Integrand size = 29, antiderivative size = 100 \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=2 \text {arctanh}\left (\frac {\tan (x)}{\sqrt {\tan (x) \tan (2 x)}}\right )-\frac {11 \text {arctanh}\left (\frac {\sqrt {2} \tan (x)}{\sqrt {\tan (x) \tan (2 x)}}\right )}{4 \sqrt {2}}+\frac {\tan (x)}{2 (\tan (x) \tan (2 x))^{3/2}}+\frac {2 \tan ^3(x)}{3 (\tan (x) \tan (2 x))^{3/2}}+\frac {3 \tan (x)}{4 \sqrt {\tan (x) \tan (2 x)}} \] Output:
2*arctanh(tan(x)/(tan(x)*tan(2*x))^(1/2))-11/8*arctanh(2^(1/2)*tan(x)/(tan (x)*tan(2*x))^(1/2))*2^(1/2)+3/4*tan(x)/(tan(x)*tan(2*x))^(1/2)+1/2*tan(x) /(tan(x)*tan(2*x))^(3/2)+2/3*tan(x)^3/(tan(x)*tan(2*x))^(3/2)
Time = 4.82 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.69 \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=\frac {\left (-\cos (2 x)+2 \tan ^2(x)\right ) \left (\frac {4 \sqrt {2} \left (-2 \text {arctanh}\left (\sqrt {\frac {\cos (2 x)}{1+\cos (2 x)}}\right )+\sqrt {2} \text {arctanh}\left (\sqrt {1-\tan ^2(x)}\right )\right ) \cos (2 x) \tan (x)}{\sqrt {1-\tan ^2(x)}}-3 \arctan \left (\sqrt {-1+\tan ^2(x)}\right ) \cos (x) \sin (x) \sqrt {-1+\tan ^2(x)}+\frac {1}{3} \left (-3 \cot (x)-4 \cos (x) \sin (x)+(5+9 \cos (2 x)) \tan ^3(x)\right )\right ) \tan ^2(2 x)}{2 (-3+6 \cos (2 x)+\cos (4 x)) (\tan (x) \tan (2 x))^{3/2}} \] Input:
Integrate[(Sec[x]^2*(-Cos[2*x] + 2*Tan[x]^2))/(Tan[x]*Tan[2*x])^(3/2),x]
Output:
((-Cos[2*x] + 2*Tan[x]^2)*((4*Sqrt[2]*(-2*ArcTanh[Sqrt[Cos[2*x]/(1 + Cos[2 *x])]] + Sqrt[2]*ArcTanh[Sqrt[1 - Tan[x]^2]])*Cos[2*x]*Tan[x])/Sqrt[1 - Ta n[x]^2] - 3*ArcTan[Sqrt[-1 + Tan[x]^2]]*Cos[x]*Sin[x]*Sqrt[-1 + Tan[x]^2] + (-3*Cot[x] - 4*Cos[x]*Sin[x] + (5 + 9*Cos[2*x])*Tan[x]^3)/3)*Tan[2*x]^2) /(2*(-3 + 6*Cos[2*x] + Cos[4*x])*(Tan[x]*Tan[2*x])^(3/2))
Time = 1.13 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {3042, 4897, 3042, 4889, 27, 2058, 34, 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.
