Integrand size = 23, antiderivative size = 155 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}} \]
-a*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(3/2)+a*arctanh((e*sin(d*x+c)) ^(1/2)/e^(1/2))/d/e^(3/2)-2*a/d/e/(e*sin(d*x+c))^(1/2)-2*a*cos(d*x+c)/d/e/ (e*sin(d*x+c))^(1/2)+2*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))*(e*sin(d*x+c))^( 1/2)/d/e^2/sin(d*x+c)^(1/2)
Time = 0.57 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {a (1+\cos (c+d x)) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\arctan \left (\sqrt {\sin (c+d x)}\right ) \sec \left (\frac {1}{2} (c+d x)\right )-\text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sec \left (\frac {1}{2} (c+d x)\right )-2 E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sec \left (\frac {1}{2} (c+d x)\right )+2 \csc \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sin (c+d x)}\right ) \sin ^{\frac {3}{2}}(c+d x)}{2 d (e \sin (c+d x))^{3/2}} \]
-1/2*(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]*(ArcTan[Sqrt[Sin[c + d*x]]]*Se c[(c + d*x)/2] - ArcTanh[Sqrt[Sin[c + d*x]]]*Sec[(c + d*x)/2] - 2*Elliptic E[(-2*c + Pi - 2*d*x)/4, 2]*Sec[(c + d*x)/2] + 2*Csc[(c + d*x)/2]*Sqrt[Sin [c + d*x]])*Sin[c + d*x]^(3/2))/(d*(e*Sin[c + d*x])^(3/2))
Time = 0.79 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 4360, 25, 25, 3042, 25, 3317, 25, 3042, 3044, 27, 264, 266, 827, 216, 219, 3116, 3042, 3121, 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.
| \(\displaystyle \int \frac {a \sec (c+d x)+a}{(e \sin (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sec (c+d x) (a (-\cos (c+d x))-a)}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {(\cos (c+d x) a+a) \sec (c+d x)}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sec (c+d x) (a \cos (c+d x)+a)}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {a-a \sin \left (c+d x-\frac {\pi }{2}\right )}{\sin \left (c+d x-\frac {\pi }{2}\right ) \left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\left (e \cos \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^{3/2} \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx-a \int -\frac {\sec (c+d x)}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx+a \int \frac {\sec (c+d x)}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx+a \int \frac {1}{\cos (c+d x) (e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {a \int \frac {e^2}{(e \sin (c+d x))^{3/2} \left (e^2-e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d e}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a e \int \frac {1}{(e \sin (c+d x))^{3/2} \left (e^2-e^2 \sin ^2(c+d x)\right )}d(e \sin (c+d x))}{d}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {a e \left (\frac {\int \frac {\sqrt {e \sin (c+d x)}}{e^2-e^2 \sin ^2(c+d x)}d(e \sin (c+d x))}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {a e \left (\frac {2 \int \frac {e^2 \sin ^2(c+d x)}{e^2-e^4 \sin ^4(c+d x)}d\sqrt {e \sin (c+d x)}}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {a e \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{e-e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}-\frac {1}{2} \int \frac {1}{e^2 \sin ^2(c+d x)+e}d\sqrt {e \sin (c+d x)}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {a e \left (\frac {2 \left (\frac {1}{2} \int \frac {1}{e-e^2 \sin ^2(c+d x)}d\sqrt {e \sin (c+d x)}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}+a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{3/2}}dx+\frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (-\frac {\int \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )+\frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {\int \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )+\frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (-\frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )+\frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )+\frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {a e \left (\frac {2 \left (\frac {\text {arctanh}\left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}-\frac {\arctan \left (\sqrt {e} \sin (c+d x)\right )}{2 \sqrt {e}}\right )}{e^2}-\frac {2}{e^2 \sqrt {e \sin (c+d x)}}\right )}{d}+a \left (-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )\) |
(a*e*((2*(-1/2*ArcTan[Sqrt[e]*Sin[c + d*x]]/Sqrt[e] + ArcTanh[Sqrt[e]*Sin[ c + d*x]]/(2*Sqrt[e])))/e^2 - 2/(e^2*Sqrt[e*Sin[c + d*x]])))/d + a*((-2*Co s[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]))
3.2.12.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(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]
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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
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]
Time = 10.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(\frac {-\frac {a \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}-\frac {2 a}{e \sqrt {e \sin \left (d x +c \right )}}+\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}+\frac {a \left (2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(201\) |
| parts | \(\frac {a \left (2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a \left (-\frac {\arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \sin \left (d x +c \right )}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}\right )}{d}\) | \(203\) |
(-a/e^(3/2)*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))-2*a/e/(e*sin(d*x+c))^(1/2 )+a/e^(3/2)*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))+a/e*(2*(-sin(d*x+c)+1)^( 1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/ 2),1/2*2^(1/2))-(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(1 /2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-2*cos(d*x+c)^2)/cos(d*x+c )/(e*sin(d*x+c))^(1/2))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.32 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\left [\frac {-8 i \, \sqrt {2} a \sqrt {-i \, e} \sin \left (d x + c\right ) {\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 ) + 8 i \, \sqrt {2} a \sqrt {i \, e} \sin \left (d x + c\right ) {\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 ) - 2 \, 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 \sin \left (d x + c\right )} \sqrt {-e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} - e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) - 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 (7 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e} + 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 ) - 16 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}}{8 \, d e^{2} \sin \left (d x + c\right )}, \frac {-8 i \, \sqrt {2} a \sqrt {-i \, e} \sin \left (d x + c\right ) {\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 ) + 8 i \, \sqrt {2} a \sqrt {i \, e} \sin \left (d x + c\right ) {\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 ) - 2 \, 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 \sin \left (d x + c\right )} \sqrt {e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} + e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) + 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 (7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e} - 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 ) - 16 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}}{8 \, d e^{2} \sin \left (d x + c\right )}\right ] \]
[1/8*(-8*I*sqrt(2)*a*sqrt(-I*e)*sin(d*x + c)*weierstrassZeta(4, 0, weierst rassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 8*I*sqrt(2)*a*sqrt(I* e)*sin(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*a*sqrt(-e)*arctan(1/4*(cos(d*x + c)^2 - 6*sin(d* x + c) - 2)*sqrt(e*sin(d*x + c))*sqrt(-e)/(e*cos(d*x + c)^2 - e*sin(d*x + c) - e))*sin(d*x + c) - a*sqrt(-e)*log((e*cos(d*x + c)^4 - 72*e*cos(d*x + c)^2 + 8*(7*cos(d*x + c)^2 - (cos(d*x + c)^2 - 8)*sin(d*x + c) - 8)*sqrt(e *sin(d*x + c))*sqrt(-e) + 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) - 16*(a*cos(d*x + c) + a)*sqrt(e*sin(d*x + c)))/(d*e^2* sin(d*x + c)), 1/8*(-8*I*sqrt(2)*a*sqrt(-I*e)*sin(d*x + c)*weierstrassZeta (4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 8*I*sqr t(2)*a*sqrt(I*e)*sin(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*a*sqrt(e)*arctan(1/4*(cos(d*x + c) ^2 + 6*sin(d*x + c) - 2)*sqrt(e*sin(d*x + c))*sqrt(e)/(e*cos(d*x + c)^2 + e*sin(d*x + c) - e))*sin(d*x + c) + a*sqrt(e)*log((e*cos(d*x + c)^4 - 72*e *cos(d*x + c)^2 - 8*(7*cos(d*x + c)^2 + (cos(d*x + c)^2 - 8)*sin(d*x + c) - 8)*sqrt(e*sin(d*x + c))*sqrt(e) - 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) - 16*(a*cos(d*x + c) + a)*sqrt(e*sin(d*x +...
\[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
a*(Integral((e*sin(c + d*x))**(-3/2), x) + Integral(sec(c + d*x)/(e*sin(c + d*x))**(3/2), x))
Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]