\(\int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx\) [286]

Optimal result

Integrand size = 23, antiderivative size = 197 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=-\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {6 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 a \sin (c+d x)}{3 d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d e^2 \sqrt {e \csc (c+d x)}} \] Output:

-a*arctan(sin(d*x+c)^(1/2))/d/e^2/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+a* 
arctanh(sin(d*x+c)^(1/2))/d/e^2/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)-6/5* 
a*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))/d/e^2/(e*csc(d*x+c))^(1/2)/ 
sin(d*x+c)^(1/2)-2/3*a*sin(d*x+c)/d/e^2/(e*csc(d*x+c))^(1/2)-2/5*a*cos(d*x 
+c)*sin(d*x+c)/d/e^2/(e*csc(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 1.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\frac {a \left (-72 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right )-2 \sqrt {-\cot ^2(c+d x)} \left (-30 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\csc (c+d x)}+15 \sqrt {\csc (c+d x)} \left (\log \left (1-\sqrt {\csc (c+d x)}\right )-\log \left (1+\sqrt {\csc (c+d x)}\right )\right )+20 \sin (c+d x)+6 \sin (2 (c+d x))\right )\right )}{60 d e^2 \sqrt {-\cot ^2(c+d x)} \sqrt {e \csc (c+d x)}} \] Input:

Integrate[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(5/2),x]
 

Output:

(a*(-72*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2] - 2 
*Sqrt[-Cot[c + d*x]^2]*(-30*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Csc[c + d*x]] 
+ 15*Sqrt[Csc[c + d*x]]*(Log[1 - Sqrt[Csc[c + d*x]]] - Log[1 + Sqrt[Csc[c 
+ d*x]]]) + 20*Sin[c + d*x] + 6*Sin[2*(c + d*x)])))/(60*d*e^2*Sqrt[-Cot[c 
+ d*x]^2]*Sqrt[e*Csc[c + d*x]])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.65, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 3317, 3042, 3044, 262, 266, 827, 216, 219, 3115, 3042, 3119}

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.

\(\displaystyle \int \frac {a \sec (c+d x)+a}{(e \csc (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4366

\(\displaystyle \frac {\int (\sec (c+d x) a+a) \sin ^{\frac {5}{2}}(c+d x)dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 4360

\(\displaystyle \frac {\int -\left ((-\cos (c+d x) a-a) \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x)\right )dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -\left ((\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x)\right )dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x)dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3317

\(\displaystyle \frac {a \int \sin ^{\frac {5}{2}}(c+d x)dx+a \int \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x)dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \int \sin (c+d x)^{5/2}dx+a \int \frac {\sin (c+d x)^{5/2}}{\cos (c+d x)}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3044

\(\displaystyle \frac {\frac {a \int \frac {\sin ^{\frac {5}{2}}(c+d x)}{1-\sin ^2(c+d x)}d\sin (c+d x)}{d}+a \int \sin (c+d x)^{5/2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {a \left (\int \frac {\sqrt {\sin (c+d x)}}{1-\sin ^2(c+d x)}d\sin (c+d x)-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}+a \int \sin (c+d x)^{5/2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {a \left (2 \int \frac {\sin (c+d x)}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}+a \int \sin (c+d x)^{5/2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}+a \int \sin (c+d x)^{5/2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}+a \int \sin (c+d x)^{5/2}dx}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a \int \sin (c+d x)^{5/2}dx+\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {a \left (\frac {3}{5} \int \sqrt {\sin (c+d x)}dx-\frac {2 \sin ^{\frac {3}{2}}(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \left (\frac {3}{5} \int \sqrt {\sin (c+d x)}dx-\frac {2 \sin ^{\frac {3}{2}}(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3} \sin ^{\frac {3}{2}}(c+d x)\right )}{d}+a \left (\frac {6 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{5 d}-\frac {2 \sin ^{\frac {3}{2}}(c+d x) \cos (c+d x)}{5 d}\right )}{e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\)

Input:

Int[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(5/2),x]
 

Output:

((a*(2*(-1/2*ArcTan[Sqrt[Sin[c + d*x]]] + ArcTanh[Sqrt[Sin[c + d*x]]]/2) - 
 (2*Sin[c + d*x]^(3/2))/3))/d + a*((6*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5 
*d) - (2*Cos[c + d*x]*Sin[c + d*x]^(3/2))/(5*d)))/(e^2*Sqrt[e*Csc[c + d*x] 
]*Sqrt[Sin[c + d*x]])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

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])
 

rule 219

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])
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

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]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3044

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])
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3317

