\(\int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx\) [191]

Optimal result

Integrand size = 25, antiderivative size = 765 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt {a} d}-\frac {\sqrt {b} \cos (c+d x) \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt {a+b} d \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )}-\frac {\sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\sqrt [4]{b} (a+b)^{3/4} \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{2 a \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac {\left (\sqrt {b}-\sqrt {a+b}\right )^2 \left (\sqrt {a+b}+\sqrt {b} \cos ^2(c+d x)\right ) \sqrt {\frac {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{\left (\sqrt {a+b}+\sqrt {b} \cos ^2(c+d x)\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (1+\frac {\sqrt {b}}{\sqrt {a+b}}\right )\right )}{8 a \sqrt [4]{b} \sqrt [4]{a+b} d \sqrt {a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}} \] Output:

-1/4*arctanh(a^(1/2)*cos(d*x+c)/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2 
))/a^(1/2)/d-1/2*b^(1/2)*cos(d*x+c)*(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^ 
(1/2)/a/(a+b)^(1/2)/d/(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/2))-1/2*(a+b-2*b*co 
s(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)*cot(d*x+c)*csc(d*x+c)/a/d+1/2*b^(1/4)*(a+ 
b)^(3/4)*(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/2))*((a+b-2*b*cos(d*x+c)^2+b*cos 
(d*x+c)^4)/(a+b)/(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/2))^2)^(1/2)*EllipticE(s 
in(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/2*(2+2*b^(1/2)/(a+b)^(1/2)) 
^(1/2))/a/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)-1/2*b^(1/4)*(a+b-b 
^(1/2)*(a+b)^(1/2))*(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/2))*((a+b-2*b*cos(d*x 
+c)^2+b*cos(d*x+c)^4)/(a+b)/(1+b^(1/2)*cos(d*x+c)^2/(a+b)^(1/2))^2)^(1/2)* 
InverseJacobiAM(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4)),1/2*(2+2*b^(1/2)/ 
(a+b)^(1/2))^(1/2))/a/(a+b)^(1/4)/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^ 
(1/2)-1/8*(b^(1/2)-(a+b)^(1/2))^2*((a+b)^(1/2)+b^(1/2)*cos(d*x+c)^2)*((a+b 
-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)/((a+b)^(1/2)+b^(1/2)*cos(d*x+c)^2)^2)^(1 
/2)*EllipticPi(sin(2*arctan(b^(1/4)*cos(d*x+c)/(a+b)^(1/4))),1/4*(b^(1/2)+ 
(a+b)^(1/2))^2/b^(1/2)/(a+b)^(1/2),1/2*(2+2*b^(1/2)/(a+b)^(1/2))^(1/2))/a/ 
b^(1/4)/(a+b)^(1/4)/d/(a+b-2*b*cos(d*x+c)^2+b*cos(d*x+c)^4)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 26.14 (sec) , antiderivative size = 1442, normalized size of antiderivative = 1.88 \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx =\text {Too large to display} \] Input:

Integrate[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]
 

Output:

-1/4*(Sqrt[8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)]]*Cot[c + 
d*x]*Csc[c + d*x])/(Sqrt[2]*a*d) + (Sec[c + d*x]*(-a - 2*a*Tan[c + d*x]^2 
- a*Tan[c + d*x]^4 - b*Tan[c + d*x]^4 - (b*(Sqrt[b]*EllipticE[ArcSin[Sqrt[ 
(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a] 
*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + I*(Sqrt[a] + I*S 
qrt[b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 
 + b*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - 
I*Sqrt[b])])*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]] 
*Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(S 
qrt[a]*Sqrt[b])]*(Sqrt[b]*Tan[c + d*x]^2 + I*Sqrt[a]*(1 + Tan[c + d*x]^2)) 
)/((a + b)*Sqrt[((-I)*(a + I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c 
+ d*x]^2))/(Sqrt[a]*Sqrt[b])]) + (a*((-I)*Sqrt[b]*EllipticE[ArcSin[Sqrt[(I 
*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[a]*S 
qrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sqrt[b])] + (Sqrt[a] + I*Sqrt[ 
b])*EllipticF[ArcSin[Sqrt[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b 
*Tan[c + d*x]^2))/(Sqrt[a]*Sqrt[b])]/Sqrt[2]], (2*Sqrt[a])/(Sqrt[a] - I*Sq 
rt[b])])*Sqrt[(((-I)*Sqrt[a] + Sqrt[b])*(1 + Tan[c + d*x]^2))/Sqrt[b]]*Sqr 
t[(I*(a - I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[c + d*x]^2))/(Sqrt[ 
a]*Sqrt[b])]*((-I)*Sqrt[b]*Tan[c + d*x]^2 + Sqrt[a]*(1 + Tan[c + d*x]^2))) 
/((a + b)*Sqrt[((-I)*(a + I*Sqrt[a]*Sqrt[b] + a*Tan[c + d*x]^2 + b*Tan[...
 

