Integrand size = 23, antiderivative size = 453 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {b c \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d^2 \sqrt {c^2 x^2}}-\frac {b c \left (12 c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{225 d x^2 \sqrt {c^2 x^2}}-\frac {b c \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}+\frac {b c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{225 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
-1/5*(e*x^2+d)^(3/2)*(a+b*arccsc(c*x))/d/x^5+2/15*e*(e*x^2+d)^(3/2)*(a+b*a rccsc(c*x))/d^2/x^3+2/15*b*c*e^2*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^ 2*x^2)^(1/2)-1/45*b*c*e*(2*c^2*d+e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d^2/ (c^2*x^2)^(1/2)-1/75*b*c*(8*c^4*d^2+3*c^2*d*e-2*e^2)*(c^2*x^2-1)^(1/2)*(e* x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)-1/25*b*c*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2 )/x^4/(c^2*x^2)^(1/2)-1/45*b*c*e*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^2/( c^2*x^2)^(1/2)-1/75*b*c*(4*c^2*d+e)*(c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/d/x^ 2/(c^2*x^2)^(1/2)-2/15*b*c^2*e^2*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2*x ^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x^2 /d)^(1/2)+1/45*b*c^2*e*(2*c^2*d+e)*x*EllipticE(c*x,(-e/c^2/d)^(1/2))*(-c^2 *x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2-1)^(1/2)/(1+e*x ^2/d)^(1/2)+1/75*b*c^2*(8*c^4*d^2+3*c^2*d*e-2*e^2)*x*EllipticE(c*x,(-e/c^2 /d)^(1/2))*(-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c^2*x^2 -1)^(1/2)/(1+e*x^2/d)^(1/2)-1/75*b*c^2*(8*c^2*d-e)*(c^2*d+e)*x*EllipticF(c *x,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d/(c^2*x^2)^(1/2 )/(c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)-2/45*b*c^2*e*(c^2*d+e)*x*EllipticF(c*x ,(-e/c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d/(c^2*x^2)^(1/2)/ (c^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)+2/15*b*e^2*(c^2*d+e)*x*EllipticF(c*x,(-e /c^2/d)^(1/2))*(-c^2*x^2+1)^(1/2)*(1+e*x^2/d)^(1/2)/d^2/(c^2*x^2)^(1/2)/(c ^2*x^2-1)^(1/2)/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 10.69 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=-\frac {\sqrt {d+e x^2} \left (15 a \left (3 d^2+d e x^2-2 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (-31 e^2 x^4+d e x^2 \left (8+19 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )+15 b \left (3 d^2+d e x^2-2 e^2 x^4\right ) \csc ^{-1}(c x)\right )}{225 d^2 x^5}+\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
-1/225*(Sqrt[d + e*x^2]*(15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-31*e^2*x^4 + d*e*x^2*(8 + 19*c^2*x^2) + 3*d^2*(3 + 4*c^2 *x^2 + 8*c^4*x^4)) + 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcCsc[c*x]))/(d^2 *x^5) + ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2 4*c^4*d^2 + 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c ^2*d))] + (-24*c^6*d^3 - 31*c^4*d^2*e + 23*c^2*d*e^2 + 30*e^3)*EllipticF[I *ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]* Sqrt[d + e*x^2])
Time = 0.86 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5762, 27, 442, 25, 442, 25, 445, 27, 399, 323, 323, 321, 331, 330, 327}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int -\frac {\left (3 d-2 e x^2\right ) \left (e x^2+d\right )^{3/2}}{15 d^2 x^6 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {\left (3 d-2 e x^2\right ) \left (e x^2+d\right )^{3/2}}{x^6 \sqrt {c^2 x^2-1}}dx}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b c x \left (\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}-\frac {1}{5} \int -\frac {\sqrt {e x^2+d} \left (\left (3 c^2 d-10 e\right ) e x^2+d \left (12 c^2 d-e\right )\right )}{x^4 \sqrt {c^2 x^2-1}}dx\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \int \frac {\sqrt {e x^2+d} \left (\left (3 c^2 d-10 e\right ) e x^2+d \left (12 c^2 d-e\right )\right )}{x^4 \sqrt {c^2 x^2-1}}dx+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 442 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}-\frac {1}{3} \int -\frac {2 e \left (6 d^2 c^4+4 d e c^2-15 e^2\right ) x^2+d \left (24 d^2 c^4+19 d e c^2-31 