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]
 

rule 4360

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]
 

rule 4366

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]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.98 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.21

method	result	size
parts	\(\frac {a \sqrt {2}\, \left (\left (3 \cos \left (d x +c \right )+3\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\left (-6 \cos \left (d x +c \right )-6\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {-i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\left (\cos \left (d x +c \right )^{3}-4 \cos \left (d x +c \right )+3\right ) \sqrt {2}\right ) \csc \left (d x +c \right )}{5 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}+\frac {a \left (-\frac {\sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \left (3 \cot \left (d x +c \right )+3 \csc \left (d x +c \right )\right )}{3}-\frac {\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\, \left (3 \cot \left (d x +c \right )+3 \csc \left (d x +c \right )\right )}{3}-\frac {2 \sin \left (d x +c \right )}{3}\right )}{d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}\)	\(436\)
default	\(\frac {a \sqrt {2}\, \left (i \left (15 \cos \left (d x +c \right )+15\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+i \left (-15 \cos \left (d x +c \right )-15\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (18 \cos \left (d x +c \right )+18\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (15 \cos \left (d x +c \right )+15\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (15 \cos \left (d x +c \right )+15\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (-36 \cos \left (d x +c \right )-36\right ) \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\left (6 \cos \left (d x +c \right )^{3}+10 \cos \left (d x +c \right )^{2}-24 \cos \left (d x +c \right )+8\right ) \sqrt {2}\right ) \csc \left (d x +c \right )}{30 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}\)	\(698\)

Input:

int((a+a*sec(d*x+c))/(e*csc(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/5*a/d*2^(1/2)*((3*cos(d*x+c)+3)*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I 
*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*(-csc(d*x+c)+cot(d*x+c)))^(1/2)*Ellipt 
icF((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))+(-6*cos(d*x+c)-6)*(1+ 
I*cot(d*x+c)-I*csc(d*x+c))^(1/2)*(1-I*cot(d*x+c)+I*csc(d*x+c))^(1/2)*(-I*( 
-csc(d*x+c)+cot(d*x+c)))^(1/2)*EllipticE((1+I*cot(d*x+c)-I*csc(d*x+c))^(1/ 
2),1/2*2^(1/2))+(cos(d*x+c)^3-4*cos(d*x+c)+3)*2^(1/2))/(e*csc(d*x+c))^(1/2 
)/e^2*csc(d*x+c)+a/d*(-1/3*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*arctan(sin( 
d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))*(3*cot(d*x+c)+ 
3*csc(d*x+c))-1/3*arctanh(sin(d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/( 
-1+cos(d*x+c)))*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)*(3*cot(d*x+c)+3*csc(d* 
x+c))-2/3*sin(d*x+c))/(e*csc(d*x+c))^(1/2)/e^2
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.46 (sec) , antiderivative size = 654, normalized size of antiderivative = 3.32 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

[-1/120*(30*a*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*s 
qrt(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e)) + 15*a*sqrt(-e)*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*co 
s(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2)*sin(d*x + c) + 8)) - 72*a*sqrt(2*I*e 
)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x 
 + c))) - 72*a*sqrt(-2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0 
, cos(d*x + c) - I*sin(d*x + c))) - 16*(3*a*cos(d*x + c)^3 + 5*a*cos(d*x + 
 c)^2 - 3*a*cos(d*x + c) - 5*a)*sqrt(e/sin(d*x + c)))/(d*e^3), 1/120*(30*a 
*sqrt(e)*arctan(1/4*(cos(d*x + c)^2 + 6*sin(d*x + c) - 2)*sqrt(e)*sqrt(e/s 
in(d*x + c))/(e*sin(d*x + c) - e)) + 15*a*sqrt(e)*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)) + 72*a*sqrt(2*I*e)*weierstrassZeta( 
4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 72*a*sqr 
t(-2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I 
*sin(d*x + c))) + 16*(3*a*cos(d*x + c)^3 + 5*a*cos(d*x + c)^2 - 3*a*cos(d* 
x + c) - 5*a)*sqrt(e/sin(d*x + c)))/(d*e^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*sec(d*x + c) + a)/(e*csc(d*x + c))^(5/2), x)
 

Giac [F]

\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+a*sec(d*x+c))/(e*csc(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((a*sec(d*x + c) + a)/(e*csc(d*x + c))^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}} \,d x \] Input:

int((a + a/cos(c + d*x))/(e/sin(c + d*x))^(5/2),x)
 

Output:

int((a + a/cos(c + d*x))/(e/sin(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\frac {\sqrt {e}\, a \left (\int \frac {\sqrt {\csc \left (d x +c \right )}}{\csc \left (d x +c \right )^{3}}d x +\int \frac {\sqrt {\csc \left (d x +c \right )}\, \sec \left (d x +c \right )}{\csc \left (d x +c \right )^{3}}d x \right )}{e^{3}} \] Input:

int((a+a*sec(d*x+c))/(e*csc(d*x+c))^(5/2),x)
 

Output:

(sqrt(e)*a*(int(sqrt(csc(c + d*x))/csc(c + d*x)**3,x) + int((sqrt(csc(c + 
d*x))*sec(c + d*x))/csc(c + d*x)**3,x)))/e**3