Integrand size = 16, antiderivative size = 109 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e x^2 \sqrt {-1+c^2 x^2}}{6 c \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \left (6 c^2 d+e\right ) x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{6 c^2 \sqrt {c^2 x^2}} \]
d*x*(a+b*arccsc(c*x))+1/3*e*x^3*(a+b*arccsc(c*x))+1/6*b*(6*c^2*d+e)*x*arct anh(c*x/(c^2*x^2-1)^(1/2))/c^2/(c^2*x^2)^(1/2)+1/6*b*e*x^2*(c^2*x^2-1)^(1/ 2)/c/(c^2*x^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.37 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x+\frac {1}{3} a e x^3+\frac {b e x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{6 c}+b d x \csc ^{-1}(c x)+\frac {1}{3} b e x^3 \csc ^{-1}(c x)+\frac {b d \sqrt {1-\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}}+\frac {b e \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{6 c^3} \]
a*d*x + (a*e*x^3)/3 + (b*e*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + b*d *x*ArcCsc[c*x] + (b*e*x^3*ArcCsc[c*x])/3 + (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*Ar cTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2] + (b*e*Log[x*(1 + Sqrt [(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5752, 27, 299, 224, 219}
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.
| \(\displaystyle \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5752 |
| \(\displaystyle \frac {b c x \int \frac {e x^2+3 d}{3 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {e x^2+3 d}{\sqrt {c^2 x^2-1}}dx}{3 \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {b c x \left (\frac {\left (6 c^2 d+e\right ) \int \frac {1}{\sqrt {c^2 x^2-1}}dx}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{3 \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {b c x \left (\frac {\left (6 c^2 d+e\right ) \int \frac {1}{1-\frac {c^2 x^2}{c^2 x^2-1}}d\frac {x}{\sqrt {c^2 x^2-1}}}{2 c^2}+\frac {e x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{3 \sqrt {c^2 x^2}}+d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d x \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \csc ^{-1}(c x)\right )+\frac {b c x \left (\frac {\text {arctanh}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right ) \left (6 c^2 d+e\right )}{2 c^3}+\frac {e x \sqrt {c^2 x^2-1}}{2 c^2}\right )}{3 \sqrt {c^2 x^2}}\) |
d*x*(a + b*ArcCsc[c*x]) + (e*x^3*(a + b*ArcCsc[c*x]))/3 + (b*c*x*((e*x*Sqr t[-1 + c^2*x^2])/(2*c^2) + ((6*c^2*d + e)*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2] ])/(2*c^3)))/(3*Sqrt[c^2*x^2])
3.1.78.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && (IGtQ[p, 0] || ILtQ[p + 1 /2, 0])
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.21
| method | result | size |
| parts | \(a \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b \left (\frac {c \,\operatorname {arccsc}\left (c x \right ) x^{3} e}{3}+\operatorname {arccsc}\left (c x \right ) d x c +\frac {\sqrt {c^{2} x^{2}-1}\, \left (6 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 c^{3} x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}\right )}{c}\) | \(132\) |
| derivativedivides | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (6 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}\right )}{c^{2}}}{c}\) | \(149\) |
| default | \(\frac {\frac {a \left (c^{3} d x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{3} x^{3}}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (6 d \,c^{2} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}\right )}{c^{2}}}{c}\) | \(149\) |
a*(1/3*x^3*e+d*x)+b/c*(1/3*c*arccsc(c*x)*x^3*e+arccsc(c*x)*d*x*c+1/6/c^3*( c^2*x^2-1)^(1/2)*(6*d*c^2*ln(c*x+(c^2*x^2-1)^(1/2))+e*c*x*(c^2*x^2-1)^(1/2 )+e*ln(c*x+(c^2*x^2-1)^(1/2)))/x/((c^2*x^2-1)/c^2/x^2)^(1/2))
Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.29 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} e x^{3} + 6 \, a c^{3} d x + \sqrt {c^{2} x^{2} - 1} b c e x + 2 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \operatorname {arccsc}\left (c x\right ) - 4 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (6 \, b c^{2} d + b e\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{6 \, c^{3}} \]
