3.1.48 \(\int \frac {x}{(a+b \csc (c+d \sqrt {x}))^2} \, dx\) [48]

3.1.48.1 Optimal result

Integrand size = 18, antiderivative size = 1565 \[ \int \frac {x}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

output

12*b^2*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a^2-b^2)^(1/2)))/a^2/(a^2-b 
^2)/d^4+12*b^2*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))/a^ 
2/(a^2-b^2)/d^4+12*b^3*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1 
/2)))/a^2/(-a^2+b^2)^(3/2)/d^4-12*b^3*polylog(4,I*a*exp(I*(c+d*x^(1/2)))/( 
b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4-24*b*polylog(4,I*a*exp(I*(c+ 
d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^4/(-a^2+b^2)^(1/2)+24*b*polylog(4, 
I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^4/(-a^2+b^2)^(1/2)-2* 
b^2*x^(3/2)*cos(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*sin(c+d*x^(1/2)))-2*I*b^3* 
x^(3/2)*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2) 
^(3/2)/d-4*I*b*x^(3/2)*ln(1-I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2))) 
/a^2/d/(-a^2+b^2)^(1/2)-12*I*b^2*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(I*b-(a 
^2-b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^3-12*I*b^2*polylog(2,-a*exp(I*(c+d 
*x^(1/2)))/(I*b+(a^2-b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^3-12*I*b^3*polyl 
og(3,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(-a^2+b^2) 
^(3/2)/d^3-24*I*b*polylog(3,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2))) 
*x^(1/2)/a^2/d^3/(-a^2+b^2)^(1/2)+1/2*x^2/a^2-6*b^3*x*polylog(2,I*a*exp(I* 
(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2+6*b^3*x*poly 
log(2,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/ 
d^2+12*b*x*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^ 
2/(-a^2+b^2)^(1/2)-12*b*x*polylog(2,I*a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b...
 

3.1.48.2 Mathematica [A] (warning: unable to verify)

Time = 15.34 (sec) , antiderivative size = 1752, normalized size of antiderivative = 1.12 \[ \int \frac {x}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

input

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

output

(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(x^2*(b + a*Sin[c + d*Sqr 
t[x]]) - ((4*I)*b*E^(I*c)*(2*b*E^(I*c)*x^(3/2) + ((-1 + E^((2*I)*c))*((3*I 
)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])) 
)/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^3*E^(I*c)*x 
^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2) 
*E^((2*I)*c)])] + I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[ 
x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (3*I)*b*d^2*Sqrt[(a 
^2 - b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) 
+ Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + 
(a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] 
 - I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I 
*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*d*(2*b*Sqrt[(a^2 - b^2)*E^((2*I) 
*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, 
 (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c) 
])] + 3*d*(2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b 
^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I* 
b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (6*I)*b*Sqrt[(a^2 - b^2)*E^ 
((2*I)*c)]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a 
^2 - b^2)*E^((2*I)*c)])] - (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^ 
(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] +...
 

3.1.48.3 Rubi [A] (verified)

Time = 2.63 (sec) , antiderivative size = 1567, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4693, 3042, 4679, 2009}

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.

\(\displaystyle \int \frac {x}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4693

\(\displaystyle 2 \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {i x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {i x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {3 x \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {3 x \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {6 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {i x^{3/2} b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {3 x \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b-\sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {3 x \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{i b+\sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b-\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{i b+\sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {x^{3/2} \cos \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \sin \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {2 i x^{3/2} \log \left (1-\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {6 x \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}-\frac {6 x \operatorname {PolyLog}\left (2,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {12 i \sqrt {x} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 i \sqrt {x} \operatorname {PolyLog}\left (3,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {12 \operatorname {PolyLog}\left (4,\frac {i a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {x^2}{4 a^2}\right )\)

input

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

output

2*(((-I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(4*a^2) + (3*b^2*x*Log[1 + 
 (a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) 
 + (3*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2])])/(a 
^2*(a^2 - b^2)*d^2) - (I*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/( 
b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^(3/2)*Log[ 
1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + 
b^2]*d) + (I*b^3*x^(3/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^ 
2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((2*I)*b*x^(3/2)*Log[1 - (I*a*E^( 
I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - (( 
6*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(I*b - Sqrt[a^2 - 
b^2]))])/(a^2*(a^2 - b^2)*d^3) - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*( 
c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (3*b^3 
*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(- 
a^2 + b^2)^(3/2)*d^2) + (6*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - 
 Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (3*b^3*x*PolyLog[2, (I*a 
*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d 
^2) - (6*b*x*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]) 
])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x] 
)))/(I*b - Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (6*b^2*PolyLog[3, - 
((a*E^(I*(c + d*Sqrt[x])))/(I*b + Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*...
 

3.1.48.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 4693

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 
 1)/n], 0] && IntegerQ[p]
 

3.1.48.4 Maple [F]

\[\int \frac {x}{\left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}}d x\]

input

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

output

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

3.1.48.5 Fricas [F]

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

input

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

output

integral(x/(b^2*csc(d*sqrt(x) + c)^2 + 2*a*b*csc(d*sqrt(x) + c) + a^2), x)
 

3.1.48.6 Sympy [F]

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

input

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

output

Integral(x/(a + b*csc(c + d*sqrt(x)))**2, x)
 

3.1.48.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x}{\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

3.1.48.8 Giac [F]

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

input

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

output

integrate(x/(b*csc(d*sqrt(x) + c) + a)^2, x)
 

3.1.48.9 Mupad [F(-1)]

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

input

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

output

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