Integrand size = 21, antiderivative size = 132 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {b c \sqrt {d} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{e \sqrt {c^2 x^2}}+\frac {b x \text {arctanh}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c^2 x^2}} \] Output:
(e*x^2+d)^(1/2)*(a+b*arccsc(c*x))/e-b*c*d^(1/2)*x*arctan((e*x^2+d)^(1/2)/d ^(1/2)/(c^2*x^2-1)^(1/2))/e/(c^2*x^2)^(1/2)+b*x*arctanh(e^(1/2)*(c^2*x^2-1 )^(1/2)/c/(e*x^2+d)^(1/2))/e^(1/2)/(c^2*x^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.81 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (a+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},1,\frac {3}{2},\frac {e-c^2 e x^2}{c^2 d+e},1-c^2 x^2\right )}{\sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}}}+b \csc ^{-1}(c x)\right )}{e} \] Input:
Integrate[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]
Output:
(Sqrt[d + e*x^2]*(a + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*AppellF1[1/2, -1/2, 1, 3/2, (e - c^2*e*x^2)/(c^2*d + e), 1 - c^2*x^2])/Sqrt[(c^2*(d + e*x^2))/(c^ 2*d + e)] + b*ArcCsc[c*x]))/e
Time = 0.40 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5760, 354, 140, 27, 66, 104, 217, 221}
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.
\(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx\) |
\(\Big \downarrow \) 5760 |
\(\displaystyle \frac {b c x \int \frac {\sqrt {e x^2+d}}{x \sqrt {c^2 x^2-1}}dx}{e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {b c x \int \frac {\sqrt {e x^2+d}}{x^2 \sqrt {c^2 x^2-1}}dx^2}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 140 |
\(\displaystyle \frac {b c x \left (e \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+\int \frac {d}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b c x \left (e \int \frac {1}{\sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+d \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {b c x \left (d \int \frac {1}{x^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {b c x \left (2 d \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {c^2 x^2-1}}+2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {b c x \left (2 e \int \frac {1}{c^2-e x^4}d\frac {\sqrt {c^2 x^2-1}}{\sqrt {e x^2+d}}-2 \sqrt {d} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )\right )}{2 e \sqrt {c^2 x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{e}+\frac {b c x \left (\frac {2 \sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}-2 \sqrt {d} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )\right )}{2 e \sqrt {c^2 x^2}}\) |
Input:
Int[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]
Output:
(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e + (b*c*x*(-2*Sqrt[d]*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 + c^2*x^2])] + (2*Sqrt[e]*ArcTanh[(Sqrt[e]*Sqrt [-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/c))/(2*e*Sqrt[c^2*x^2])
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_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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]
\[\int \frac {x \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}d x\]
Input:
int(x*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x)
Output:
int(x*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x)
Time = 0.21 (sec) , antiderivative size = 870, normalized size of antiderivative = 6.59 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx =\text {Too large to display} \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")
Output:
[1/4*(b*c*sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e )*x^2 + 4*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(- d) + 8*d^2)/x^4) + b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*( c^4*d*e - c^2*e^2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*s qrt(e*x^2 + d)*sqrt(e) + e^2) + 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c)) /(c*e), -1/4*(2*b*c*sqrt(d)*arctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d)/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) - b*sqrt(e)*log(8*c^4*e^2*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^ 2)*x^2 + 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*s qrt(e) + e^2) - 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c))/(c*e), 1/4*(b*c *sqrt(-d)*log(((c^4*d^2 - 6*c^2*d*e + e^2)*x^4 - 8*(c^2*d^2 - d*e)*x^2 + 4 *sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(-d) + 8*d^ 2)/x^4) - 2*b*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2) ) + 4*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c))/(c*e), -1/2*(b*c*sqrt(d)*ar ctan(-1/2*sqrt(c^2*x^2 - 1)*((c^2*d - e)*x^2 - 2*d)*sqrt(e*x^2 + d)*sqrt(d )/(c^2*d*e*x^4 + (c^2*d^2 - d*e)*x^2 - d^2)) + b*sqrt(-e)*arctan(1/2*(2*c^ 2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sqrt(-e)/(c^3*e^2*x ^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) - 2*sqrt(e*x^2 + d)*(b*c*arccsc(c*x) + a*c))/(c*e)]
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \] Input:
integrate(x*(a+b*acsc(c*x))/(e*x**2+d)**(1/2),x)
Output:
Integral(x*(a + b*acsc(c*x))/sqrt(d + e*x**2), x)
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")
Output:
(e*integrate((c^2*e*x^3 + c^2*d*x)*e^(-1/2*log(e*x^2 + d) + 1/2*log(c*x + 1) + 1/2*log(c*x - 1))/(c^2*e*x^2 + (c^2*e*x^2 - e)*e^(log(c*x + 1) + log( c*x - 1)) - e), x) + sqrt(e*x^2 + d)*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1 )))*b/e + sqrt(e*x^2 + d)*a/e
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{\sqrt {e x^{2} + d}} \,d x } \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")
Output:
integrate((b*arccsc(c*x) + a)*x/sqrt(e*x^2 + d), x)
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \] Input:
int((x*(a + b*asin(1/(c*x))))/(d + e*x^2)^(1/2),x)
Output:
int((x*(a + b*asin(1/(c*x))))/(d + e*x^2)^(1/2), x)
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, a +\left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e \,x^{2}+d}}d x \right ) b e}{e} \] Input:
int(x*(a+b*acsc(c*x))/(e*x^2+d)^(1/2),x)
Output:
(sqrt(d + e*x**2)*a + int((acsc(c*x)*x)/sqrt(d + e*x**2),x)*b*e)/e