e^2\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {2 e \left (6 d^2 c^4+4 d e c^2-15 e^2\right ) x^2+d \left (24 d^2 c^4+19 d e c^2-31 e^2\right )}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {\int \frac {d e \left (2 \left (6 d^2 c^4+4 d e c^2-15 e^2\right )-c^2 \left (24 d^2 c^4+19 d e c^2-31 e^2\right ) x^2\right )}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{d}+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \int \frac {2 \left (6 d^2 c^4+4 d e c^2-15 e^2\right )-c^2 \left (24 d^2 c^4+19 d e c^2-31 e^2\right ) x^2}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}dx}{e}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c^2 \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}+\frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 e \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{5 d x^5}-\frac {b c x \left (\frac {1}{5} \left (\frac {1}{3} \left (e \left (\frac {\sqrt {1-c^2 x^2} \left (c^2 d+e\right ) \left (24 c^4 d^2+7 c^2 d e-30 e^2\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {c \sqrt {1-c^2 x^2} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}\right )+\frac {\sqrt {c^2 x^2-1} \left (24 c^4 d^2+19 c^2 d e-31 e^2\right ) \sqrt {d+e x^2}}{x}\right )+\frac {d \sqrt {c^2 x^2-1} \left (12 c^2 d-e\right ) \sqrt {d+e x^2}}{3 x^3}\right )+\frac {3 d \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{5 x^5}\right )}{15 d^2 \sqrt {c^2 x^2}}\) |
-1/5*((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/(d*x^5) + (2*e*(d + e*x^2)^(3 /2)*(a + b*ArcCsc[c*x]))/(15*d^2*x^3) - (b*c*x*((3*d*Sqrt[-1 + c^2*x^2]*(d + e*x^2)^(3/2))/(5*x^5) + ((d*(12*c^2*d - e)*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])/(3*x^3) + (((24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[-1 + c^2*x^2]* Sqrt[d + e*x^2])/x + e*(-((c*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*Sqrt[1 - c ^2*x^2]*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[-1 + c^2*x^2]*Sqrt[1 + (e*x^2)/d])) + ((c^2*d + e)*(24*c^4*d^2 + 7*c^2*d*e - 3 0*e^2)*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c ^2*d))])/(c*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])))/3)/5))/(15*d^2*Sqrt[c^ 2*x^2])
3.2.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1)) Int[(g*x) ^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 *(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x ], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) | | (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
\[\int \frac {\left (a +b \,\operatorname {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{6}}d x\]
Time = 0.11 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\frac {{\left (30 \, a c d e^{2} x^{4} - 15 \, a c d^{2} e x^{2} - 45 \, a c d^{3} + 15 \, {\left (2 \, b c d e^{2} x^{4} - b c d^{2} e x^{2} - 3 \, b c d^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (9 \, b c d^{3} + {\left (24 \, b c^{5} d^{3} + 19 \, b c^{3} d^{2} e - 31 \, b c d e^{2}\right )} x^{4} + 4 \, {\left (3 \, b c^{3} d^{3} + 2 \, b c d^{2} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (24 \, b c^{8} d^{3} + 19 \, b c^{6} d^{2} e - 31 \, b c^{4} d e^{2}\right )} x^{5} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (24 \, b c^{8} d^{3} + {\left (19 \, b c^{6} + 12 \, b c^{4}\right )} d^{2} e - {\left (31 \, b c^{4} - 8 \, b c^{2}\right )} d e^{2} - 30 \, b e^{3}\right )} x^{5} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{225 \, c d^{3} x^{5}} \]
1/225*((30*a*c*d*e^2*x^4 - 15*a*c*d^2*e*x^2 - 45*a*c*d^3 + 15*(2*b*c*d*e^2 *x^4 - b*c*d^2*e*x^2 - 3*b*c*d^3)*arccsc(c*x) - (9*b*c*d^3 + (24*b*c^5*d^3 + 19*b*c^3*d^2*e - 31*b*c*d*e^2)*x^4 + 4*(3*b*c^3*d^3 + 2*b*c*d^2*e)*x^2) *sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d) - ((24*b*c^8*d^3 + 19*b*c^6*d^2*e - 31 *b*c^4*d*e^2)*x^5*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (24*b*c^8*d^3 + (1 9*b*c^6 + 12*b*c^4)*d^2*e - (31*b*c^4 - 8*b*c^2)*d*e^2 - 30*b*e^3)*x^5*ell iptic_f(arcsin(c*x), -e/(c^2*d)))*sqrt(-d))/(c*d^3*x^5)
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{6}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^6} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \]