\(\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx\) [397]

Optimal result

Integrand size = 33, antiderivative size = 240 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=-\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \] Output:

-2*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(a,c)),2^(1/2)*((a^2+c^2)^(1/2)/( 
b+(a^2+c^2)^(1/2)))^(1/2))*(b+c*cos(e*x+d)+a*sin(e*x+d))^2/(a^2-b^2+c^2)/e 
/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2)/((b+c*cos(e*x+d)+a*s 
in(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)-2*(b+c*cos(e*x+d)+a*sin(e*x+d))*(a*c 
os(e*x+d)-c*sin(e*x+d))/(a^2-b^2+c^2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2 
)/sin(e*x+d)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.58 (sec) , antiderivative size = 5959, normalized size of antiderivative = 24.83 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2 
)),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3042, 3643, 3042, 3607, 3042, 3598, 3042, 3132}

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.

\(\displaystyle \int \frac {1}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (d+e x)^{3/2} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}dx\)

\(\Big \downarrow \) 3643

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3607

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {\sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{3/2} \left (-\frac {2 \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}\right )}{\sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}\)

Input:

Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)),x]
 

Output:

((b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*((-2*EllipticE[(d + e*x - Arc 
Tan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[b + c*Cos[d 
+ e*x] + a*Sin[d + e*x]])/((a^2 - b^2 + c^2)*e*Sqrt[(b + c*Cos[d + e*x] + 
a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*(a*Cos[d + e*x] - c*Sin[d + e 
*x]))/((a^2 - b^2 + c^2)*e*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]])))/(( 
a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + 
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]   Int[Sqrt[a/(a + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] 
 && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3607

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(-3/2), x_Symbol] :> Simp[2*((c*Cos[d + e*x] - b*Sin[d + e*x])/(e*(a^2 - b^ 
2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] + Simp[1/(a^2 - b^ 
2 - c^2)   Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{ 
a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
 

rule 3643

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.)) 
^(n_)*sin[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[Sin[d + e*x]^n*((a + 
b*Csc[d + e*x] + c*Cot[d + e*x])^n/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n) 
   Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d 
, e}, x] &&  !IntegerQ[n]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.45 (sec) , antiderivative size = 12778, normalized size of antiderivative = 53.24

method	result	size
default	\(\text {Expression too large to display}\)	\(12778\)

Input:

int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x,method=_RETUR 
NVERBOSE)
 

Output:

result too large to display
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 1710, normalized size of antiderivative = 7.12 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x, algori 
thm="fricas")
 

Output:

-2/3*((a*b^2 + I*b^2*c + (a*b*c + I*b*c^2)*cos(e*x + d) + (a^2*b + I*a*b*c 
)*sin(e*x + d))*sqrt(1/2*I*a + 1/2*c)*weierstrassPInverse(4/3*(3*a^4 - 4*a 
^2*b^2 + 4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2 
*a^2*c^2 + c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 
+ 2*(9*a^2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a 
^2*b^3)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(-2*I*a*b + 2*b*c + 3* 
(a^2 + c^2)*cos(e*x + d) - 3*(I*a^2 + I*c^2)*sin(e*x + d))/(a^2 + c^2)) + 
(a*b^2 - I*b^2*c + (a*b*c - I*b*c^2)*cos(e*x + d) + (a^2*b - I*a*b*c)*sin( 
e*x + d))*sqrt(-1/2*I*a + 1/2*c)*weierstrassPInverse(4/3*(3*a^4 - 4*a^2*b^ 
2 + 4*b^2*c^2 - 6*I*a*c^3 - 3*c^4 - 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2* 
c^2 + c^4), -8/27*(9*I*a^5*b - 8*I*a^3*b^3 - 27*I*a*b*c^4 - 9*b*c^5 + 2*(9 
*a^2*b + 4*b^3)*c^3 - 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3 
)*c)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), 1/3*(2*I*a*b + 2*b*c + 3*(a^2 + 
c^2)*cos(e*x + d) - 3*(-I*a^2 - I*c^2)*sin(e*x + d))/(a^2 + c^2)) + 3*(-I* 
a^2*b - I*b*c^2 + (-I*a^2*c - I*c^3)*cos(e*x + d) + (-I*a^3 - I*a*c^2)*sin 
(e*x + d))*sqrt(1/2*I*a + 1/2*c)*weierstrassZeta(4/3*(3*a^4 - 4*a^2*b^2 + 
4*b^2*c^2 + 6*I*a*c^3 - 3*c^4 + 2*I*(3*a^3 - 4*a*b^2)*c)/(a^4 + 2*a^2*c^2 
+ c^4), -8/27*(-9*I*a^5*b + 8*I*a^3*b^3 + 27*I*a*b*c^4 - 9*b*c^5 + 2*(9*a^ 
2*b + 4*b^3)*c^3 + 6*I*(3*a^3*b - 4*a*b^3)*c^2 + 3*(9*a^4*b - 8*a^2*b^3)*c 
)/(a^6 + 3*a^4*c^2 + 3*a^2*c^4 + c^6), weierstrassPInverse(4/3*(3*a^4 -...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\text {Timed out} \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))**(3/2)/sin(e*x+d)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x, algori 
thm="maxima")
 

Output:

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(3/2)*sin(e*x + d)^(3/2 
)), x)
 

Giac [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x, algori 
thm="giac")
 

Output:

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(3/2)*sin(e*x + d)^(3/2 
)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int \frac {1}{{\sin \left (d+e\,x\right )}^{3/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(1/(sin(d + e*x)^(3/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(3/2)),x)
 

Output:

int(1/(sin(d + e*x)^(3/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx=\int \frac {\sqrt {\sin \left (e x +d \right )}\, \sqrt {a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )}}{\cot \left (e x +d \right )^{2} \sin \left (e x +d \right )^{2} c^{2}+2 \cot \left (e x +d \right ) \csc \left (e x +d \right ) \sin \left (e x +d \right )^{2} b c +2 \cot \left (e x +d \right ) \sin \left (e x +d \right )^{2} a c +\csc \left (e x +d \right )^{2} \sin \left (e x +d \right )^{2} b^{2}+2 \csc \left (e x +d \right ) \sin \left (e x +d \right )^{2} a b +\sin \left (e x +d \right )^{2} a^{2}}d x \] Input:

int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x)
 

Output:

int((sqrt(sin(d + e*x))*sqrt(cot(d + e*x)*c + csc(d + e*x)*b + a))/(cot(d 
+ e*x)**2*sin(d + e*x)**2*c**2 + 2*cot(d + e*x)*csc(d + e*x)*sin(d + e*x)* 
*2*b*c + 2*cot(d + e*x)*sin(d + e*x)**2*a*c + csc(d + e*x)**2*sin(d + e*x) 
**2*b**2 + 2*csc(d + e*x)*sin(d + e*x)**2*a*b + sin(d + e*x)**2*a**2),x)