| \(\displaystyle \int \frac {\sec ^2(x) \left (2 \tan ^2(x)-\cos (2 x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (x)^2 \left (2 \tan (x)^2-\cos (2 x)\right )}{(\tan (x) \tan (2 x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {\sec ^2(x) \left (2 \tan ^2(x)-\cos (2 x)\right )}{(\sec (2 x)-1)^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (x)^2 \left (2 \tan (x)^2-\cos (2 x)\right )}{(\sec (2 x)-1)^{3/2}}dx\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \int -\frac {-2 \tan ^4(x)-3 \tan ^2(x)+1}{2 \sqrt {2} \left (\frac {\tan ^2(x)}{1-\tan ^2(x)}\right )^{3/2} \left (\tan ^2(x)+1\right )}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {-2 \tan ^4(x)-3 \tan ^2(x)+1}{\left (\frac {\tan ^2(x)}{1-\tan ^2(x)}\right )^{3/2} \left (\tan ^2(x)+1\right )}d\tan (x)}{2 \sqrt {2}}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle -\frac {\sqrt {\tan ^2(x)} \int \frac {\left (1-\tan ^2(x)\right )^{3/2} \left (-2 \tan ^4(x)-3 \tan ^2(x)+1\right )}{\tan ^2(x)^{3/2} \left (\tan ^2(x)+1\right )}d\tan (x)}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\) |
| \(\Big \downarrow \) 34 |
| \(\displaystyle -\frac {\tan (x) \int \frac {\cot ^3(x) \left (1-\tan ^2(x)\right )^{3/2} \left (-2 \tan ^4(x)-3 \tan ^2(x)+1\right )}{\tan ^2(x)+1}d\tan (x)}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {\tan (x) \int \left (\left (1-\tan ^2(x)\right )^{3/2} \cot ^3(x)-4 \left (1-\tan ^2(x)\right )^{3/2} \cot (x)+\frac {2 \tan (x) \left (1-\tan ^2(x)\right )^{3/2}}{\tan ^2(x)+1}\right )d\tan (x)}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\tan (x) \left (\frac {11}{2} \text {arctanh}\left (\sqrt {1-\tan ^2(x)}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1-\tan ^2(x)}}{\sqrt {2}}\right )-\frac {2}{3} \left (1-\tan ^2(x)\right )^{3/2}-\frac {3}{2} \sqrt {1-\tan ^2(x)}-\frac {1}{2} \left (1-\tan ^2(x)\right )^{3/2} \cot ^2(x)\right )}{2 \sqrt {2} \sqrt {\frac {\tan ^2(x)}{1-\tan ^2(x)}} \sqrt {1-\tan ^2(x)}}\) |
Input:
Int[(Sec[x]^2*(-Cos[2*x] + 2*Tan[x]^2))/(Tan[x]*Tan[2*x])^(3/2),x]
Output:
-1/2*(Tan[x]*((11*ArcTanh[Sqrt[1 - Tan[x]^2]])/2 - 4*Sqrt[2]*ArcTanh[Sqrt[ 1 - Tan[x]^2]/Sqrt[2]] - (3*Sqrt[1 - Tan[x]^2])/2 - (2*(1 - Tan[x]^2)^(3/2 ))/3 - (Cot[x]^2*(1 - Tan[x]^2)^(3/2))/2))/(Sqrt[2]*Sqrt[Tan[x]^2/(1 - Tan [x]^2)]*Sqrt[1 - Tan[x]^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
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. \(498\) vs. \(2(78)=156\).
Time = 3.97 (sec) , antiderivative size = 499, normalized size of antiderivative = 4.99
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (-9 \sin \left (2 x \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\right )-9 \sin \left (2 x \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}+4 \left (5+\cos \left (2 x \right )\right ) \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}\, \cot \left (2 x \right )\right )}{12 \left (1+\cos \left (2 x \right )\right ) \sqrt {2 \sec \left (2 x \right )-2}\, \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}-\frac {5 \sqrt {2}\, \csc \left (2 x \right ) \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\right ) \cos \left (2 x \right )+\sqrt {2}\, \cos \left (2 x \right ) \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\right )-\sqrt {2}\, \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}+4 \cos \left (2 x \right ) \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}\right )}{8 \sqrt {2 \sec \left (2 x \right )-2}\, \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}-\frac {\sqrt {2}\, \csc \left (2 x \right ) \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\right ) \cos \left (2 x \right )+\sqrt {2}\, \cos \left (2 x \right ) \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}+4 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}{2}\right ) \cos \left (2 x \right )-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}}{2 \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\right )-\sqrt {2}\, \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}-4 \cos \left (2 x \right ) \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}-4 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {2 \cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}{2}\right )\right )}{2 \sqrt {2 \sec \left (2 x \right )-2}\, \sqrt {-\frac {\cos \left (2 x \right )}{1+\cos \left (2 x \right )}}}\) | \(499\) |