Rubi [A] (warning: unable to verify)

Time = 1.36 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3694, 1551, 25, 2232, 25, 27, 1509, 2226, 1416, 2222}

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.

\(\displaystyle \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (c+d x)^3 \sqrt {a+b \sin (c+d x)^4}}dx\)

\(\Big \downarrow \) 3694

\(\displaystyle -\frac {\int \frac {1}{\left (1-\cos ^2(c+d x)\right )^2 \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{d}\)

\(\Big \downarrow \) 1551

\(\displaystyle -\frac {\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}-\frac {\int -\frac {-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)+a-b}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 2232

\(\displaystyle -\frac {\frac {-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\frac {\int -\frac {b \left (\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}\right )}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{b}}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\int \frac {b \left (\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}\right )}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{b}-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \int \frac {1-\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\frac {\int \frac {\sqrt {b} \left (\sqrt {b}-\sqrt {a+b}\right ) \cos ^2(c+d x)+a-b+\sqrt {b} \sqrt {a+b}}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 2226

\(\displaystyle -\frac {\frac {\frac {2 \sqrt {b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \int \frac {1}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)}{\sqrt {a+b}}-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\frac {-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \int \frac {\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1}{\left (1-\cos ^2(c+d x)\right ) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}d\cos (c+d x)+\frac {\sqrt [4]{b} \left (-\sqrt {b} \sqrt {a+b}+a+b\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{a+b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )}{2 a}+\frac {\cos (c+d x) \sqrt {a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a \left (1-\cos ^2(c+d x)\right )}}{d}\)

\(\Big \downarrow \) 2222

\(\displaystyle -\frac {\frac {\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b} \cos (c+d x)}{2 a \left (1-\cos ^2(c+d x)\right )}+\frac {-\sqrt {b} \sqrt {a+b} \left (\frac {\sqrt [4]{a+b} \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}-\frac {\cos (c+d x) \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )}\right )+\frac {\sqrt [4]{b} \left (a+b-\sqrt {b} \sqrt {a+b}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{\sqrt [4]{a+b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}-\sqrt {a+b} \left (\sqrt {b}-\sqrt {a+b}\right ) \left (\frac {\left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right ) \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}\right )}{2 \sqrt {a}}+\frac {\sqrt [4]{a+b} \left (1-\frac {\sqrt {b}}{\sqrt {a+b}}\right ) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right ) \sqrt {\frac {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}{(a+b) \left (\frac {\sqrt {b} \cos ^2(c+d x)}{\sqrt {a+b}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b}+\sqrt {a+b}\right )^2}{4 \sqrt {b} \sqrt {a+b}},2 \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right ),\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a+b}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt {b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+a+b}}\right )}{2 a}}{d}\)

Input:

Int[Csc[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]^4],x]
 

Output:

