Integrand size = 25, antiderivative size = 117 \[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\frac {\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}+\frac {(3 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 a f}-\frac {\cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{3 f} \]
arctan((a-b)^(1/2)*tan(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2))*(a-b)^(1/2)/f+1/3* (3*a-b)*cot(f*x+e)*(a+b*tan(f*x+e)^2)^(1/2)/a/f-1/3*cot(f*x+e)^3*(a+b*tan( f*x+e)^2)^(1/2)/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 9.67 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.06 \[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)} \left (1+\frac {b \tan ^2(e+f x)}{a}\right ) \left (\frac {\sec ^2(e+f x) \left (\arcsin \left (\sqrt {\frac {(a-b) \sin ^2(e+f x)}{a}}\right ) \sqrt {\frac {(a-b) \sin ^2(e+f x)}{a}}+\sqrt {\cos ^2(e+f x)+\frac {b \sin ^2(e+f x)}{a}}\right ) \left (a-2 b \tan ^2(e+f x)\right )}{\sqrt {\cos ^2(e+f x)+\frac {b \sin ^2(e+f x)}{a}} \left (a+b \tan ^2(e+f x)\right )}-\frac {4 (a-b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {3}{2},\frac {(a-b) \sin ^2(e+f x)}{a}\right ) \sin ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )}{a^2}\right )}{3 f} \]
-1/3*(Cos[e + f*x]^2*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2]*(1 + (b*Tan [e + f*x]^2)/a)*((Sec[e + f*x]^2*(ArcSin[Sqrt[((a - b)*Sin[e + f*x]^2)/a]] *Sqrt[((a - b)*Sin[e + f*x]^2)/a] + Sqrt[Cos[e + f*x]^2 + (b*Sin[e + f*x]^ 2)/a])*(a - 2*b*Tan[e + f*x]^2))/(Sqrt[Cos[e + f*x]^2 + (b*Sin[e + f*x]^2) /a]*(a + b*Tan[e + f*x]^2)) - (4*(a - b)*Hypergeometric2F1[2, 2, 3/2, ((a - b)*Sin[e + f*x]^2)/a]*Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2))/a^2))/f
Time = 0.33 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4153, 377, 25, 445, 27, 291, 216}
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.
\(\displaystyle \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \tan (e+f x)^2}}{\tan (e+f x)^4}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \frac {\int \frac {\cot ^4(e+f x) \sqrt {b \tan ^2(e+f x)+a}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 377 |
\(\displaystyle \frac {\frac {1}{3} \int -\frac {\cot ^2(e+f x) \left (2 b \tan ^2(e+f x)+3 a-b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\frac {1}{3} \int \frac {\cot ^2(e+f x) \left (2 b \tan ^2(e+f x)+3 a-b\right )}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
\(\Big \downarrow \) 445 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {\int \frac {3 a (a-b)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{a}+\frac {(3 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{3} \left (3 (a-b) \int \frac {1}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)+\frac {(3 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {1}{3} \left (3 (a-b) \int \frac {1}{1-\frac {(b-a) \tan ^2(e+f x)}{b \tan ^2(e+f x)+a}}d\frac {\tan (e+f x)}{\sqrt {b \tan ^2(e+f x)+a}}+\frac {(3 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{3} \left (3 \sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )+\frac {(3 a-b) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}\right )-\frac {1}{3} \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{f}\) |
(-1/3*(Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2]) + (3*Sqrt[a - b]*ArcTan[ (Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]] + ((3*a - b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/3)/f
3.4.4.3.1 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]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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. \(355\) vs. \(2(103)=206\).
Time = 4.76 (sec) , antiderivative size = 356, normalized size of antiderivative = 3.04
method | result | size |
default | \(-\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}\, \left (-3 \sqrt {a -b}\, \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) a \cot \left (f x +e \right )^{2}+\sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, b \cot \left (f x +e \right )+3 \sqrt {a -b}\, \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{\sqrt {a -b}}\right ) a \cot \left (f x +e \right ) \csc \left (f x +e \right )+4 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, a \cot \left (f x +e \right )^{3}-3 \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, a \cot \left (f x +e \right ) \csc \left (f x +e \right )^{2}\right )}{3 f a \sqrt {\frac {a \cos \left (f x +e \right )^{2}-b \cos \left (f x +e \right )^{2}+b}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(356\) |
-1/3/f/a*(a+b*tan(f*x+e)^2)^(1/2)/((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos( f*x+e)+1)^2)^(1/2)*(-3*(a-b)^(1/2)*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b *sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*cot(f*x+ e)^2+((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*b*cot(f*x+ e)+3*(a-b)^(1/2)*arctan(1/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(co s(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)+csc(f*x+e)))*a*cot(f*x+e)*csc(f*x+e)+4*(( a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a*cot(f*x+e)^3-3* ((a*cos(f*x+e)^2-b*cos(f*x+e)^2+b)/(cos(f*x+e)+1)^2)^(1/2)*a*cot(f*x+e)*cs c(f*x+e)^2)
Time = 0.34 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.66 \[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\left [\frac {3 \, a \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} + 4 \, {\left ({\left (3 \, a - b\right )} \tan \left (f x + e\right )^{2} - a\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{12 \, a f \tan \left (f x + e\right )^{3}}, \frac {3 \, \sqrt {a - b} a \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right )^{3} + 2 \, {\left ({\left (3 \, a - b\right )} \tan \left (f x + e\right )^{2} - a\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, a f \tan \left (f x + e\right )^{3}}\right ] \]
[1/12*(3*a*sqrt(-a + b)*log(-((a^2 - 8*a*b + 8*b^2)*tan(f*x + e)^4 - 2*(3* a^2 - 4*a*b)*tan(f*x + e)^2 + a^2 + 4*((a - 2*b)*tan(f*x + e)^3 - a*tan(f* x + e))*sqrt(b*tan(f*x + e)^2 + a)*sqrt(-a + b))/(tan(f*x + e)^4 + 2*tan(f *x + e)^2 + 1))*tan(f*x + e)^3 + 4*((3*a - b)*tan(f*x + e)^2 - a)*sqrt(b*t an(f*x + e)^2 + a))/(a*f*tan(f*x + e)^3), 1/6*(3*sqrt(a - b)*a*arctan(-2*s qrt(b*tan(f*x + e)^2 + a)*sqrt(a - b)*tan(f*x + e)/((a - 2*b)*tan(f*x + e) ^2 - a))*tan(f*x + e)^3 + 2*((3*a - b)*tan(f*x + e)^2 - a)*sqrt(b*tan(f*x + e)^2 + a))/(a*f*tan(f*x + e)^3)]
\[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int \sqrt {a + b \tan ^{2}{\left (e + f x \right )}} \cot ^{4}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int { \sqrt {b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{4} \,d x } \]
\[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int { \sqrt {b \tan \left (f x + e\right )^{2} + a} \cot \left (f x + e\right )^{4} \,d x } \]
Timed out. \[ \int \cot ^4(e+f x) \sqrt {a+b \tan ^2(e+f x)} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a} \,d x \]