Integrand size = 23, antiderivative size = 190 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d} \]
-2/3*a*e^2*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/d-2/3*a*e^2*csc(d*x+c)*(e*csc(d *x+c))^(1/2)/d+a*e^2*arctan(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x +c)^(1/2)/d+a*e^2*arctanh(sin(d*x+c)^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x+c )^(1/2)/d-2/3*a*e^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1 /2*d*x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)* sin(d*x+c)^(1/2)/d
Time = 2.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {a (e \csc (c+d x))^{5/2} \left (6 \arctan \left (\sqrt {\csc (c+d x)}\right )+4 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\csc (c+d x)}+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (1+\sqrt {\csc (c+d x)}\right )+4 \sqrt {\csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}\right )}{6 d \csc ^{\frac {5}{2}}(c+d x)} \]
-1/6*(a*(e*Csc[c + d*x])^(5/2)*(6*ArcTan[Sqrt[Csc[c + d*x]]] + 4*Cot[(c + d*x)/2]*Sqrt[Csc[c + d*x]] + 3*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Log[1 + Sqr t[Csc[c + d*x]]] + 4*Sqrt[Csc[c + d*x]]*EllipticF[(-2*c + Pi - 2*d*x)/4, 2 ]*Sqrt[Sin[c + d*x]]))/(d*Csc[c + d*x]^(5/2))
Time = 0.68 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 25, 3317, 25, 3042, 3044, 264, 266, 756, 216, 219, 3116, 3042, 3120}
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.
| \(\displaystyle \int (a \sec (c+d x)+a) (e \csc (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right ) \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sec (c+d x) a+a}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {(-\cos (c+d x) a-a) \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {(\cos (c+d x) a+a) \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {(\cos (c+d x) a+a) \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \sin \left (c+d x-\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^{5/2} \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (a \int -\frac {\sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx-a \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)}dx-a \int \frac {\sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin (c+d x)^{5/2}}dx-a \int \frac {1}{\cos (c+d x) \sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x) \left (1-\sin ^2(c+d x)\right )}d\sin (c+d x)}{d}-a \int \frac {1}{\sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (\int \frac {1}{\sqrt {\sin (c+d x)} \left (1-\sin ^2(c+d x)\right )}d\sin (c+d x)-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \int \frac {1}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \int \frac {1}{\sin (c+d x)^{5/2}}dx\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \int \frac {1}{\sin (c+d x)^{5/2}}dx-\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )-\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )-\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )\right )\) |
-(e^2*Sqrt[e*Csc[c + d*x]]*(-((a*(2*(ArcTan[Sqrt[Sin[c + d*x]]]/2 + ArcTan h[Sqrt[Sin[c + d*x]]]/2) - 2/(3*Sin[c + d*x]^(3/2))))/d) - a*((2*EllipticF [(c - Pi/2 + d*x)/2, 2])/(3*d) - (2*Cos[c + d*x])/(3*d*Sin[c + d*x]^(3/2)) ))*Sqrt[Sin[c + d*x]])
3.3.81.3.1 Defintions of rubi rules used
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[((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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 3.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.82
| method | result | size |
| default | \(\frac {a \sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-1\right ) \csc \left (d x +c \right )^{2}}{6 d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2}}+\frac {a \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right ) \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-3 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-3 \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 \,\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )\right ) \csc \left (d x +c \right )}{3 d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(536\) |
| parts | \(\frac {a \sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-1\right ) \csc \left (d x +c \right )^{2}}{6 d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2}}+\frac {a \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right ) \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-3 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-3 \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 \,\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )\right ) \csc \left (d x +c \right )}{3 d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(536\) |
1/6*a/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+c)) )^(5/2)*(1-cos(d*x+c))^2*(2*I*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*2^(1/2) *(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-cot(d*x+c)+csc(d*x+c)))^(1/2)*E llipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))*(-cot(d*x+c)+cs c(d*x+c))+(1-cos(d*x+c))^4*csc(d*x+c)^4-1)/((1-cos(d*x+c))^3*csc(d*x+c)^3+ csc(d*x+c)-cot(d*x+c))^(1/2)/((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^ 2+1)*csc(d*x+c))^(1/2)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^2*csc(d*x+c)^2+1/ 3*a/d*e^2*(e*csc(d*x+c))^(1/2)*(3*cos(d*x+c)*arctan((sin(d*x+c)/(cos(d*x+c )+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))-3*cos(d*x+c)*arctanh((sin(d*x+c)/(c os(d*x+c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))-2*(sin(d*x+c)/(cos(d*x+c)+1 )^2)^(1/2)-3*arctan((sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d* x+c)))+3*arctanh((sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(cot(d*x+c)+csc(d*x+c ))))/(sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*csc(d*x+c)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 682, normalized size of antiderivative = 3.59 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\left [-\frac {6 \, a \sqrt {-e} e^{2} \arctan \left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) + e\right )}}\right ) \sin \left (d x + c\right ) - 3 \, a \sqrt {-e} e^{2} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 8 i \, a \sqrt {-2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 16 \, {\left (a e^{2} \cos \left (d x + c\right ) + a e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{24 \, d \sin \left (d x + c\right )}, -\frac {6 \, a e^{\frac {5}{2}} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) - 3 \, a e^{\frac {5}{2}} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 8 i \, a \sqrt {-2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 16 \, {\left (a e^{2} \cos \left (d x + c\right ) + a e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{24 \, d \sin \left (d x + c\right )}\right ] \]
[-1/24*(6*a*sqrt(-e)*e^2*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2) *sqrt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e))*sin(d*x + c) - 3*a*sq rt(-e)*e^2*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 + (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt(-e)*sq rt(e/sin(d*x + c)) + 28*(e*cos(d*x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos (d*x + c)^4 - 8*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) *sin(d*x + c) + 8*I*a*sqrt(2*I*e)*e^2*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*e^2*sin(d*x + c)*we ierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 16*(a*e^2*cos(d*x + c) + a*e^2)*sqrt(e/sin(d*x + c)))/(d*sin(d*x + c)), -1/24*(6*a*e^(5/2)* arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) - e))*sin(d*x + c) - 3*a*e^(5/2)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(cos(d*x + c)^4 - 9*cos(d*x + c)^2 - (7*cos(d*x + c)^2 - 8)*sin(d*x + c) + 8)*sqrt(e)*sqrt(e/sin(d*x + c)) - 28*(e*cos(d* x + c)^2 - 2*e)*sin(d*x + c) + 72*e)/(cos(d*x + c)^4 - 8*cos(d*x + c)^2 + 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8))*sin(d*x + c) + 8*I*a*sqrt(2*I*e) *e^2*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*e^2*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + 16*(a*e^2*cos(d*x + c) + a*e^2)*sqrt(e/sin(d*x + c )))/(d*sin(d*x + c))]
Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int \left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2} \,d x \]