Integrand size = 23, antiderivative size = 182 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
-2*a/d/e/(e*csc(d*x+c))^(1/2)-2/3*a*cos(d*x+c)/d/e/(e*csc(d*x+c))^(1/2)+a* arctan(sin(d*x+c)^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+a*arcta nh(sin(d*x+c)^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)-2/3*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)*EllipticF(cos(1/2 *c+1/4*Pi+1/2*d*x),2^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)
Time = 9.74 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.74 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {a \left (12+4 \cos (c+d x)+6 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\csc (c+d x)}+3 \sqrt {\csc (c+d x)} \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \sqrt {\csc (c+d x)} \log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {4 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sqrt {\sin (c+d x)}}\right )}{6 d e \sqrt {e \csc (c+d x)}} \]
-1/6*(a*(12 + 4*Cos[c + d*x] + 6*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Csc[c + d *x]] + 3*Sqrt[Csc[c + d*x]]*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Sqrt[Csc[c + d *x]]*Log[1 + Sqrt[Csc[c + d*x]]] + (4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]) /Sqrt[Sin[c + d*x]]))/(d*e*Sqrt[e*Csc[c + d*x]])
Time = 0.65 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.70, 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, 756, 216, 219, 3115, 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.
| \(\displaystyle \int \frac {a \sec (c+d x)+a}{(e \csc (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \cos \left (c+d x-\frac {\pi }{2}\right )^{3/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int -\left ((-\cos (c+d x) a-a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\left ((\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)\right )dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {a \int \sin ^{\frac {3}{2}}(c+d x)dx+a \int \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+a \int \frac {\sin (c+d x)^{3/2}}{\cos (c+d x)}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\frac {a \int \frac {\sin ^{\frac {3}{2}}(c+d x)}{1-\sin ^2(c+d x)}d\sin (c+d x)}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {a \left (\int \frac {1}{\sqrt {\sin (c+d x)} \left (1-\sin ^2(c+d x)\right )}d\sin (c+d x)-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {a \left (2 \int \frac {1}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}+\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}+a \int \sin (c+d x)^{3/2}dx}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \int \sin (c+d x)^{3/2}dx+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin (c+d x)}}dx-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )+\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {a \left (2 \left (\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )\right )-2 \sqrt {\sin (c+d x)}\right )}{d}+a \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d}-\frac {2 \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}\right )}{e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
((a*(2*(ArcTan[Sqrt[Sin[c + d*x]]]/2 + ArcTanh[Sqrt[Sin[c + d*x]]]/2) - 2* Sqrt[Sin[c + d*x]]))/d + a*((2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) - ( 2*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(3*d)))/(e*Sqrt[e*Csc[c + d*x]]*Sqrt[Si n[c + d*x]])
3.3.85.3.1 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*(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]
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[(-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]
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 = 10.88 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.01
| method | result | size |
| parts | \(-\frac {a \sqrt {2}\, \left (i \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 )+i \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 ) \sin \left (d x +c \right ) \sqrt {2}\right ) \sin \left (d x +c \right )}{3 d \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) e \left (\cos \left (d x +c \right )+1\right )}+\frac {a \left (\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}}}\, \cos \left (d x +c \right )-\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 ) \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 ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-\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 \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{d \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) e \left (\cos \left (d x +c \right )+1\right )}\) | \(547\) |
| default | \(\text {Expression too large to display}\) | \(1058\) |
-1/3*a/d*2^(1/2)*(I*(-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)+I*(-I*(I-cot(d*x+c)+cs c(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)) -cos(d*x+c)*sin(d*x+c)*2^(1/2))/(e*csc(d*x+c))^(1/2)/(cos(d*x+c)-1)/e/(cos (d*x+c)+1)*sin(d*x+c)+a/d*(arctan((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)*cos(d*x+c)-(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)))*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)))*(sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)-(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)*(c ot(d*x+c)+csc(d*x+c)))+2*sin(d*x+c))/(e*csc(d*x+c))^(1/2)/(cos(d*x+c)-1)/e /(cos(d*x+c)+1)*sin(d*x+c)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.34 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\left [-\frac {6 \, 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 ) + 3 \, 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 ) + 16 \, {\left (a \cos \left (d x + c\right ) + 3 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} {\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} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{24 \, d e^{2}}, -\frac {6 \, 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 ) - 3 \, 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 ) + 16 \, {\left (a \cos \left (d x + c\right ) + 3 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} {\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} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{24 \, d e^{2}}\right ] \]
[-1/24*(6*a*sqrt(-e)*arctan(-1/4*(cos(d*x + c)^2 - 6*sin(d*x + c) - 2)*sqr t(-e)*sqrt(e/sin(d*x + c))/(e*sin(d*x + c) + e)) + 3*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)) + 16*(a*cos(d*x + c) + 3*a)*sqrt(e/sin(d*x + c))*sin(d*x + c) + 8*I*a*sqrt(2*I*e)*weierstrassP Inverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*weierstr assPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)))/(d*e^2), -1/24*(6*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)) - 3*a*sqrt(e)*log((e*cos(d*x + c)^4 - 72*e*c os(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)) + 16*(a*cos(d*x + c) + 3*a)*sqrt(e/sin(d* x + c))*sin(d*x + c) + 8*I*a*sqrt(2*I*e)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) - 8*I*a*sqrt(-2*I*e)*weierstrassPInverse(4, 0, cos (d*x + c) - I*sin(d*x + c)))/(d*e^2)]
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
a*(Integral((e*csc(c + d*x))**(-3/2), x) + Integral(sec(c + d*x)/(e*csc(c + d*x))**(3/2), x))
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]