\(\int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx\) [281]

Optimal result

Integrand size = 23, antiderivative size = 190 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d} \] Output:

-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*(e*csc(d*x+c))^(1/2)*InverseJacobiAM(1/2*c-1/4*Pi+1/2* 
d*x,2^(1/2))*sin(d*x+c)^(1/2)/d
 

Mathematica [A] (verified)

Time = 2.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {a (e \csc (c+d x))^{5/2} \left (6 \arctan \left (\sqrt {\csc (c+d x)}\right )+4 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\csc (c+d x)}+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (1+\sqrt {\csc (c+d x)}\right )+4 \sqrt {\csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}\right )}{6 d \csc ^{\frac {5}{2}}(c+d x)} \] Input:

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

Output:

-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))
 

Rubi [A] (verified)

Time = 0.69 (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

\(\displaystyle -e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-\frac {2}{3 \sin ^{\frac {3}{2}}(c+d x)}\right )}{d}-a \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \cos (c+d x)}{3 d \sin ^{\frac {3}{2}}(c+d x)}\right )\right )\)

Input:

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

Output:

-(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]])
 

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 264

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]
 

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 756

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]
 

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 3116

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]
 

rule 3120

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]
 

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 = 2.05 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.59

method	result	size
parts	\(-\frac {a \sqrt {2}\, e^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (i \left (-1-\cos \left (d x +c \right )\right ) \sqrt {1-i \cot \left (d x +c \right )+i \csc \left (d x +c \right )}\, \sqrt {i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+i \cot \left (d x +c \right )-i \csc \left (d x +c \right )}+\sqrt {2}\, \cot \left (d x +c \right )\right )}{3 d}+\frac {a \left (\left (3 \cos \left (d x +c \right )-3\right ) \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )+\left (-3 \cos \left (d x +c \right )+3\right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}{-1+\cos \left (d x +c \right )}\right )-2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}\right ) \sqrt {e \csc \left (d x +c \right )}\, e^{2} \csc \left (d x +c \right )}{3 d \sqrt {\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right )^{2}}}}\)	\(302\)
default	\(-\frac {a \sqrt {2}\, \left (1+\cos \left (d x +c \right )\right ) \left (4 i \sin \left (d x +c \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 )-3 i \sin \left (d x +c \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 {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 )-3 i \sin \left (d x +c \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 {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 )-3 \sin \left (d x +c \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 {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 )+3 \sin \left (d x +c \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 {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 )+2 \sqrt {2}\right ) e^{2} \sqrt {e \csc \left (d x +c \right )}\, \csc \left (d x +c \right )}{6 d}\)	\(558\)

Input:

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

Output:

-1/3*a/d*2^(1/2)*e^2*(e*csc(d*x+c))^(1/2)*(I*(-1-cos(d*x+c))*(1-I*cot(d*x+ 
c)+I*csc(d*x+c))^(1/2)*(I*(csc(d*x+c)-cot(d*x+c)))^(1/2)*EllipticF((1+I*co 
t(d*x+c)-I*csc(d*x+c))^(1/2),1/2*2^(1/2))*(1+I*cot(d*x+c)-I*csc(d*x+c))^(1 
/2)+2^(1/2)*cot(d*x+c))+1/3*a/d*((3*cos(d*x+c)-3)*arctanh(sin(d*x+c)*(sin( 
d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))+(-3*cos(d*x+c)+3)*arctan(s 
in(d*x+c)*(sin(d*x+c)/(1+cos(d*x+c))^2)^(1/2)/(-1+cos(d*x+c)))-2*(sin(d*x+ 
c)/(1+cos(d*x+c))^2)^(1/2))*(e*csc(d*x+c))^(1/2)*e^2/(sin(d*x+c)/(1+cos(d* 
x+c))^2)^(1/2)*csc(d*x+c)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 682, normalized size of antiderivative = 3.59 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

[-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))]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

sqrt(e)*a*e**2*(int(sqrt(csc(c + d*x))*csc(c + d*x)**2*sec(c + d*x),x) + i 
nt(sqrt(csc(c + d*x))*csc(c + d*x)**2,x))