Integrand size = 23, antiderivative size = 106 \[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \]
-arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f+arctanh((a+b*tan(f*x+ e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/f-b/a/(a-b)/f/(a+b*tan(f*x+e)^2)^(1/2 )
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\frac {-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \tan ^2(e+f x)}{a}\right )}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \]
(-(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[e + f*x]^2)/(a - b)]) + (a - b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Tan[e + f*x]^2)/a])/(a*(a - b )*f*Sqrt[a + b*Tan[e + f*x]^2])
Time = 0.33 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4153, 354, 96, 174, 73, 221}
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.
| \(\displaystyle \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x) \left (a+b \tan (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle \frac {\int \frac {\cot (e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\cot (e+f x)}{\left (\tan ^2(e+f x)+1\right ) \left (b \tan ^2(e+f x)+a\right )^{3/2}}d\tan ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 96 |
| \(\displaystyle \frac {\frac {\int \frac {\cot (e+f x) \left (-b \tan ^2(e+f x)+a-b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}-\frac {2 b}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {\frac {(a-b) \int \frac {\cot (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)-a \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan ^2(e+f x)}{a (a-b)}-\frac {2 b}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {2 (a-b) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {2 a \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{a (a-b)}-\frac {2 b}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {2 a \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (a-b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a-b)}-\frac {2 b}{a (a-b) \sqrt {a+b \tan ^2(e+f x)}}}{2 f}\) |
(((-2*(a - b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] + (2*a* ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/(a*(a - b)) - (2*b)/(a*(a - b)*Sqrt[a + b*Tan[e + f*x]^2]))/(2*f)
3.4.36.3.1 Defintions of rubi rules used
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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && 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. \(16006\) vs. \(2(92)=184\).
Time = 1.23 (sec) , antiderivative size = 16007, normalized size of antiderivative = 151.01
Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (92) = 184\).
Time = 0.32 (sec) , antiderivative size = 920, normalized size of antiderivative = 8.68 \[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\left [-\frac {{\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) + 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) + {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) - {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {{\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) + {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) - {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f}\right ] \]
[-1/2*((a^2*b*tan(f*x + e)^2 + a^3)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2* sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(a)*l og((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a) + 2*a)/tan(f*x + e)^2) + 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b ^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), 1/2*(2*(a^2 *b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan(f*x + e)^2 + a)*s qrt(-a + b)/(a - b)) + (a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*ta n(f*x + e)^2)*sqrt(a)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a) *sqrt(a) + 2*a)/tan(f*x + e)^2) - 2*(a^2*b - a*b^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a ^3*b^2)*f), 1/2*(2*(a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2*a*b^2 + b^3)*tan(f* x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a)/a) - (a^2*b* tan(f*x + e)^2 + a^3)*sqrt(a - b)*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1)) - 2*(a^2*b - a*b ^2)*sqrt(b*tan(f*x + e)^2 + a))/((a^4*b - 2*a^3*b^2 + a^2*b^3)*f*tan(f*x + e)^2 + (a^5 - 2*a^4*b + a^3*b^2)*f), ((a^3 - 2*a^2*b + a*b^2 + (a^2*b - 2 *a*b^2 + b^3)*tan(f*x + e)^2)*sqrt(-a)*arctan(sqrt(b*tan(f*x + e)^2 + a)*s qrt(-a)/a) + (a^2*b*tan(f*x + e)^2 + a^3)*sqrt(-a + b)*arctan(-sqrt(b*tan( f*x + e)^2 + a)*sqrt(-a + b)/(a - b)) - (a^2*b - a*b^2)*sqrt(b*tan(f*x ...
\[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\cot {\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cot \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
Time = 11.85 (sec) , antiderivative size = 1922, normalized size of antiderivative = 18.13 \[ \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
b/(f*(a + b*tan(e + f*x)^2)^(1/2)*(a*b - a^2)) - atanh((2*a^2*b^8*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30* a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) - (12*a^3* b^7*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b ^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2) ) + (30*a^4*b^6*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^ 2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6* a^6*b^3*f^2)) - (38*a^5*b^5*f^2*(a + b*tan(e + f*x)^2)^(1/2))/((a^3)^(1/2) *(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f^2 + 24*a^5* b^4*f^2 - 6*a^6*b^3*f^2)) + (24*a^6*b^4*f^2*(a + b*tan(e + f*x)^2)^(1/2))/ ((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 38*a^4*b^5*f ^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)) - (6*a^7*b^3*f^2*(a + b*tan(e + f*x) ^2)^(1/2))/((a^3)^(1/2)*(2*a*b^8*f^2 - 12*a^2*b^7*f^2 + 30*a^3*b^6*f^2 - 3 8*a^4*b^5*f^2 + 24*a^5*b^4*f^2 - 6*a^6*b^3*f^2)))/(f*(a^3)^(1/2)) + (atan( (((((a + b*tan(e + f*x)^2)^(1/2)*(2*a^3*b^7*f^3 - 10*a^4*b^6*f^3 + 22*a^5* b^5*f^3 - 26*a^6*b^4*f^3 + 16*a^7*b^3*f^3 - 4*a^8*b^2*f^3))/2 + (((a - b)^ 3)^(1/2)*(12*a^5*b^7*f^4 - 2*a^4*b^8*f^4 - 28*a^6*b^6*f^4 + 32*a^7*b^5*f^4 - 18*a^8*b^4*f^4 + 4*a^9*b^3*f^4 + ((a + b*tan(e + f*x)^2)^(1/2)*((a - b) ^3)^(1/2)*(8*a^5*b^8*f^5 - 56*a^6*b^7*f^5 + 160*a^7*b^6*f^5 - 240*a^8*b^5* f^5 + 200*a^9*b^4*f^5 - 88*a^10*b^3*f^5 + 16*a^11*b^2*f^5))/(4*f*(a - b...