Integrand size = 15, antiderivative size = 103 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {b \left (1-\tan ^2(x)\right )}{2 a (a+b) \sqrt {a+b \tan ^4(x)}} \] Output:
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*arctanh((a+b*tan(x)^4)^(1/2)/a^(1/2))/a^(3/2)+1/2*b*(1-tan(x)^2)/a/(a+b )/(a+b*tan(x)^4)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.05 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \tan ^4(x)}{a}\right )}{a \sqrt {a+b \tan ^4(x)}}-\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right ) \] Input:
Integrate[Cot[x]/(a + b*Tan[x]^4)^(3/2),x]
Output:
(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3/2 ) + Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[x]^4)/a]/(a*Sqrt[a + b*Tan[ x]^4]) - (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2
Time = 0.39 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4153, 1579, 617, 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.
\(\displaystyle \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (x) \left (a+b \tan (x)^4\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int \frac {\cot (x)}{\left (\tan ^2(x)+1\right ) \left (a+b \tan ^4(x)\right )^{3/2}}d\tan (x)\) |
\(\Big \downarrow \) 1579 |
\(\displaystyle \frac {1}{2} \int \frac {\cot (x)}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{3/2}}d\tan ^2(x)\) |
\(\Big \downarrow \) 617 |
\(\displaystyle \frac {1}{2} \int \left (\frac {\cot (x)}{\left (b \tan ^4(x)+a\right )^{3/2}}+\frac {1}{\left (-\tan ^2(x)-1\right ) \left (b \tan ^4(x)+a\right )^{3/2}}\right )d\tan ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{3/2}}+\frac {1}{a \sqrt {a+b \tan ^4(x)}}-\frac {a+b \tan ^2(x)}{a (a+b) \sqrt {a+b \tan ^4(x)}}\right )\) |
Input:
Int[Cot[x]/(a + b*Tan[x]^4)^(3/2),x]
Output:
(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(3/2 ) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/a^(3/2) + 1/(a*Sqrt[a + b*Tan[x] ^4]) - (a + b*Tan[x]^2)/(a*(a + b)*Sqrt[a + b*Tan[x]^4]))/2
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
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]))
\[\int \frac {\cot \left (x \right )}{\left (a +b \tan \left (x \right )^{4}\right )^{\frac {3}{2}}}d x\]
Input:
int(cot(x)/(a+b*tan(x)^4)^(3/2),x)
Output:
int(cot(x)/(a+b*tan(x)^4)^(3/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (83) = 166\).
Time = 0.33 (sec) , antiderivative size = 954, normalized size of antiderivative = 9.26 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cot(x)/(a+b*tan(x)^4)^(3/2),x, algorithm="fricas")
Output:
[1/4*((a^2*b*tan(x)^4 + a^3)*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)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(a)*log(-(b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*s qrt(a) + 2*a)/tan(x)^4) + 2*sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3 )*tan(x)^4), 1/4*(2*((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a* b^2)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a)/a) + (a^2*b*tan(x)^4 + a^3)*sqrt(a + b)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*t an(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^2*b + a*b^2 - (a^2*b + a*b^2)*tan( x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4), 1/4*(2*(a^2*b*tan(x)^4 + a^3)*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)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 + 2*a^2*b + a*b^2)*sqrt(a)*log(-(b*tan(x)^ 4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) + 2*sqrt(b*tan(x)^4 + a)*(a^2*b + a*b^2 - (a^2*b + a*b^2)*tan(x)^2))/(a^5 + 2*a^4*b + a^3*b^2 + (a^4*b + 2*a^3*b^2 + a^2*b^3)*tan(x)^4), 1/2*((a^2*b*tan(x)^4 + a^3)*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)) + ((a^2*b + 2*a*b^2 + b^3)*tan(x)^4 + a^3 +...
\[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(cot(x)/(a+b*tan(x)**4)**(3/2),x)
Output:
Integral(cot(x)/(a + b*tan(x)**4)**(3/2), x)
Exception generated. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(cot(x)/(a+b*tan(x)^4)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(x)/(a+b*tan(x)^4)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{3/2}} \,d x \] Input:
int(cot(x)/(a + b*tan(x)^4)^(3/2),x)
Output:
int(cot(x)/(a + b*tan(x)^4)^(3/2), x)
\[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\tan \left (x \right )^{4} b +a}\, \cot \left (x \right )}{\tan \left (x \right )^{8} b^{2}+2 \tan \left (x \right )^{4} a b +a^{2}}d x \] Input:
int(cot(x)/(a+b*tan(x)^4)^(3/2),x)
Output:
int((sqrt(tan(x)**4*b + a)*cot(x))/(tan(x)**8*b**2 + 2*tan(x)**4*a*b + a** 2),x)