-4 \sec (x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4873 |
| \(\displaystyle -\int -\frac {\left (1-\cos ^2(x)\right ) \left (4-\cos ^2(x)\right ) \sec ^3(x)}{\left (2-\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}}d\cos (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\left (1-\cos ^2(x)\right ) \left (4-\cos ^2(x)\right ) \sec ^3(x)}{\left (2-\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}}d\cos (x)\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {3 \sec (x)}{2 \left (5-4 \sec ^2(x)\right )^{3/2}}+\frac {2 \sec ^3(x)}{\left (5-4 \sec ^2(x)\right )^{3/2}}+\frac {\cos (x)}{2 \left (\cos ^2(x)-2\right ) \left (5-4 \sec ^2(x)\right )^{3/2}}\right )d\cos (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {5-4 \sec ^2(x)}}{\sqrt {3}}\right )}{6 \sqrt {3}}-\frac {\text {arctanh}\left (\frac {\sqrt {5-4 \sec ^2(x)}}{\sqrt {5}}\right )}{5 \sqrt {5}}-\frac {2}{15 \sqrt {5-4 \sec ^2(x)}}\) |
Input:
Int[((3 + Sin[x]^2)*Tan[x]^3)/((-2 + Cos[x]^2)*(5 - 4*Sec[x]^2)^(3/2)),x]
Output:
-1/6*ArcTanh[Sqrt[5 - 4*Sec[x]^2]/Sqrt[3]]/Sqrt[3] - ArcTanh[Sqrt[5 - 4*Se c[x]^2]/Sqrt[5]]/(5*Sqrt[5]) - 2/(15*Sqrt[5 - 4*Sec[x]^2])
Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = Free Factors[Cos[c*(a + b*x)], x]}, Simp[-(b*c*d^(n - 1))^(-1) Subst[Int[Subst For[(1 - d^2*x^2)^((n - 1)/2)/x^n, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c* (a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Tan] || EqQ[F, tan] )
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(523\) vs. \(2(55)=110\).
Time = 2.60 (sec) , antiderivative size = 524, normalized size of antiderivative = 7.18
| method | result | size |
| default | \(\frac {\frac {3 \sec \left (x \right )^{3} \left (\left (250 \cos \left (x \right )^{3}+250 \cos \left (x \right )^{2}-200 \cos \left (x \right )-200\right ) \sqrt {3}\, \sqrt {2}+\left (-500 \cos \left (x \right )^{3}-500 \cos \left (x \right )^{2}+400 \cos \left (x \right )+400\right ) \sqrt {3}+\left (125 \cos \left (x \right )^{3}+125 \cos \left (x \right )^{2}-100 \cos \left (x \right )-100\right ) \sqrt {6}\, \sqrt {2}+\left (-250 \cos \left (x \right )^{3}-250 \cos \left (x \right )^{2}+200 \cos \left (x \right )+200\right ) \sqrt {6}\right ) \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}\, \operatorname {arctanh}\left (\frac {5 \cos \left (x \right ) \sqrt {2}+4 \sqrt {2}-10 \cos \left (x \right )-4}{\left (1+\cos \left (x \right )\right ) \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}\, \left (2 \sqrt {3}-\sqrt {6}\right )}\right )}{5}+\frac {3 \sec \left (x \right )^{3} \left (\left (250 \cos \left (x \right )^{3}+250 \cos \left (x \right )^{2}-200 \cos \left (x \right )-200\right ) \sqrt {3}\, \sqrt {2}+\left (500 \cos \left (x \right )^{3}+500 \cos \left (x \right )^{2}-400 \cos \left (x \right )-400\right ) \sqrt {3}+\left (-125 \cos \left (x \right )^{3}-125 \cos \left (x \right )^{2}+100 \cos \left (x \right )+100\right ) \sqrt {6}\, \sqrt {2}+\left (-250 \cos \left (x \right )^{3}-250 \cos \left (x \right )^{2}+200 \cos \left (x \right )+200\right ) \sqrt {6}\right ) \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}\, \operatorname {arctanh}\left (\frac {5 \cos \left (x \right ) \sqrt {2}+4 \sqrt {2}+10 \cos \left (x \right )+4}{\left (1+\cos \left (x \right )\right ) \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}\, \left (2 \sqrt {3}+\sqrt {6}\right )}\right )}{5}+\frac {3 \sqrt {5}\, \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}\, \operatorname {arctanh}\left (\frac {\cos \left (x \right ) \sqrt {5}}{\left (1+\cos \left (x \right )\right ) \sqrt {\frac {5 \cos \left (x \right )^{2}-4}{\left (1+\cos \left (x \right )\right )^{2}}}}\right ) \left (360+360 \sec \left (x \right )-288 \sec \left (x \right )^{2}-288 \sec \left (x \right )^{3}\right )}{5}+720-576 \sec \left (x \right )^{2}}{\left (5-4 \sec \left (x \right )^{2}\right )^{\frac {3}{2}} \left (5+2 \sqrt {5}\right ) \left (-5+2 \sqrt {5}\right ) \left (6+2 \sqrt {5}+\sqrt {2}\right ) \left (-6-2 \sqrt {5}+\sqrt {2}\right ) \left (-6+2 \sqrt {5}+\sqrt {2}\right ) \left (6-2 \sqrt {5}+\sqrt {2}\right ) \left (2 \sqrt {3}-\sqrt {6}\right ) \left (2 \sqrt {3}+\sqrt {6}\right )}\) | \(524\) |
Input:
int((3+sin(x)^2)*tan(x)^3/(cos(x)^2-2)/(5-4*sec(x)^2)^(3/2),x,method=_RETU RNVERBOSE)
Output:
3/5/(5-4*sec(x)^2)^(3/2)*(sec(x)^3*((250*cos(x)^3+250*cos(x)^2-200*cos(x)- 200)*3^(1/2)*2^(1/2)+(-500*cos(x)^3-500*cos(x)^2+400*cos(x)+400)*3^(1/2)+( 125*cos(x)^3+125*cos(x)^2-100*cos(x)-100)*6^(1/2)*2^(1/2)+(-250*cos(x)^3-2 50*cos(x)^2+200*cos(x)+200)*6^(1/2))*((5*cos(x)^2-4)/(1+cos(x))^2)^(1/2)*a rctanh((5*cos(x)*2^(1/2)+4*2^(1/2)-10*cos(x)-4)/(1+cos(x))/((5*cos(x)^2-4) /(1+cos(x))^2)^(1/2)/(2*3^(1/2)-6^(1/2)))+sec(x)^3*((250*cos(x)^3+250*cos( x)^2-200*cos(x)-200)*3^(1/2)*2^(1/2)+(500*cos(x)^3+500*cos(x)^2-400*cos(x) -400)*3^(1/2)+(-125*cos(x)^3-125*cos(x)^2+100*cos(x)+100)*6^(1/2)*2^(1/2)+ (-250*cos(x)^3-250*cos(x)^2+200*cos(x)+200)*6^(1/2))*((5*cos(x)^2-4)/(1+co s(x))^2)^(1/2)*arctanh((5*cos(x)*2^(1/2)+4*2^(1/2)+10*cos(x)+4)/(1+cos(x)) /((5*cos(x)^2-4)/(1+cos(x))^2)^(1/2)/(2*3^(1/2)+6^(1/2)))+5^(1/2)*((5*cos( x)^2-4)/(1+cos(x))^2)^(1/2)*arctanh(cos(x)/(1+cos(x))/((5*cos(x)^2-4)/(1+c os(x))^2)^(1/2)*5^(1/2))*(360+360*sec(x)-288*sec(x)^2-288*sec(x)^3)+1200-9 60*sec(x)^2)/(5+2*5^(1/2))/(-5+2*5^(1/2))/(6+2*5^(1/2)+2^(1/2))/(-6-2*5^(1 /2)+2^(1/2))/(-6+2*5^(1/2)+2^(1/2))/(6-2*5^(1/2)+2^(1/2))/(2*3^(1/2)-6^(1/ 2))/(2*3^(1/2)+6^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (55) = 110\).
