Integrand size = 19, antiderivative size = 134 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-a-b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{2 d e \sqrt {c^2 x^2}}+\frac {b c x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e} \sqrt {c^2 x^2}} \]
1/2*(-a-b*arccsc(c*x))/e/(e*x^2+d)-1/2*b*c*x*arctan((c^2*x^2-1)^(1/2))/d/e /(c^2*x^2)^(1/2)+1/2*b*c*x*arctan(e^(1/2)*(c^2*x^2-1)^(1/2)/(c^2*d+e)^(1/2 ))/d/e^(1/2)/(c^2*d+e)^(1/2)/(c^2*x^2)^(1/2)
Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.13 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a}{d+e x^2}+\frac {2 b \csc ^{-1}(c x)}{d+e x^2}-\frac {2 b \arcsin \left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \log \left (\frac {4 i d e-4 c d \sqrt {e} \left (c \sqrt {d}+i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x}{b \sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (-d e+c d \sqrt {e} \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}}{4 e} \]
-1/4*((2*a)/(d + e*x^2) + (2*b*ArcCsc[c*x])/(d + e*x^2) - (2*b*ArcSin[1/(c *x)])/d + (b*Sqrt[e]*Log[((4*I)*d*e - 4*c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-( c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x)/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*S qrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]) + (b*Sqrt[e]*Log[((4*I)*(-(d*e) + c*d* Sqrt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*Sq rt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]))/e
Time = 0.32 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5760, 354, 97, 73, 218}
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.
\(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 5760 |
\(\displaystyle -\frac {b c x \int \frac {1}{x \sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx}{2 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {b c x \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx^2}{4 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 97 |
\(\displaystyle -\frac {b c x \left (\frac {\int \frac {1}{x^2 \sqrt {c^2 x^2-1}}dx^2}{d}-\frac {e \int \frac {1}{\sqrt {c^2 x^2-1} \left (e x^2+d\right )}dx^2}{d}\right )}{4 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {b c x \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}+\frac {1}{c^2}}d\sqrt {c^2 x^2-1}}{c^2 d}-\frac {2 e \int \frac {1}{\frac {e x^4}{c^2}+d+\frac {e}{c^2}}d\sqrt {c^2 x^2-1}}{c^2 d}\right )}{4 e \sqrt {c^2 x^2}}-\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \left (\frac {2 \arctan \left (\sqrt {c^2 x^2-1}\right )}{d}-\frac {2 \sqrt {e} \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{d \sqrt {c^2 d+e}}\right )}{4 e \sqrt {c^2 x^2}}\) |
-1/2*(a + b*ArcCsc[c*x])/(e*(d + e*x^2)) - (b*c*x*((2*ArcTan[Sqrt[-1 + c^2 *x^2]])/d - (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e] ])/(d*Sqrt[c^2*d + e])))/(4*e*Sqrt[c^2*x^2])
3.2.5.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[b/(b*c - a*d) Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCsc[c*x])/(2*e*(p + 1))), x ] + Simp[b*c*(x/(2*e*(p + 1)*Sqrt[c^2*x^2])) Int[(d + e*x^2)^(p + 1)/(x*S qrt[c^2*x^2 - 1]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(112)=224\).
Time = 8.61 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.98
method | result | size |
parts | \(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arccsc}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c \sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(265\) |
derivativedivides | \(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(278\) |
default | \(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\) | \(278\) |
-1/2*a/e/(e*x^2+d)+b/c^2*(-1/2*c^4/e/(c^2*e*x^2+c^2*d)*arccsc(c*x)+1/4*c/e *(c^2*x^2-1)^(1/2)*(2*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)-ln( 2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(c*e*x +(-c^2*d*e)^(1/2)))-ln(-2*((c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*e+(-c^2* d*e)^(1/2)*c*x-e)/(-c*e*x+(-c^2*d*e)^(1/2))))/((c^2*x^2-1)/c^2/x^2)^(1/2)/ x/d/(-(c^2*d+e)/e)^(1/2))
Time = 0.32 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.87 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e - \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]
[-1/4*(2*a*c^2*d^2 + 2*a*d*e + sqrt(-c^2*d*e - e^2)*(b*e*x^2 + b*d)*log((c ^2*e*x^2 - c^2*d - 2*sqrt(-c^2*d*e - e^2)*sqrt(c^2*x^2 - 1) - 2*e)/(e*x^2 + d)) + 2*(b*c^2*d^2 + b*d*e)*arccsc(c*x) + 4*(b*c^2*d^2 + b*d*e + (b*c^2* d*e + b*e^2)*x^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2), -1/2*(a*c^2*d^2 + a*d*e - sqrt(c^2*d*e + e^2) *(b*e*x^2 + b*d)*arctan(sqrt(c^2*d*e + e^2)*sqrt(c^2*x^2 - 1)/(c^2*d + e)) + (b*c^2*d^2 + b*d*e)*arccsc(c*x) + 2*(b*c^2*d^2 + b*d*e + (b*c^2*d*e + b *e^2)*x^2)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 + (c^2*d ^2*e^2 + d*e^3)*x^2)]
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]
-1/2*(2*(c^2*e^2*x^2 + c^2*d*e)*integrate(1/2*x*e^(1/2*log(c*x + 1) + 1/2* log(c*x - 1))/(c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 - d*e + (c^2*e^2*x^4 + (c ^2*d*e - e^2)*x^2 - d*e)*e^(log(c*x + 1) + log(c*x - 1))), x) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*b/(e^2*x^2 + d*e) - 1/2*a/(e^2*x^2 + d*e)
Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]