\(\int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx\) [303]

Optimal result

Integrand size = 25, antiderivative size = 199 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {16 \cot (c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {4 \csc (c+d x)}{a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {4 \csc ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {28 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 a^2 d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]

[Out]

16/5*cot(d*x+c)/a^2/d/(e*csc(d*x+c))^(1/2)-2/5*cot(d*x+c)^3/a^2/d/(e*csc(d*x+c))^(1/2)-4*csc(d*x+c)/a^2/d/(e*c
sc(d*x+c))^(1/2)-2/5*cot(d*x+c)*csc(d*x+c)^2/a^2/d/(e*csc(d*x+c))^(1/2)+4/5*csc(d*x+c)^3/a^2/d/(e*csc(d*x+c))^
(1/2)-28/5*(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))/a^2/d/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3963, 3957, 2954, 2952, 2647, 2716, 2719, 2644, 14} \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {4 \csc ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {4 \csc (c+d x)}{a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {16 \cot (c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {28 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{5 a^2 d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]

[In]

Int[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2),x]

[Out]

(16*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) - (4*Csc[
c + d*x])/(a^2*d*Sqrt[e*Csc[c + d*x]]) - (2*Cot[c + d*x]*Csc[c + d*x]^2)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (4*C
sc[c + d*x]^3)/(5*a^2*d*Sqrt[e*Csc[c + d*x]]) + (28*EllipticE[(c - Pi/2 + d*x)/2, 2])/(5*a^2*d*Sqrt[e*Csc[c +
d*x]]*Sqrt[Sin[c + d*x]])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

Rule 2716

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] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2954

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Dist[(a/g)^(2*m), Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e +
f*x])^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]

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 \frac {\sqrt {\sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x) \sqrt {\sin (c+d x)}}{(-a-a \cos (c+d x))^2} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a^4 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)}\right ) \, dx}{a^4 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\int \frac {\cos ^4(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{5 a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {6 \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{5 a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \text {Subst}\left (\int \frac {1-x^2}{x^{7/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {16 \cot (c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {2 \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {12 \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {2 \text {Subst}\left (\int \left (\frac {1}{x^{7/2}}-\frac {1}{x^{3/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {16 \cot (c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}-\frac {4 \csc (c+d x)}{a^2 d \sqrt {e \csc (c+d x)}}-\frac {2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {4 \csc ^3(c+d x)}{5 a^2 d \sqrt {e \csc (c+d x)}}+\frac {28 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 a^2 d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ \end{align*}

Mathematica [C] (verified)

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

Time = 2.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {\csc (c+d x)} \sec ^2(c+d x) \left (-\frac {28 \sqrt {2} e^{i (c-d x)} \sqrt {\frac {i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left (3-3 e^{2 i (c+d x)}+e^{2 i d x} \left (1+e^{2 i c}\right ) \sqrt {1-e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )\right )}{1+e^{2 i c}}-3 \sqrt {\csc (c+d x)} \left ((-23+5 \cos (2 c)) \cos (d x) \sec (c)-2 \left (-10+\sec ^2\left (\frac {1}{2} (c+d x)\right )+5 \sin (c) \sin (d x)\right )\right )\right )}{15 a^2 d \sqrt {e \csc (c+d x)} (1+\sec (c+d x))^2} \]

[In]

Integrate[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2),x]

[Out]

(4*Cos[(c + d*x)/2]^4*Sqrt[Csc[c + d*x]]*Sec[c + d*x]^2*((-28*Sqrt[2]*E^(I*(c - d*x))*Sqrt[(I*E^(I*(c + d*x)))
/(-1 + E^((2*I)*(c + d*x)))]*(3 - 3*E^((2*I)*(c + d*x)) + E^((2*I)*d*x)*(1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c
+ d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))]))/(1 + E^((2*I)*c)) - 3*Sqrt[Csc[c + d*x]]*((-
23 + 5*Cos[2*c])*Cos[d*x]*Sec[c] - 2*(-10 + Sec[(c + d*x)/2]^2 + 5*Sin[c]*Sin[d*x]))))/(15*a^2*d*Sqrt[e*Csc[c
+ d*x]]*(1 + Sec[c + d*x])^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.70 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.31

method	result	size
default	\(-\frac {\sqrt {2}\, \left (28 \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 )}\, \cos \left (d x +c \right )^{2}-14 \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 )}\, \sqrt {i \left (-i+\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}+56 \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 )}\, \cos \left (d x +c \right )-28 \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 )}\, \sqrt {i \left (-i+\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 )+28 \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 )}-14 \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 )}\, \sqrt {i \left (-i+\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 )+5 \sqrt {2}\, \cos \left (d x +c \right )^{2}+\sqrt {2}\, \cos \left (d x +c \right )-6 \sqrt {2}\right ) \csc \left (d x +c \right )}{5 a^{2} d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \csc \left (d x +c \right )}}\)	\(659\)

[In]

int(1/(a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/a^2/d*2^(1/2)*(28*(-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)*cos(d*x+c)^2-14*(-I*(I+cot(d
*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-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+56*(-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)*cos(d*x+c)-28*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-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)+28*(-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)-14*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(-I*(cot(d*x+c)-csc(d*
x+c)))^(1/2)*(I*(-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))+
5*2^(1/2)*cos(d*x+c)^2+2^(1/2)*cos(d*x+c)-6*2^(1/2))/(cos(d*x+c)+1)/(e*csc(d*x+c))^(1/2)*csc(d*x+c)

Fricas [C] (verification not implemented)

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

Time = 0.10 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {2 \, {\left (7 \, \sqrt {2 i \, e} {\left (\cos \left (d x + c\right ) + 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 7 \, \sqrt {-2 i \, e} {\left (\cos \left (d x + c\right ) + 1\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (9 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 8\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{5 \, {\left (a^{2} d e \cos \left (d x + c\right ) + a^{2} d e\right )}} \]

[In]

integrate(1/(a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

2/5*(7*sqrt(2*I*e)*(cos(d*x + c) + 1)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x
+ c))) + 7*sqrt(-2*I*e)*(cos(d*x + c) + 1)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*s
in(d*x + c))) + (9*cos(d*x + c)^2 - cos(d*x + c) - 8)*sqrt(e/sin(d*x + c)))/(a^2*d*e*cos(d*x + c) + a^2*d*e)

Sympy [F]

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

[In]

integrate(1/(a+a*sec(d*x+c))**2/(e*csc(d*x+c))**(1/2),x)

[Out]

Integral(1/(sqrt(e*csc(c + d*x))*sec(c + d*x)**2 + 2*sqrt(e*csc(c + d*x))*sec(c + d*x) + sqrt(e*csc(c + d*x)))
, x)/a**2

Maxima [F]

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

[In]

integrate(1/(a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(e*csc(d*x + c))*(a*sec(d*x + c) + a)^2), x)

Giac [F]

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

[In]

integrate(1/(a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(e*csc(d*x + c))*(a*sec(d*x + c) + a)^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]

[In]

int(1/((a + a/cos(c + d*x))^2*(e/sin(c + d*x))^(1/2)),x)

[Out]

int(cos(c + d*x)^2/(a^2*(e/sin(c + d*x))^(1/2)*(cos(c + d*x) + 1)^2), x)