Integrand size = 23, antiderivative size = 122 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=-\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
-a*arctan(sin(d*x+c)^(1/2))/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+a*arct anh(sin(d*x+c)^(1/2))/d/(e*csc(d*x+c))^(1/2)/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))/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.76 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\frac {a \left (-4 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right )+\sqrt {-\cot ^2(c+d x)} \sqrt {\csc (c+d x)} \left (2 \arctan \left (\sqrt {\csc (c+d x)}\right )-\log \left (1-\sqrt {\csc (c+d x)}\right )+\log \left (1+\sqrt {\csc (c+d x)}\right )\right )\right )}{2 d \sqrt {-\cot ^2(c+d x)} \sqrt {e \csc (c+d x)}} \]
(a*(-4*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2] + Sq rt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*(2*ArcTan[Sqrt[Csc[c + d*x]]] - Log [1 - Sqrt[Csc[c + d*x]]] + Log[1 + Sqrt[Csc[c + d*x]]])))/(2*d*Sqrt[-Cot[c + d*x]^2]*Sqrt[e*Csc[c + d*x]])
Time = 0.54 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.68, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 4366, 3042, 4360, 25, 25, 3042, 3317, 3042, 3044, 266, 827, 216, 219, 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.
| \(\displaystyle \int \frac {a \sec (c+d x)+a}{\sqrt {e \csc (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-a \csc \left (c+d x-\frac {\pi }{2}\right )}{\sqrt {e \sec \left (c+d x-\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int (\sec (c+d x) a+a) \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int -\left ((-\cos (c+d x) a-a) \sec (c+d x) \sqrt {\sin (c+d x)}\right )dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\left ((\cos (c+d x) a+a) \sec (c+d x) \sqrt {\sin (c+d x)}\right )dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (\cos (c+d x) a+a) \sec (c+d x) \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {-\cos \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle \frac {a \int \sqrt {\sin (c+d x)}dx+a \int \sec (c+d x) \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sqrt {\sin (c+d x)}dx+a \int \frac {\sqrt {\sin (c+d x)}}{\cos (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle \frac {\frac {a \int \frac {\sqrt {\sin (c+d x)}}{1-\sin ^2(c+d x)}d\sin (c+d x)}{d}+a \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {2 a \int \frac {\sin (c+d x)}{1-\sin ^2(c+d x)}d\sqrt {\sin (c+d x)}}{d}+a \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {a \int \sqrt {\sin (c+d x)}dx+\frac {2 a \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \int \frac {1}{\sin (c+d x)+1}d\sqrt {\sin (c+d x)}\right )}{d}}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2 a \left (\frac {1}{2} \int \frac {1}{1-\sin (c+d x)}d\sqrt {\sin (c+d x)}-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )}{d}+a \int \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {a \int \sqrt {\sin (c+d x)}dx+\frac {2 a \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )}{d}}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2 a \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-\frac {1}{2} \arctan \left (\sqrt {\sin (c+d x)}\right )\right )}{d}+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}}{\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
((2*a*(-1/2*ArcTan[Sqrt[Sin[c + d*x]]] + ArcTanh[Sqrt[Sin[c + d*x]]]/2))/d + (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2])/d)/(Sqrt[e*Csc[c + d*x]]*Sqrt[Si n[c + d*x]])
3.3.84.3.1 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] :> 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[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[(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 = 9.39 (sec) , antiderivative size = 544, normalized size of antiderivative = 4.46
| method | result | size |
| parts | \(-\frac {a \sqrt {2}\, \left (2 \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 )-\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 )+2 \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 )-\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 )+\sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \csc \left (d x +c \right )}{d \sqrt {e \csc \left (d x +c \right )}}+\frac {a \left (\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 )+\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 )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(544\) |
| default | \(\text {Expression too large to display}\) | \(1235\) |
-a/d*2^(1/2)*(2*(-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)-(-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))*cos(d *x+c)+2*(-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))-(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*(-I*(I+c ot(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))+2^(1/2)*cos(d*x+c)-2^(1/ 2))/(e*csc(d*x+c))^(1/2)*csc(d*x+c)+a/d*(arctanh((sin(d*x+c)/(cos(d*x+c)+1 )^2)^(1/2)*(cot(d*x+c)+csc(d*x+c)))+arctan((sin(d*x+c)/(cos(d*x+c)+1)^2)^( 1/2)*(cot(d*x+c)+csc(d*x+c))))/(cos(d*x+c)+1)/(e*csc(d*x+c))^(1/2)/(sin(d* x+c)/(cos(d*x+c)+1)^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 554, normalized size of antiderivative = 4.54 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\left [-\frac {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} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) + e\right )}}\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 (\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 ) - 8 \, 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 ) - 8 \, 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 )}{8 \, d e}, \frac {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} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) - e\right )}}\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 (\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 ) + 8 \, 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 ) + 8 \, 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 )}{8 \, d e}\right ] \]
[-1/8*(2*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)) + 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)) - 8*a*sqrt(2*I*e)*weier strassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) - 8*a*sqrt(-2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d* x + c) - I*sin(d*x + c))))/(d*e), 1/8*(2*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 )) + 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)) + 8*a*sqrt(2*I*e)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos( d*x + c) + I*sin(d*x + c))) + 8*a*sqrt(-2*I*e)*weierstrassZeta(4, 0, weier strassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))))/(d*e)]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=a \left (\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx\right ) \]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \]
\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}} \,d x \]