\(\int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx\) [398]

Optimal result

Integrand size = 33, antiderivative size = 492 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\frac {8 b E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)}+\frac {8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \] Output:

8/3*b*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))^3/(a^2-b^2+c^2 
)^2/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2)/((b+c*cos(e*x+d 
)+a*sin(e*x+d))/(b+(a^2+c^2)^(1/2)))^(1/2)+2/3*InverseJacobiAM(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*((b+c*cos(e*x+d)+a*sin(e*x+d))/(b+(a^2+c^2)^ 
(1/2)))^(1/2)/(a^2-b^2+c^2)/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+ 
d)^(5/2)-2/3*(b+c*cos(e*x+d)+a*sin(e*x+d))*(a*cos(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))^(5/2)/sin(e*x+d)^(5/2)+8/3*(b+c 
*cos(e*x+d)+a*sin(e*x+d))^2*(a*b*cos(e*x+d)-b*c*sin(e*x+d))/(a^2-b^2+c^2)^ 
2/e/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 17.34 (sec) , antiderivative size = 6066, normalized size of antiderivative = 12.33 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2 
)),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.85 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.90, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 3643, 3042, 3608, 27, 3042, 3635, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}

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.

\(\displaystyle \int \frac {1}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (d+e x)^{5/2} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}dx\)

\(\Big \downarrow \) 3643

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}}dx}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}}dx}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3608

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (\frac {2 \int -\frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{2 (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\int \frac {3 b-c \cos (d+e x)-a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}}dx}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3635

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {\frac {2 \int -\frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{2 \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\int \frac {a^2+4 b \sin (d+e x) a+3 b^2+c^2+4 b c \cos (d+e x)}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3628

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+4 b \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)}dx}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3598

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {4 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}dx}{\sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\left (a^2-b^2+c^2\right ) \int \frac {1}{\sqrt {b+c \cos (d+e x)+a \sin (d+e x)}}dx+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {\left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \sin \left (d+e x-\tan ^{-1}(c,a)+\frac {\pi }{2}\right )}{b+\sqrt {a^2+c^2}}}}dx}{\sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {(a \sin (d+e x)+b+c \cos (d+e x))^{5/2} \left (-\frac {-\frac {\frac {2 \left (a^2-b^2+c^2\right ) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right ),\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}+\frac {8 b \sqrt {a \sin (d+e x)+b+c \cos (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}}}}{a^2-b^2+c^2}-\frac {8 (a b \cos (d+e x)-b c \sin (d+e x))}{e \left (a^2-b^2+c^2\right ) \sqrt {a \sin (d+e x)+b+c \cos (d+e x)}}}{3 \left (a^2-b^2+c^2\right )}-\frac {2 (a \cos (d+e x)-c \sin (d+e x))}{3 e \left (a^2-b^2+c^2\right ) (a \sin (d+e x)+b+c \cos (d+e x))^{3/2}}\right )}{\sin ^{\frac {5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}\)

Input:

Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)),x]
 

Output:

((b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*((-2*(a*Cos[d + e*x] - c*Sin[ 
d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3 
/2)) - ((-8*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*Sq 
rt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]) - ((8*b*EllipticE[(d + e*x - ArcT 
an[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]])/(e*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b 
+ Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*EllipticF[(d + e*x - ArcTan[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])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[b + c*Cos[d + e*x] + a 
*Sin[d + e*x]]))/(a^2 - b^2 + c^2))/(3*(a^2 - b^2 + c^2))))/((a + c*Cot[d 
+ e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2))
 

Defintions of rubi rules used

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 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 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(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 3606

Int[1/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])/(a + Sq 
rt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]   Int[1/Sqrt[a/(a 
 + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - 
 ArcTan[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 3608

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

rule 3628

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

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

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 = 1.89 (sec) , antiderivative size = 41817, normalized size of antiderivative = 84.99

method	result	size
default	\(\text {Expression too large to display}\)	\(41817\)

Input:

int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x,method=_RETUR 
NVERBOSE)
                                                                                    
                                                                                    
 

Output:

result too large to display
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 2730, normalized size of antiderivative = 5.55 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algori 
thm="fricas")
 

Output:

