Integrand size = 15, antiderivative size = 74 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{3/2}}+\frac {a+b \tan ^2(x)}{2 a (a+b) \sqrt {a+b \tan ^4(x)}} \]
-1/2*arctanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))/(a+b)^(3/2)+ 1/2*(a+b*tan(x)^2)/a/(a+b)/(a+b*tan(x)^4)^(1/2)
Time = 0.32 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}+\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right ) \]
(-(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3 /2)) + (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 4153, 1577, 496, 25, 27, 488, 219}
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.
\(\displaystyle \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (x)}{\left (a+b \tan (x)^4\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int \frac {\tan (x)}{\left (\tan ^2(x)+1\right ) \left (a+b \tan ^4(x)\right )^{3/2}}d\tan (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{3/2}}d\tan ^2(x)\) |
\(\Big \downarrow \) 496 |
\(\displaystyle \frac {1}{2} \left (\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\int -\frac {a}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{a (a+b)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {a}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{a (a+b)}+\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{a+b}+\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{2} \left (\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\int \frac {1}{-\tan ^4(x)+a+b}d\frac {a-b \tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}}{a+b}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}\right )\) |
(-(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3 /2)) + (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2
3.5.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(62)=124\).
Time = 0.08 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.35
method | result | size |
derivativedivides | \(\frac {b \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}+\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(248\) |
default | \(\frac {b \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right ) \left (\sqrt {-a b}-b \right ) \sqrt {a +b}}+\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}+b \right ) a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{4 \left (\sqrt {-a b}-b \right ) a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(248\) |
1/2*b/((-a*b)^(1/2)+b)/((-a*b)^(1/2)-b)/(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+tan (x)^2)+2*(a+b)^(1/2)*(b*(1+tan(x)^2)^2-2*b*(1+tan(x)^2)+a+b)^(1/2))/(1+tan (x)^2))+1/4/((-a*b)^(1/2)+b)/a/(tan(x)^2-(-a*b)^(1/2)/b)*(b*(tan(x)^2-(-a* b)^(1/2)/b)^2+2*(-a*b)^(1/2)*(tan(x)^2-(-a*b)^(1/2)/b))^(1/2)-1/4/((-a*b)^ (1/2)-b)/a/(tan(x)^2+(-a*b)^(1/2)/b)*(b*(tan(x)^2+(-a*b)^(1/2)/b)^2-2*(-a* b)^(1/2)*(tan(x)^2+(-a*b)^(1/2)/b))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (64) = 128\).
Time = 0.43 (sec) , antiderivative size = 319, normalized size of antiderivative = 4.31 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\left [\frac {{\left (a b \tan \left (x\right )^{4} + a^{2}\right )} \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left ({\left (a b + b^{2}\right )} \tan \left (x\right )^{2} + a^{2} + a b\right )}}{4 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )}}, -\frac {{\left (a b \tan \left (x\right )^{4} + a^{2}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) - \sqrt {b \tan \left (x\right )^{4} + a} {\left ({\left (a b + b^{2}\right )} \tan \left (x\right )^{2} + a^{2} + a b\right )}}{2 \, {\left ({\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )}}\right ] \]
[1/4*((a*b*tan(x)^4 + a^2)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b *tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 2*sqrt(b*tan(x)^4 + a)*((a*b + b^2)*ta n(x)^2 + a^2 + a*b))/((a^3*b + 2*a^2*b^2 + a*b^3)*tan(x)^4 + a^4 + 2*a^3*b + a^2*b^2), -1/2*((a*b*tan(x)^4 + a^2)*sqrt(-a - b)*arctan(sqrt(b*tan(x)^ 4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) - sqrt(b*tan(x)^4 + a)*((a*b + b^2)*tan(x)^2 + a^2 + a*b))/((a^3*b + 2*a^2* b^2 + a*b^3)*tan(x)^4 + a^4 + 2*a^3*b + a^2*b^2)]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.61 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\frac {\frac {{\left (a b + b^{2}\right )} \tan \left (x\right )^{2}}{a^{3} + 2 \, a^{2} b + a b^{2}} + \frac {a^{2} + a b}{a^{3} + 2 \, a^{2} b + a b^{2}}}{2 \, \sqrt {b \tan \left (x\right )^{4} + a}} - \frac {\arctan \left (\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{{\left (a + b\right )} \sqrt {-a - b}} \]
1/2*((a*b + b^2)*tan(x)^2/(a^3 + 2*a^2*b + a*b^2) + (a^2 + a*b)/(a^3 + 2*a ^2*b + a*b^2))/sqrt(b*tan(x)^4 + a) - arctan((sqrt(b)*tan(x)^2 - sqrt(b*ta n(x)^4 + a) + sqrt(b))/sqrt(-a - b))/((a + b)*sqrt(-a - b))
Timed out. \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int \frac {\mathrm {tan}\left (x\right )}{{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{3/2}} \,d x \]