Input:
int((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x,method=_RETU RNVERBOSE)
Output:
1/12*2^(1/2)/(1+cos(2*x))/(2*sec(2*x)-2)^(1/2)/(-1/(1+cos(2*x))*cos(2*x))^ (1/2)*(-9*sin(2*x)*2^(1/2)*arctan(1/2*2^(1/2)/(-1/(1+cos(2*x))*cos(2*x))^( 1/2))-9*sin(2*x)*2^(1/2)*(-2/(1+cos(2*x))*cos(2*x))^(1/2)+4*(5+cos(2*x))*( -1/(1+cos(2*x))*cos(2*x))^(1/2)*cot(2*x))-5/8*2^(1/2)*csc(2*x)*(2^(1/2)*ar ctan(1/2*2^(1/2)/(-1/(1+cos(2*x))*cos(2*x))^(1/2))*cos(2*x)+2^(1/2)*cos(2* x)*(-2/(1+cos(2*x))*cos(2*x))^(1/2)-2^(1/2)*arctan(1/2*2^(1/2)/(-1/(1+cos( 2*x))*cos(2*x))^(1/2))-2^(1/2)*(-2/(1+cos(2*x))*cos(2*x))^(1/2)+4*cos(2*x) *(-1/(1+cos(2*x))*cos(2*x))^(1/2))/(2*sec(2*x)-2)^(1/2)/(-1/(1+cos(2*x))*c os(2*x))^(1/2)-1/2*2^(1/2)*csc(2*x)*(3*2^(1/2)*arctan(1/2*2^(1/2)/(-1/(1+c os(2*x))*cos(2*x))^(1/2))*cos(2*x)+2^(1/2)*cos(2*x)*(-2/(1+cos(2*x))*cos(2 *x))^(1/2)+4*arctan(1/2*2^(1/2)*(-2/(1+cos(2*x))*cos(2*x))^(1/2))*cos(2*x) -3*2^(1/2)*arctan(1/2*2^(1/2)/(-1/(1+cos(2*x))*cos(2*x))^(1/2))-2^(1/2)*(- 2/(1+cos(2*x))*cos(2*x))^(1/2)-4*cos(2*x)*(-1/(1+cos(2*x))*cos(2*x))^(1/2) -4*arctan(1/2*2^(1/2)*(-2/(1+cos(2*x))*cos(2*x))^(1/2)))/(2*sec(2*x)-2)^(1 /2)/(-1/(1+cos(2*x))*cos(2*x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (78) = 156\).
Time = 0.10 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.71 \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=-\frac {24 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (8 \, \cos \left (x\right )^{5} - 6 \, \cos \left (x\right )^{3} + \cos \left (x\right )\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - {\left (32 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{\sin \left (x\right )}\right ) \sin \left (x\right ) - 33 \, {\left (\sqrt {2} \cos \left (x\right )^{5} - \sqrt {2} \cos \left (x\right )^{3}\right )} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (2 \, {\left (3 \, \sqrt {2} - 4\right )} \cos \left (x\right )^{3} - {\left (3 \, \sqrt {2} - 4\right )} \cos \left (x\right )\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} + {\left (3 \, {\left (2 \, \sqrt {2} - 3\right )} \cos \left (x\right )^{2} - 2 \, \sqrt {2} + 3\right )} \sin \left (x\right )\right )}}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 2 \, \sqrt {2} {\left (22 \, \cos \left (x\right )^{6} - 47 \, \cos \left (x\right )^{4} + 26 \, \cos \left (x\right )^{2} - 4\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{2 \, \cos \left (x\right )^{2} - 1}} - 44 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )}{48 \, {\left (\cos \left (x\right )^{5} - \cos \left (x\right )^{3}\right )} \sin \left (x\right )} \] Input:
integrate((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x, algor ithm="fricas")
Output:
-1/48*(24*(cos(x)^5 - cos(x)^3)*log(-(4*sqrt(2)*(8*cos(x)^5 - 6*cos(x)^3 + cos(x))*sqrt(-(cos(x)^2 - 1)/(2*cos(x)^2 - 1)) - (32*cos(x)^4 - 16*cos(x) ^2 + 1)*sin(x))/sin(x))*sin(x) - 33*(sqrt(2)*cos(x)^5 - sqrt(2)*cos(x)^3)* log(4*(sqrt(2)*(2*(3*sqrt(2) - 4)*cos(x)^3 - (3*sqrt(2) - 4)*cos(x))*sqrt( -(cos(x)^2 - 1)/(2*cos(x)^2 - 1)) + (3*(2*sqrt(2) - 3)*cos(x)^2 - 2*sqrt(2 ) + 3)*sin(x))/((cos(x)^2 - 1)*sin(x)))*sin(x) - 2*sqrt(2)*(22*cos(x)^6 - 47*cos(x)^4 + 26*cos(x)^2 - 4)*sqrt(-(cos(x)^2 - 1)/(2*cos(x)^2 - 1)) - 44 *(cos(x)^5 - cos(x)^3)*sin(x))/((cos(x)^5 - cos(x)^3)*sin(x))
Timed out. \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((-cos(2*x)+2*tan(x)**2)/cos(x)**2/(tan(x)*tan(2*x))**(3/2),x)
Output:
Timed out
\[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=\int { \frac {2 \, \tan \left (x\right )^{2} - \cos \left (2 \, x\right )}{\left (\tan \left (2 \, x\right ) \tan \left (x\right )\right )^{\frac {3}{2}} \cos \left (x\right )^{2}} \,d x } \] Input:
integrate((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x, algor ithm="maxima")
Output:
integrate((2*tan(x)^2 - cos(2*x))/((tan(2*x)*tan(x))^(3/2)*cos(x)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (78) = 156\).
