Integrand size = 12, antiderivative size = 55 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.62 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {a c x \left (2 b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right )+2 b c x \left (b \sqrt {1-\frac {1}{c^2 x^2}}+a c x\right ) \csc ^{-1}(c x)+b^2 c^2 x^2 \csc ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]
(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) + 2*b*c*x*(b*Sqrt[1 - 1/(c^2*x^ 2)] + a*c*x)*ArcCsc[c*x] + b^2*c^2*x^2*ArcCsc[c*x]^2 + 2*b^2*Log[c*x])/(2* c^2)
Time = 0.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5746, 4910, 3042, 4672, 3042, 25, 3956}
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.
\(\displaystyle \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx\) |
\(\Big \downarrow \) 5746 |
\(\displaystyle -\frac {\int c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)}{c^2}\) |
\(\Big \downarrow \) 4910 |
\(\displaystyle -\frac {b \int c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \int \left (a+b \csc ^{-1}(c x)\right ) \csc \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle -\frac {b \left (b \int c \sqrt {1-\frac {1}{c^2 x^2}} xd\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \left (b \int -\tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b \left (-b \int \tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle -\frac {b \left (b \log \left (\frac {1}{c x}\right )-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{2} c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{c^2}\) |
-((-1/2*(c^2*x^2*(a + b*ArcCsc[c*x])^2) + b*(-(c*Sqrt[1 - 1/(c^2*x^2)]*x*( a + b*ArcCsc[c*x])) + b*Log[1/(c*x)]))/c^2)
3.1.17.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x ] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n , 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(121\) vs. \(2(51)=102\).
Time = 1.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.22
method | result | size |
parts | \(\frac {a^{2} x^{2}}{2}+\frac {b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )}{c^{2}}+\frac {2 a b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(122\) |
derivativedivides | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(123\) |
default | \(\frac {\frac {c^{2} x^{2} a^{2}}{2}+b^{2} \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}}{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-\ln \left (\frac {1}{c x}\right )\right )+2 a b \left (\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\) | \(123\) |
1/2*a^2*x^2+b^2/c^2*(1/2*c^2*x^2*arccsc(c*x)^2+arccsc(c*x)*c*x*((c^2*x^2-1 )/c^2/x^2)^(1/2)-ln(1/c/x))+2*a*b/c^2*(1/2*c^2*x^2*arccsc(c*x)+1/2/((c^2*x ^2-1)/c^2/x^2)^(1/2)/c/x*(c^2*x^2-1))
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (51) = 102\).
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.02 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {b^{2} c^{2} x^{2} \operatorname {arccsc}\left (c x\right )^{2} + a^{2} c^{2} x^{2} - 4 \, a b c^{2} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \left (x\right ) + 2 \, {\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \]
1/2*(b^2*c^2*x^2*arccsc(c*x)^2 + a^2*c^2*x^2 - 4*a*b*c^2*arctan(-c*x + sqr t(c^2*x^2 - 1)) + 2*b^2*log(x) + 2*(a*b*c^2*x^2 - a*b*c^2)*arccsc(c*x) + 2 *sqrt(c^2*x^2 - 1)*(b^2*arccsc(c*x) + a*b))/c^2
\[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \]
Time = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.53 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {1}{2} \, b^{2} x^{2} \operatorname {arccsc}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a b + {\left (\frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right )}{c} + \frac {\log \left (x\right )}{c^{2}}\right )} b^{2} \]
1/2*b^2*x^2*arccsc(c*x)^2 + 1/2*a^2*x^2 + (x^2*arccsc(c*x) + x*sqrt(-1/(c^ 2*x^2) + 1)/c)*a*b + (x*sqrt(-1/(c^2*x^2) + 1)*arccsc(c*x)/c + log(x)/c^2) *b^2
Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (51) = 102\).
Time = 0.35 (sec) , antiderivative size = 427, normalized size of antiderivative = 7.76 \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\frac {1}{8} \, {\left (\frac {b^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c} + \frac {2 \, a b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {4 \, b^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{2}} + \frac {4 \, a b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {2 \, b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{3}} + \frac {4 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{3}} - \frac {16 \, b^{2} \log \left (2\right )}{c^{3}} + \frac {8 \, b^{2} \log \left (2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 2\right )}{c^{3}} - \frac {8 \, b^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{3}} - \frac {8 \, b^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{3}} + \frac {2 \, a^{2}}{c^{3}} - \frac {4 \, b^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {4 \, a b}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {2 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a^{2}}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \]
1/8*(b^2*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))^2/c + 2*a*b*x^ 2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c + a^2*x^2*(sqrt(-1/(c^2 *x^2) + 1) + 1)^2/c + 4*b^2*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcsin(1/(c*x)) /c^2 + 4*a*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 + 2*b^2*arcsin(1/(c*x))^2/ c^3 + 4*a*b*arcsin(1/(c*x))/c^3 - 16*b^2*log(2)/c^3 + 8*b^2*log(2*sqrt(-1/ (c^2*x^2) + 1) + 2)/c^3 - 8*b^2*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^3 - 8*b^ 2*log(1/(abs(c)*abs(x)))/c^3 + 2*a^2/c^3 - 4*b^2*arcsin(1/(c*x))/(c^4*x*(s qrt(-1/(c^2*x^2) + 1) + 1)) - 4*a*b/(c^4*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + b^2*arcsin(1/(c*x))^2/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + 2*a*b*ar csin(1/(c*x))/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + a^2/(c^5*x^2*(sqr t(-1/(c^2*x^2) + 1) + 1)^2))*c
Timed out. \[ \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx=\int x\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]