Integrand size = 17, antiderivative size = 103 \[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{4 \sqrt {b}}+\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{4} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)} \]
1/4*(a+2*b)*arctanh(b^(1/2)*tan(x)^2/(a+b*tan(x)^4)^(1/2))/b^(1/2)+1/2*arc tanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))*(a+b)^(1/2)-1/4*(a+b *tan(x)^4)^(1/2)*(2-tan(x)^2)
Time = 3.85 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.41 \[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{4} \left (2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+2 \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {\left (-2+\tan ^2(x)\right ) \left (a+b \tan ^4(x)\right )+\frac {a^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan ^4(x)}{a}}}{\sqrt {b}}}{\sqrt {a+b \tan ^4(x)}}\right ) \]
(2*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2*Sqrt[a + b ]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + ((-2 + Ta n[x]^2)*(a + b*Tan[x]^4) + (a^(3/2)*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sq rt[1 + (b*Tan[x]^4)/a])/Sqrt[b])/Sqrt[a + b*Tan[x]^4])/4
Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {3042, 4153, 1579, 591, 25, 719, 224, 219, 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.
| \(\displaystyle \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (x)^3 \sqrt {a+b \tan (x)^4}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int \frac {\tan ^3(x) \sqrt {a+b \tan ^4(x)}}{\tan ^2(x)+1}d\tan (x)\) |
| \(\Big \downarrow \) 1579 |
| \(\displaystyle \frac {1}{2} \int \frac {\tan ^2(x) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\) |
| \(\Big \downarrow \) 591 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int -\frac {a-(a+2 b) \tan ^2(x)}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {a-(a+2 b) \tan ^2(x)}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left ((a+2 b) \int \frac {1}{\sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-2 (a+b) \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left ((a+2 b) \int \frac {1}{1-b \tan ^4(x)}d\frac {\tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}-2 (a+b) \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{\sqrt {b}}-2 (a+b) \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b) \int \frac {1}{-\tan ^4(x)+a+b}d\frac {a-b \tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}+\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{\sqrt {b}}\right )-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {(a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{\sqrt {b}}+2 \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )\right )-\frac {1}{2} \left (2-\tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
((((a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/Sqrt[b] + 2 *Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]) /2 - ((2 - Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/2)/2
3.4.89.3.1 Defintions of rubi rules used
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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 2*p + 1)*x)/ (d^2*(n + 2*p + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 2*p + 1)*(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*Simp[a*c*d*n + (b*c^2*(2*p + 1) + a*d^2*(n + 2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && LeQ[-1, n, 0] && !ILtQ[n + 2*p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
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. \(180\) vs. \(2(83)=166\).
Time = 0.18 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.76
| method | result | size |
| derivativedivides | \(\frac {\sqrt {a +b \tan \left (x \right )^{4}}\, \tan \left (x \right )^{2}}{4}+\frac {a \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{4 \sqrt {b}}-\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\) | \(181\) |
| default | \(\frac {\sqrt {a +b \tan \left (x \right )^{4}}\, \tan \left (x \right )^{2}}{4}+\frac {a \ln \left (\sqrt {b}\, \tan \left (x \right )^{2}+\sqrt {a +b \tan \left (x \right )^{4}}\right )}{4 \sqrt {b}}-\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\) | \(181\) |
1/4*(a+b*tan(x)^4)^(1/2)*tan(x)^2+1/4*a/b^(1/2)*ln(b^(1/2)*tan(x)^2+(a+b*t an(x)^4)^(1/2))-1/2*(b*(1+tan(x)^2)^2-2*b*(1+tan(x)^2)+a+b)^(1/2)+1/2*b^(1 /2)*ln((b*(1+tan(x)^2)-b)/b^(1/2)+(b*(1+tan(x)^2)^2-2*b*(1+tan(x)^2)+a+b)^ (1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+tan(x)^2)+2*(a+b)^(1/2)*(b*(1+ta n(x)^2)^2-2*b*(1+tan(x)^2)+a+b)^(1/2))/(1+tan(x)^2))
Time = 0.41 (sec) , antiderivative size = 555, normalized size of antiderivative = 5.39 \[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\left [\frac {{\left (a + 2 \, b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + 2 \, \sqrt {a + b} 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 (b \tan \left (x\right )^{2} - 2 \, b\right )}}{8 \, b}, -\frac {{\left (a + 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) - \sqrt {a + b} 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 ) - \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - 2 \, b\right )}}{4 \, b}, \frac {4 \, \sqrt {-a - b} 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 ) + {\left (a + 2 \, b\right )} \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - 2 \, b\right )}}{8 \, b}, \frac {2 \, \sqrt {-a - b} 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 ) - {\left (a + 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - 2 \, b\right )}}{4 \, b}\right ] \]
[1/8*((a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b) *tan(x)^2 - a) + 2*sqrt(a + b)*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)*(b*tan(x)^2 - 2*b))/b , -1/4*((a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^ 2)) - sqrt(a + b)*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)) - sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b, 1/8*(4*sqrt(- a - b)*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)) + (a + 2*b)*sqrt(b)*log(-2*b*tan(x)^4 - 2*sqr t(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 2*sqrt(b*tan(x)^4 + a)*(b*tan(x) ^2 - 2*b))/b, 1/4*(2*sqrt(-a - b)*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)) - (a + 2*b)*sqrt(- b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b))/b]
\[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\int \sqrt {a + b \tan ^{4}{\left (x \right )}} \tan ^{3}{\left (x \right )}\, dx \]
\[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\int { \sqrt {b \tan \left (x\right )^{4} + a} \tan \left (x\right )^{3} \,d x } \]
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.17 \[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{4} \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (\tan \left (x\right )^{2} - 2\right )} \]
Timed out. \[ \int \tan ^3(x) \sqrt {a+b \tan ^4(x)} \, dx=\int {\mathrm {tan}\left (x\right )}^3\,\sqrt {b\,{\mathrm {tan}\left (x\right )}^4+a} \,d x \]