Time = 0.22 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.93 \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=-\frac {\sqrt {2} {\left (2 \, {\left (-\tan \left (x\right )^{2} + 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {-\tan \left (x\right )^{2} + 1}\right )}}{12 \, \mathrm {sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (x\right )\right )} + \frac {11 \, \sqrt {2} \log \left (\sqrt {-\tan \left (x\right )^{2} + 1} + 1\right )}{16 \, \mathrm {sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (x\right )\right )} - \frac {11 \, \sqrt {2} \log \left (-\sqrt {-\tan \left (x\right )^{2} + 1} + 1\right )}{16 \, \mathrm {sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (x\right )\right )} + \frac {\log \left (\frac {\sqrt {2} - \sqrt {-\tan \left (x\right )^{2} + 1}}{\sqrt {2} + \sqrt {-\tan \left (x\right )^{2} + 1}}\right )}{\mathrm {sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (x\right )\right )} - \frac {\sqrt {2} \sqrt {-\tan \left (x\right )^{2} + 1}}{8 \, \mathrm {sgn}\left (\tan \left (x\right )^{2} - 1\right ) \mathrm {sgn}\left (\tan \left (x\right )\right ) \tan \left (x\right )^{2}} \] Input:
integrate((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x, algor ithm="giac")
Output:
-1/12*sqrt(2)*(2*(-tan(x)^2 + 1)^(3/2) + 3*sqrt(-tan(x)^2 + 1))/(sgn(tan(x )^2 - 1)*sgn(tan(x))) + 11/16*sqrt(2)*log(sqrt(-tan(x)^2 + 1) + 1)/(sgn(ta n(x)^2 - 1)*sgn(tan(x))) - 11/16*sqrt(2)*log(-sqrt(-tan(x)^2 + 1) + 1)/(sg n(tan(x)^2 - 1)*sgn(tan(x))) + log((sqrt(2) - sqrt(-tan(x)^2 + 1))/(sqrt(2 ) + sqrt(-tan(x)^2 + 1)))/(sgn(tan(x)^2 - 1)*sgn(tan(x))) - 1/8*sqrt(2)*sq rt(-tan(x)^2 + 1)/(sgn(tan(x)^2 - 1)*sgn(tan(x))*tan(x)^2)
Timed out. \[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=-\int \frac {\cos \left (2\,x\right )-2\,{\mathrm {tan}\left (x\right )}^2}{{\cos \left (x\right )}^2\,{\left (\mathrm {tan}\left (2\,x\right )\,\mathrm {tan}\left (x\right )\right )}^{3/2}} \,d x \] Input:
int(-(cos(2*x) - 2*tan(x)^2)/(cos(x)^2*(tan(2*x)*tan(x))^(3/2)),x)
Output:
-int((cos(2*x) - 2*tan(x)^2)/(cos(x)^2*(tan(2*x)*tan(x))^(3/2)), x)
\[ \int \frac {\sec ^2(x) \left (-\cos (2 x)+2 \tan ^2(x)\right )}{(\tan (x) \tan (2 x))^{3/2}} \, dx=-\left (\int \frac {\sqrt {\tan \left (x \right )}\, \sqrt {\tan \left (2 x \right )}\, \cos \left (2 x \right )}{\cos \left (x \right )^{2} \tan \left (2 x \right )^{2} \tan \left (x \right )^{2}}d x \right )+2 \left (\int \frac {\sqrt {\tan \left (x \right )}\, \sqrt {\tan \left (2 x \right )}}{\cos \left (x \right )^{2} \tan \left (2 x \right )^{2}}d x \right ) \] Input:
int((-cos(2*x)+2*tan(x)^2)/cos(x)^2/(tan(x)*tan(2*x))^(3/2),x)
Output:
- int((sqrt(tan(x))*sqrt(tan(2*x))*cos(2*x))/(cos(x)**2*tan(2*x)**2*tan(x )**2),x) + 2*int((sqrt(tan(x))*sqrt(tan(2*x)))/(cos(x)**2*tan(2*x)**2),x)