Integrand size = 15, antiderivative size = 155 \[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\frac {1}{4} \sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )+\frac {1}{2} a \sqrt {a+b \tan ^4(x)}-\frac {1}{4} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)} \]
-1/2*a^(3/2)*arctanh((a+b*tan(x)^4)^(1/2)/a^(1/2))+1/2*(a+b)^(3/2)*arctanh ((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))+1/4*(3*a+2*b)*arctanh(b^ (1/2)*tan(x)^2/(a+b*tan(x)^4)^(1/2))*b^(1/2)+1/2*a*(a+b*tan(x)^4)^(1/2)-1/ 4*(a+b*tan(x)^4)^(1/2)*(2*a+2*b-b*tan(x)^2)
Time = 3.25 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.23 \[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\frac {1}{4} \left (2 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+2 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )-2 b \sqrt {a+b \tan ^4(x)}+b \tan ^2(x) \sqrt {a+b \tan ^4(x)}+\frac {\sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \tan ^4(x)}}{\sqrt {1+\frac {b \tan ^4(x)}{a}}}\right ) \]
(2*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2*(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - 2*a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]] - 2*b*Sqrt[a + b*Tan[x]^4 ] + b*Tan[x]^2*Sqrt[a + b*Tan[x]^4] + (Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Ta n[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]^4])/Sqrt[1 + (b*Tan[x]^4)/a])/4
Time = 0.44 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4153, 1579, 606, 243, 60, 73, 221, 682, 27, 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.
| \(\displaystyle \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \tan (x)^4\right )^{3/2}}{\tan (x)}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \int \frac {\cot (x) \left (a+b \tan ^4(x)\right )^{3/2}}{\tan ^2(x)+1}d\tan (x)\) |
| \(\Big \downarrow \) 1579 |
| \(\displaystyle \frac {1}{2} \int \frac {\cot (x) \left (b \tan ^4(x)+a\right )^{3/2}}{\tan ^2(x)+1}d\tan ^2(x)\) |
| \(\Big \downarrow \) 606 |
| \(\displaystyle \frac {1}{2} \left (a \int \cot (x) \sqrt {b \tan ^4(x)+a}d\tan ^2(x)-\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} a \int \cot (x) \sqrt {b \tan ^4(x)+a}d\tan ^4(x)-\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} a \left (a \int \frac {\cot (x)}{\sqrt {b \tan ^4(x)+a}}d\tan ^4(x)+2 \sqrt {a+b \tan ^4(x)}\right )-\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} a \left (\frac {2 a \int \frac {1}{\frac {\sqrt {b \tan ^4(x)+a}}{b}-\frac {a}{b}}d\sqrt {b \tan ^4(x)+a}}{b}+2 \sqrt {a+b \tan ^4(x)}\right )-\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\int \frac {\left (a-b \tan ^2(x)\right ) \sqrt {b \tan ^4(x)+a}}{\tan ^2(x)+1}d\tan ^2(x)\right )\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {b \left (a (2 a+b)-b (3 a+2 b) \tan ^2(x)\right )}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)}{2 b}+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {a (2 a+b)-b (3 a+2 b) \tan ^2(x)}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (b (3 a+2 b) \int \frac {1}{\sqrt {b \tan ^4(x)+a}}d\tan ^2(x)-2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (b (3 a+2 b) \int \frac {1}{1-b \tan ^4(x)}d\frac {\tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}-2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-2 (a+b)^2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan ^2(x)\right )+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (2 (a+b)^2 \int \frac {1}{-\tan ^4(x)+a+b}d\frac {a-b \tan ^2(x)}{\sqrt {b \tan ^4(x)+a}}+\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\right )+\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )-\frac {1}{2} \sqrt {a+b \tan ^4(x)} \left (2 (a+b)-b \tan ^2(x)\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} a \left (2 \sqrt {a+b \tan ^4(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right )+\frac {1}{2} \left (2 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\sqrt {b} (3 a+2 b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )\right )-\frac {1}{2} \left (2 (a+b)-b \tan ^2(x)\right ) \sqrt {a+b \tan ^4(x)}\right )\) |
((Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + 2 *(a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4]) ])/2 - ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/2 + (a*(-2*Sqrt[a]* ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]] + 2*Sqrt[a + b*Tan[x]^4]))/2)/2
3.4.95.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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_), x_Symbol] : > Simp[a/c Int[(c + d*x)^(n + 1)*((a + b*x^2)^(p - 1)/x), x], x] - Simp[1 /c Int[(c + d*x)^n*(a*d - b*c*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && ILtQ[n, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]))
\[\int \cot \left (x \right ) \left (a +b \tan \left (x \right )^{4}\right )^{\frac {3}{2}}d x\]
Time = 21.20 (sec) , antiderivative size = 1269, normalized size of antiderivative = 8.19 \[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b) *tan(x)^2 + a) + 1/4*(a + b)^(3/2)*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)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan (x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), -1/4*(3*a + 2*b)*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b *tan(x)^2)) + 1/4*(a + b)^(3/2)*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)/(t an(x)^4 + 2*tan(x)^2 + 1)) + 1/4*a^(3/2)*log((b*tan(x)^4 - 2*sqrt(b*tan(x) ^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt(-a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + 1/8*(3*a + 2*b)*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + a) + 1/4*(a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sq rt(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)) + 1/4*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*sqrt( -a)*a*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) - 1/4*(3*a + 2*b)*sqrt(-b)*a rctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*(a + b)^(3/2)*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)) + 1/4*sqr t(b*tan(x)^4 + a)*(b*tan(x)^2 - 2*b), 1/2*(a + b)*sqrt(-a - b)*arctan(s...
\[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int \left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {3}{2}} \cot {\left (x \right )}\, dx \]
\[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int { {\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {3}{2}} \cot \left (x\right ) \,d x } \]
Exception generated. \[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \cot (x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\int \mathrm {cot}\left (x\right )\,{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{3/2} \,d x \]