Integrand size = 23, antiderivative size = 144 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{3/2} f}-\frac {\cot (e+f x)}{8 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f} \]
-1/8*arctanh(cos(f*x+e)*a^(1/2)/(a+a*sin(f*x+e))^(1/2))/a^(3/2)/f-1/8*cot( f*x+e)/a/f/(a+a*sin(f*x+e))^(1/2)+11/12*cot(f*x+e)*csc(f*x+e)/a/f/(a+a*sin (f*x+e))^(1/2)-1/3*cot(f*x+e)*csc(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/a^2/f
Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(144)=288\).
Time = 0.93 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (-132 \cos \left (\frac {1}{2} (e+f x)\right )+62 \cos \left (\frac {3}{2} (e+f x)\right )+6 \cos \left (\frac {5}{2} (e+f x)\right )+132 \sin \left (\frac {1}{2} (e+f x)\right )-9 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+9 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+62 \sin \left (\frac {3}{2} (e+f x)\right )-6 \sin \left (\frac {5}{2} (e+f x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-3 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{24 f \left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{3/2}} \]
(Csc[(e + f*x)/2]^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-132*Cos[(e + f*x)/2] + 62*Cos[(3*(e + f*x))/2] + 6*Cos[(5*(e + f*x))/2] + 132*Sin[(e + f*x)/2] - 9*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + 9 *Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] + 62*Sin[(3*(e + f*x))/2] - 6*Sin[(5*(e + f*x))/2] + 3*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[3*(e + f*x)] - 3*Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ]]*Sin[3*(e + f*x)]))/(24*f*(Csc[(e + f*x)/4]^2 - Sec[(e + f*x)/4]^2)^3*(a *(1 + Sin[e + f*x]))^(3/2))
Time = 1.64 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.82, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3196, 3042, 3251, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 3042, 3251, 3042, 3252, 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.
| \(\displaystyle \int \frac {\cot ^4(e+f x)}{(a \sin (e+f x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^4 (a \sin (e+f x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3196 |
| \(\displaystyle \frac {\int \csc ^4(e+f x) \sqrt {\sin (e+f x) a+a} \left (\sin ^2(e+f x)+1\right )dx}{a^2}-\frac {2 \int \csc ^3(e+f x) \sqrt {\sin (e+f x) a+a}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)^3}dx}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \int \csc ^2(e+f x) \sqrt {\sin (e+f x) a+a}dx-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)^2}dx-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \left (\frac {1}{2} \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin (e+f x) a+a} \left (\sin (e+f x)^2+1\right )}{\sin (e+f x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3523 |
| \(\displaystyle \frac {\frac {\int \frac {1}{2} \csc ^3(e+f x) \sqrt {\sin (e+f x) a+a} (9 \sin (e+f x) a+a)dx}{3 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \csc ^3(e+f x) \sqrt {\sin (e+f x) a+a} (9 \sin (e+f x) a+a)dx}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin (e+f x) a+a} (9 \sin (e+f x) a+a)}{\sin (e+f x)^3}dx}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3459 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \int \csc ^2(e+f x) \sqrt {\sin (e+f x) a+a}dx-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)^2}dx-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3251 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \left (\frac {1}{2} \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \left (\frac {1}{2} \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3252 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {39}{4} a \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a^2 \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}}{6 a}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}-\frac {a \cot (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )-\frac {a \cot (e+f x) \csc (e+f x)}{2 f \sqrt {a \sin (e+f x)+a}}\right )}{a^2}\) |
(-2*(-1/2*(a*Cot[e + f*x]*Csc[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) + (3* (-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) - (a*Cot[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]])))/4))/a^2 + (-1/3*(Cot[e + f*x]*Csc[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/f + (-1/2*(a^2*Cot[e + f*x]* Csc[e + f*x])/(f*Sqrt[a + a*Sin[e + f*x]]) + (39*a*(-((Sqrt[a]*ArcTanh[(Sq rt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) - (a*Cot[e + f*x])/(f*Sq rt[a + a*Sin[e + f*x]])))/4)/(6*a))/a^2
3.2.10.3.1 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]))* 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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[-2/(a*b) Int[(a + b*Sin[e + f*x])^(m + 2)/Sin[e + f*x]^ 3, x], x] + Simp[1/a^2 Int[(a + b*Sin[e + f*x])^(m + 2)*((1 + Sin[e + f*x ]^2)/Sin[e + f*x]^4), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] && LtQ[m, -1]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2))) Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f) Subst[Int[1/(b*c + a*d - d*x^2), x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp [(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) *(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d)) Int[Sqrt[a + b*Sin[e + f*x] ]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x ] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a *d*m + b*c*(n + 1)) + c*C*(a*c*m + b*d*(n + 1)) - b*(A*d^2*(m + n + 2) + C* (c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Time = 0.77 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00
| method | result | size |
| default | \(-\frac {\left (\sin \left (f x +e \right )+1\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (3 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (f x +e \right )\right )+8 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-3 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {11}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(144\) |
-1/24/a^(11/2)*(sin(f*x+e)+1)*(-a*(sin(f*x+e)-1))^(1/2)*(3*(-a*(sin(f*x+e) -1))^(5/2)*a^(3/2)+3*arctanh((-a*(sin(f*x+e)-1))^(1/2)/a^(1/2))*a^4*sin(f* x+e)^3+8*(-a*(sin(f*x+e)-1))^(3/2)*a^(5/2)-3*(-a*(sin(f*x+e)-1))^(1/2)*a^( 7/2))/sin(f*x+e)^3/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (124) = 248\).
Time = 0.31 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.66 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} - {\left (3 \, \cos \left (f x + e\right )^{2} - 14 \, \cos \left (f x + e\right ) - 25\right )} \sin \left (f x + e\right ) - 11 \, \cos \left (f x + e\right ) - 25\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f - {\left (a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
1/96*(3*(cos(f*x + e)^4 - 2*cos(f*x + e)^2 - (cos(f*x + e)^3 + cos(f*x + e )^2 - cos(f*x + e) - 1)*sin(f*x + e) + 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e) ^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1 )) + 4*(3*cos(f*x + e)^3 + 17*cos(f*x + e)^2 - (3*cos(f*x + e)^2 - 14*cos( f*x + e) - 25)*sin(f*x + e) - 11*cos(f*x + e) - 25)*sqrt(a*sin(f*x + e) + a))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f - (a^2*f*cos(f* x + e)^3 + a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - a^2*f)*sin(f*x + e) )
\[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Timed out. \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
Time = 0.47 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, {\left (12 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 16 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{96 \, f} \]
-1/96*sqrt(2)*sqrt(a)*(3*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2* f*x + 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e)))/(a^2*sgn( cos(-1/4*pi + 1/2*f*x + 1/2*e))) + 4*(12*sin(-1/4*pi + 1/2*f*x + 1/2*e)^5 + 16*sin(-1/4*pi + 1/2*f*x + 1/2*e)^3 - 3*sin(-1/4*pi + 1/2*f*x + 1/2*e))/ ((2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^3*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))/f
Timed out. \[ \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]