-(((Cos[c + d*x]*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/(2*a 
*(1 - Cos[c + d*x]^2)) + (-(Sqrt[b]*Sqrt[a + b]*(-((Cos[c + d*x]*Sqrt[a + 
b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])/((a + b)*(1 + (Sqrt[b]*Cos[c + 
 d*x]^2)/Sqrt[a + b]))) + ((a + b)^(1/4)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqr 
t[a + b])*Sqrt[(a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 
 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])^2)]*EllipticE[2*ArcTan[(b^(1/4)*C 
os[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + b])/2])/(b^(1/4)*Sqrt[a 
 + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4]))) + (b^(1/4)*(a + b - Sqrt[ 
b]*Sqrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2 
*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^ 
2)/Sqrt[a + b])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4 
)], (1 + Sqrt[b]/Sqrt[a + b])/2])/((a + b)^(1/4)*Sqrt[a + b - 2*b*Cos[c + 
d*x]^2 + b*Cos[c + d*x]^4]) - Sqrt[a + b]*(Sqrt[b] - Sqrt[a + b])*(((1 + S 
qrt[b]/Sqrt[a + b])*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + b - 2*b*Cos[c 
+ d*x]^2 + b*Cos[c + d*x]^4]])/(2*Sqrt[a]) + ((a + b)^(1/4)*(1 - Sqrt[b]/S 
qrt[a + b])*(1 + (Sqrt[b]*Cos[c + d*x]^2)/Sqrt[a + b])*Sqrt[(a + b - 2*b*C 
os[c + d*x]^2 + b*Cos[c + d*x]^4)/((a + b)*(1 + (Sqrt[b]*Cos[c + d*x]^2)/S 
qrt[a + b])^2)]*EllipticPi[(Sqrt[b] + Sqrt[a + b])^2/(4*Sqrt[b]*Sqrt[a + b 
]), 2*ArcTan[(b^(1/4)*Cos[c + d*x])/(a + b)^(1/4)], (1 + Sqrt[b]/Sqrt[a + 
b])/2])/(4*b^(1/4)*Sqrt[a + b - 2*b*Cos[c + d*x]^2 + b*Cos[c + d*x]^4])...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1551

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_ 
Symbol] :> Simp[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + b*x^2 + c*x^4]/(2*d* 
(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 - b*d*e 
+ a*e^2))   Int[((d + e*x^2)^(q + 1)/Sqrt[a + b*x^2 + c*x^4])*Simp[a*e^2*(2 
*q + 3) + 2*d*(c*d - b*e)*(q + 1) - 2*e*(c*d*(q + 1) - b*e*(q + 2))*x^2 + c 
*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4* 
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[q, -1]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 2226

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(A*(c*d + a*e*q) 
- a*B*(e + d*q))/(c*d^2 - a*e^2)   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + 
Simp[a*(B*d - A*e)*((e + d*q)/(c*d^2 - a*e^2))   Int[(1 + q*x^2)/((d + e*x^ 
2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
 

rule 2232

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x 
, 2], C = Coeff[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + b*x^2 
 + c*x^4], x], x] + Simp[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - 
 a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] 
&&  !GtQ[b^2 - 4*a*c, 0]
 

rule 3042

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

rule 3694

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, 
 x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 
 1)/2]
 

Maple [F]

\[\int \frac {\csc \left (d x +c \right )^{3}}{\sqrt {a +b \sin \left (d x +c \right )^{4}}}d x\]

Input:

int(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x)
 

Output:

int(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x)
 

Fricas [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \] Input:

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="fricas")
 

Output:

integral(csc(d*x + c)^3/sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b 
), x)
 

Sympy [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \] Input:

integrate(csc(d*x+c)**3/(a+b*sin(d*x+c)**4)**(1/2),x)
 

Output:

Integral(csc(c + d*x)**3/sqrt(a + b*sin(c + d*x)**4), x)
 

Maxima [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \] Input:

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="maxima")
 

Output:

integrate(csc(d*x + c)^3/sqrt(b*sin(d*x + c)^4 + a), x)
 

Giac [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{\sqrt {b \sin \left (d x + c\right )^{4} + a}} \,d x } \] Input:

integrate(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")
 

Output:

sage0*x
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a}} \,d x \] Input:

int(1/(sin(c + d*x)^3*(a + b*sin(c + d*x)^4)^(1/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/(sin(c + d*x)^3*(a + b*sin(c + d*x)^4)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {\csc ^3(c+d x)}{\sqrt {a+b \sin ^4(c+d x)}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right )^{4} b +a}\, \csc \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{4} b +a}d x \] Input:

int(csc(d*x+c)^3/(a+b*sin(d*x+c)^4)^(1/2),x)
 

Output:

int((sqrt(sin(c + d*x)**4*b + a)*csc(c + d*x)**3)/(sin(c + d*x)**4*b + a), 
x)