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

Optimal result

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

[Out]

-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)-a*arctan(s
in(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*(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)*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)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3963, 3957, 2917, 2644, 327, 335, 304, 209, 212, 2715, 2719} \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=-\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {2 a \sin (c+d x)}{3 d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 a \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt {e \csc (c+d x)}}+\frac {6 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{5 d e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

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

[Out]

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

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

Rule 327

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[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] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

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 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 3963

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sec (c+d x)) \sin ^{\frac {5}{2}}(c+d x) \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int (-a-a \cos (c+d x)) \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x) \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {a \int \sin ^{\frac {5}{2}}(c+d x) \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \int \sec (c+d x) \sin ^{\frac {5}{2}}(c+d x) \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a \cos (c+d x) \sin (c+d x)}{5 d e^2 \sqrt {e \csc (c+d x)}}+\frac {(3 a) \int \sqrt {\sin (c+d x)} \, dx}{5 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {x^{5/2}}{1-x^2} \, dx,x,\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)}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\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)}}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\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)}}+\frac {a \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\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)}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 1.27 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.84 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=\frac {a \left (-72 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right )-2 \sqrt {-\cot ^2(c+d x)} \left (-30 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\csc (c+d x)}+15 \sqrt {\csc (c+d x)} \left (\log \left (1-\sqrt {\csc (c+d x)}\right )-\log \left (1+\sqrt {\csc (c+d x)}\right )\right )+20 \sin (c+d x)+6 \sin (2 (c+d x))\right )\right )}{60 d e^2 \sqrt {-\cot ^2(c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

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

[Out]

(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[S
qrt[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]])

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.35 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.06

method	result	size
default	\(\frac {a \sqrt {2}\, \left (-6 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+3 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 ) \cos \left (d x +c \right )+\cos \left (d x +c \right )^{3} \sqrt {2}-6 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 )-4 \sqrt {2}\, \cos \left (d x +c \right )+3 \sqrt {2}\right ) \csc \left (d x +c \right )}{5 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}-\frac {a \left (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 ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+3 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \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 \cos \left (d x +c \right )-2\right ) \sin \left (d x +c \right )}{3 d \left (\cos \left (d x +c \right )-1\right ) e^{2} \sqrt {e \csc \left (d x +c \right )}}\)	\(602\)
parts	\(\frac {a \sqrt {2}\, \left (-6 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+3 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 ) \cos \left (d x +c \right )+\cos \left (d x +c \right )^{3} \sqrt {2}-6 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \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 )-4 \sqrt {2}\, \cos \left (d x +c \right )+3 \sqrt {2}\right ) \csc \left (d x +c \right )}{5 d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}-\frac {a \left (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 ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+3 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \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 \cos \left (d x +c \right )-2\right ) \sin \left (d x +c \right )}{3 d \left (\cos \left (d x +c \right )-1\right ) e^{2} \sqrt {e \csc \left (d x +c \right )}}\)	\(602\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.42 (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=\left [-\frac {30 \, a \sqrt {-e} \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 ) + 15 \, a \sqrt {-e} \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 ) - 72 \, a \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 72 \, a \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 16 \, {\left (3 \, a \cos \left (d x + c\right )^{3} + 5 \, a \cos \left (d x + c\right )^{2} - 3 \, a \cos \left (d x + c\right ) - 5 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{120 \, d e^{3}}, \frac {30 \, a \sqrt {e} \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 ) + 15 \, a \sqrt {e} \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 ) + 72 \, a \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 72 \, a \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 16 \, {\left (3 \, a \cos \left (d x + c\right )^{3} + 5 \, a \cos \left (d x + c\right )^{2} - 3 \, a \cos \left (d x + c\right ) - 5 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{120 \, d e^{3}}\right ] \]

[In]

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

[Out]

[-1/120*(30*a*sqrt(-e)*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)) + 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*sqrt(-2*I*e)*weie
rstrassZeta(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/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*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*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)]

Sympy [F]

\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx=a \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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