1/6*(2*a*c^3*e*x^3 + 6*a*c^3*d*x + sqrt(c^2*x^2 - 1)*b*c*e*x + 2*(b*c^3*e* x^3 + 3*b*c^3*d*x - 3*b*c^3*d - b*c^3*e)*arccsc(c*x) - 4*(3*b*c^3*d + b*c^ 3*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)) - (6*b*c^2*d + b*e)*log(-c*x + sqrt( c^2*x^2 - 1)))/c^3
Time = 3.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x + \frac {a e x^{3}}{3} + b d x \operatorname {acsc}{\left (c x \right )} + \frac {b e x^{3} \operatorname {acsc}{\left (c x \right )}}{3} + \frac {b d \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} - 1}}{2 c} + \frac {\operatorname {acosh}{\left (c x \right )}}{2 c^{2}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- \frac {i c x^{3}}{2 \sqrt {- c^{2} x^{2} + 1}} + \frac {i x}{2 c \sqrt {- c^{2} x^{2} + 1}} - \frac {i \operatorname {asin}{\left (c x \right )}}{2 c^{2}} & \text {otherwise} \end {cases}\right )}{3 c} \]
a*d*x + a*e*x**3/3 + b*d*x*acsc(c*x) + b*e*x**3*acsc(c*x)/3 + b*d*Piecewis e((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c + b*e*Piecewis e((x*sqrt(c**2*x**2 - 1)/(2*c) + acosh(c*x)/(2*c**2), Abs(c**2*x**2) > 1), (-I*c*x**3/(2*sqrt(-c**2*x**2 + 1)) + I*x/(2*c*sqrt(-c**2*x**2 + 1)) - I* asin(c*x)/(2*c**2), True))/(3*c)
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.40 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arccsc}\left (c x\right ) + \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d}{2 \, c} \]
1/3*a*e*x^3 + 1/12*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/ (c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - log(sqrt(-1/ (c^2*x^2) + 1) - 1)/c^2)/c)*b*e + a*d*x + 1/2*(2*c*x*arccsc(c*x) + log(sqr t(-1/(c^2*x^2) + 1) + 1) - log(-sqrt(-1/(c^2*x^2) + 1) + 1))*b*d/c
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (95) = 190\).
Time = 0.94 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.34 \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{24} \, {\left (\frac {b e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c} + \frac {b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{2}} + \frac {12 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {12 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {3 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {3 \, a e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{3}} + \frac {24 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {24 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {4 \, b e \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} - \frac {4 \, b e \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{4}} + \frac {12 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, a d}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {3 \, a e}{c^{5} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {b e}{c^{6} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {b e \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {a e}{c^{7} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}\right )} c \]
1/24*(b*e*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3*arcsin(1/(c*x))/c + a*e*x^3*( sqrt(-1/(c^2*x^2) + 1) + 1)^3/c + b*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c ^2 + 12*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x))/c + 12*a*d*x*(s qrt(-1/(c^2*x^2) + 1) + 1)/c + 3*b*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin (1/(c*x))/c^3 + 3*a*e*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^3 + 24*b*d*log(sqrt (-1/(c^2*x^2) + 1) + 1)/c^2 - 24*b*d*log(1/(abs(c)*abs(x)))/c^2 + 4*b*e*lo g(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 - 4*b*e*log(1/(abs(c)*abs(x)))/c^4 + 12* b*d*arcsin(1/(c*x))/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 12*a*d/(c^3*x*( sqrt(-1/(c^2*x^2) + 1) + 1)) + 3*b*e*arcsin(1/(c*x))/(c^5*x*(sqrt(-1/(c^2* x^2) + 1) + 1)) + 3*a*e/(c^5*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) - b*e/(c^6*x^ 2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + b*e*arcsin(1/(c*x))/(c^7*x^3*(sqrt(-1/ (c^2*x^2) + 1) + 1)^3) + a*e/(c^7*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3))*c
Timed out. \[ \int \left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]