Time = 0.13 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.52 \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=-\frac {480 \, \sqrt {\frac {5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} \cos \left (x\right )^{2} - 18 \, {\left (5 \, \sqrt {5} \cos \left (x\right )^{2} - 4 \, \sqrt {5}\right )} \log \left (625 \, \cos \left (x\right )^{8} - 1000 \, \cos \left (x\right )^{6} + 500 \, \cos \left (x\right )^{4} - 80 \, \cos \left (x\right )^{2} - {\left (125 \, \sqrt {5} \cos \left (x\right )^{8} - 150 \, \sqrt {5} \cos \left (x\right )^{6} + 50 \, \sqrt {5} \cos \left (x\right )^{4} - 4 \, \sqrt {5} \cos \left (x\right )^{2}\right )} \sqrt {\frac {5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} + 2\right ) - 25 \, {\left (5 \, \sqrt {3} \cos \left (x\right )^{2} - 4 \, \sqrt {3}\right )} \log \left (\frac {1921 \, \cos \left (x\right )^{8} - 3464 \, \cos \left (x\right )^{6} + 2040 \, \cos \left (x\right )^{4} - 416 \, \cos \left (x\right )^{2} - 8 \, {\left (62 \, \sqrt {3} \cos \left (x\right )^{8} - 87 \, \sqrt {3} \cos \left (x\right )^{6} + 36 \, \sqrt {3} \cos \left (x\right )^{4} - 4 \, \sqrt {3} \cos \left (x\right )^{2}\right )} \sqrt {\frac {5 \, \cos \left (x\right )^{2} - 4}{\cos \left (x\right )^{2}}} + 16}{\cos \left (x\right )^{8} - 8 \, \cos \left (x\right )^{6} + 24 \, \cos \left (x\right )^{4} - 32 \, \cos \left (x\right )^{2} + 16}\right )}{3600 \, {\left (5 \, \cos \left (x\right )^{2} - 4\right )}} \] Input:
integrate((3+sin(x)^2)*tan(x)^3/(-2+cos(x)^2)/(5-4*sec(x)^2)^(3/2),x, algo rithm="fricas")
Output:
-1/3600*(480*sqrt((5*cos(x)^2 - 4)/cos(x)^2)*cos(x)^2 - 18*(5*sqrt(5)*cos( x)^2 - 4*sqrt(5))*log(625*cos(x)^8 - 1000*cos(x)^6 + 500*cos(x)^4 - 80*cos (x)^2 - (125*sqrt(5)*cos(x)^8 - 150*sqrt(5)*cos(x)^6 + 50*sqrt(5)*cos(x)^4 - 4*sqrt(5)*cos(x)^2)*sqrt((5*cos(x)^2 - 4)/cos(x)^2) + 2) - 25*(5*sqrt(3 )*cos(x)^2 - 4*sqrt(3))*log((1921*cos(x)^8 - 3464*cos(x)^6 + 2040*cos(x)^4 - 416*cos(x)^2 - 8*(62*sqrt(3)*cos(x)^8 - 87*sqrt(3)*cos(x)^6 + 36*sqrt(3 )*cos(x)^4 - 4*sqrt(3)*cos(x)^2)*sqrt((5*cos(x)^2 - 4)/cos(x)^2) + 16)/(co s(x)^8 - 8*cos(x)^6 + 24*cos(x)^4 - 32*cos(x)^2 + 16)))/(5*cos(x)^2 - 4)
Timed out. \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((3+sin(x)**2)*tan(x)**3/(-2+cos(x)**2)/(5-4*sec(x)**2)**(3/2),x)
Output:
Timed out
\[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=\int { \frac {{\left (\sin \left (x\right )^{2} + 3\right )} \tan \left (x\right )^{3}}{{\left (\cos \left (x\right )^{2} - 2\right )} {\left (-4 \, \sec \left (x\right )^{2} + 5\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((3+sin(x)^2)*tan(x)^3/(-2+cos(x)^2)/(5-4*sec(x)^2)^(3/2),x, algo rithm="maxima")
Output:
integrate((sin(x)^2 + 3)*tan(x)^3/((cos(x)^2 - 2)*(-4*sec(x)^2 + 5)^(3/2)) , x)
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (55) = 110\).
Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=-\frac {5 \, \sqrt {15} \sqrt {5} \log \left (-\frac {2 \, {\left ({\left (\sqrt {5} \cos \left (x\right ) - \sqrt {5 \, \cos \left (x\right )^{2} - 4}\right )}^{2} - 4 \, \sqrt {15} - 16\right )}}{{\left | 2 \, {\left (\sqrt {5} \cos \left (x\right ) - \sqrt {5 \, \cos \left (x\right )^{2} - 4}\right )}^{2} + 8 \, \sqrt {15} - 32 \right |}}\right ) - 18 \, \sqrt {5} \log \left ({\left (\sqrt {5} \cos \left (x\right ) - \sqrt {5 \, \cos \left (x\right )^{2} - 4}\right )}^{2}\right ) + \frac {120 \, \cos \left (x\right )}{\sqrt {5 \, \cos \left (x\right )^{2} - 4}}}{900 \, \mathrm {sgn}\left (\cos \left (x\right )\right )} \] Input:
integrate((3+sin(x)^2)*tan(x)^3/(-2+cos(x)^2)/(5-4*sec(x)^2)^(3/2),x, algo rithm="giac")
Output:
-1/900*(5*sqrt(15)*sqrt(5)*log(-2*((sqrt(5)*cos(x) - sqrt(5*cos(x)^2 - 4)) ^2 - 4*sqrt(15) - 16)/abs(2*(sqrt(5)*cos(x) - sqrt(5*cos(x)^2 - 4))^2 + 8* sqrt(15) - 32)) - 18*sqrt(5)*log((sqrt(5)*cos(x) - sqrt(5*cos(x)^2 - 4))^2 ) + 120*cos(x)/sqrt(5*cos(x)^2 - 4))/sgn(cos(x))
Timed out. \[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tan}\left (x\right )}^3\,\left ({\sin \left (x\right )}^2+3\right )}{\left ({\cos \left (x\right )}^2-2\right )\,{\left (5-\frac {4}{{\cos \left (x\right )}^2}\right )}^{3/2}} \,d x \] Input:
int((tan(x)^3*(sin(x)^2 + 3))/((cos(x)^2 - 2)*(5 - 4/cos(x)^2)^(3/2)),x)
Output:
int((tan(x)^3*(sin(x)^2 + 3))/((cos(x)^2 - 2)*(5 - 4/cos(x)^2)^(3/2)), x)
\[ \int \frac {\left (3+\sin ^2(x)\right ) \tan ^3(x)}{\left (-2+\cos ^2(x)\right ) \left (5-4 \sec ^2(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {-4 \sec \left (x \right )^{2}+5}\, \sin \left (x \right )^{2} \tan \left (x \right )^{3}}{16 \cos \left (x \right )^{2} \sec \left (x \right )^{4}-40 \cos \left (x \right )^{2} \sec \left (x \right )^{2}+25 \cos \left (x \right )^{2}-32 \sec \left (x \right )^{4}+80 \sec \left (x \right )^{2}-50}d x +3 \left (\int \frac {\sqrt {-4 \sec \left (x \right )^{2}+5}\, \tan \left (x \right )^{3}}{16 \cos \left (x \right )^{2} \sec \left (x \right )^{4}-40 \cos \left (x \right )^{2} \sec \left (x \right )^{2}+25 \cos \left (x \right )^{2}-32 \sec \left (x \right )^{4}+80 \sec \left (x \right )^{2}-50}d x \right ) \] Input:
int((3+sin(x)^2)*tan(x)^3/(-2+cos(x)^2)/(5-4*sec(x)^2)^(3/2),x)
Output:
int((sqrt( - 4*sec(x)**2 + 5)*sin(x)**2*tan(x)**3)/(16*cos(x)**2*sec(x)**4 - 40*cos(x)**2*sec(x)**2 + 25*cos(x)**2 - 32*sec(x)**4 + 80*sec(x)**2 - 5 0),x) + 3*int((sqrt( - 4*sec(x)**2 + 5)*tan(x)**3)/(16*cos(x)**2*sec(x)**4 - 40*cos(x)**2*sec(x)**2 + 25*cos(x)**2 - 32*sec(x)**4 + 80*sec(x)**2 - 5 0),x)