-2/9*((3*a^5 + 4*a^3*b^2 + a*b^4 + 3*I*(a^2 + b^2)*c^3 + 3*(a^3 + a*b^2)*c 
^2 - (3*a^5 + a^3*b^2 - a*b^2*c^2 - I*b^2*c^3 - 3*a*c^4 - 3*I*c^5 + I*(3*a 
^4 + a^2*b^2)*c)*cos(e*x + d)^2 + I*(3*a^4 + 4*a^2*b^2 + b^4)*c + 2*(3*a*b 
*c^3 + 3*I*b*c^4 + I*(3*a^2*b + b^3)*c^2 + (3*a^3*b + a*b^3)*c)*cos(e*x + 
d) + 2*(3*a^4*b + a^2*b^3 + 3*a^2*b*c^2 + 3*I*a*b*c^3 + I*(3*a^3*b + a*b^3 
)*c + (3*a^2*c^3 + 3*I*a*c^4 + I*(3*a^3 + a*b^2)*c^2 + (3*a^4 + a^2*b^2)*c 
)*cos(e*x + d))*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*a^5 + 4*a^3*b^2 + a*b^4 - 3*I*(a^2 + b^2)*c^3 + 3*(a^3 + 
a*b^2)*c^2 - (3*a^5 + a^3*b^2 - a*b^2*c^2 + I*b^2*c^3 - 3*a*c^4 + 3*I*c^5 
- I*(3*a^4 + a^2*b^2)*c)*cos(e*x + d)^2 - I*(3*a^4 + 4*a^2*b^2 + b^4)*c + 
2*(3*a*b*c^3 - 3*I*b*c^4 - I*(3*a^2*b + b^3)*c^2 + (3*a^3*b + a*b^3)*c)*co 
s(e*x + d) + 2*(3*a^4*b + a^2*b^3 + 3*a^2*b*c^2 - 3*I*a*b*c^3 - I*(3*a^3*b 
 + a*b^3)*c + (3*a^2*c^3 - 3*I*a*c^4 - I*(3*a^3 + a*b^2)*c^2 + (3*a^4 + a^ 
2*b^2)*c)*cos(e*x + d))*sin(e*x + d))*sqrt(-1/2*I*a + 1/2*c)*weierstrassPI 
nverse(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*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\text {Timed out} \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))**(5/2)/sin(e*x+d)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {5}{2}} \sin \left (e x + d\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algori 
thm="maxima")
 

Output:

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(5/2)*sin(e*x + d)^(5/2 
)), x)
 

Giac [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int { \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {5}{2}} \sin \left (e x + d\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x, algori 
thm="giac")
 

Output:

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(5/2)*sin(e*x + d)^(5/2 
)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int \frac {1}{{\sin \left (d+e\,x\right )}^{5/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{5/2}} \,d x \] Input:

int(1/(sin(d + e*x)^(5/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(5/2)),x)
 

Output:

int(1/(sin(d + e*x)^(5/2)*(a + c*cot(d + e*x) + b/sin(d + e*x))^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac {5}{2}}(d+e x)} \, dx=\int \frac {\sqrt {\sin \left (e x +d \right )}\, \sqrt {a +c \cot \left (e x +d \right )+b \csc \left (e x +d \right )}}{\cot \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3} c^{3}+3 \cot \left (e x +d \right )^{2} \csc \left (e x +d \right ) \sin \left (e x +d \right )^{3} b \,c^{2}+3 \cot \left (e x +d \right )^{2} \sin \left (e x +d \right )^{3} a \,c^{2}+3 \cot \left (e x +d \right ) \csc \left (e x +d \right )^{2} \sin \left (e x +d \right )^{3} b^{2} c +6 \cot \left (e x +d \right ) \csc \left (e x +d \right ) \sin \left (e x +d \right )^{3} a b c +3 \cot \left (e x +d \right ) \sin \left (e x +d \right )^{3} a^{2} c +\csc \left (e x +d \right )^{3} \sin \left (e x +d \right )^{3} b^{3}+3 \csc \left (e x +d \right )^{2} \sin \left (e x +d \right )^{3} a \,b^{2}+3 \csc \left (e x +d \right ) \sin \left (e x +d \right )^{3} a^{2} b +\sin \left (e x +d \right )^{3} a^{3}}d x \] Input:

int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(5/2)/sin(e*x+d)^(5/2),x)
 

Output:

int((sqrt(sin(d + e*x))*sqrt(cot(d + e*x)*c + csc(d + e*x)*b + a))/(cot(d 
+ e*x)**3*sin(d + e*x)**3*c**3 + 3*cot(d + e*x)**2*csc(d + e*x)*sin(d + e* 
x)**3*b*c**2 + 3*cot(d + e*x)**2*sin(d + e*x)**3*a*c**2 + 3*cot(d + e*x)*c 
sc(d + e*x)**2*sin(d + e*x)**3*b**2*c + 6*cot(d + e*x)*csc(d + e*x)*sin(d 
+ e*x)**3*a*b*c + 3*cot(d + e*x)*sin(d + e*x)**3*a**2*c + csc(d + e*x)**3* 
sin(d + e*x)**3*b**3 + 3*csc(d + e*x)**2*sin(d + e*x)**3*a*b**2 + 3*csc(d 
+ e*x)*sin(d + e*x)**3*a**2*b + sin(d + e*x)**3*a**3),x)