Integrand size = 14, antiderivative size = 83 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}-\frac {b d^2 \csc ^{-1}(c x)}{2 e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}+\frac {b d \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \]
-1/2*b*d^2*arccsc(c*x)/e+1/2*(e*x+d)^2*(a+b*arccsc(c*x))/e+b*d*arctanh((1- 1/c^2/x^2)^(1/2))/c+1/2*b*e*x*(1-1/c^2/x^2)^(1/2)/c
Time = 0.14 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.36 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x+\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+b d x \csc ^{-1}(c x)+\frac {1}{2} b e x^2 \csc ^{-1}(c x)+\frac {b d \sqrt {1-\frac {1}{c^2 x^2}} x \text {arctanh}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \]
a*d*x + (a*e*x^2)/2 + (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + b*d*x *ArcCsc[c*x] + (b*e*x^2*ArcCsc[c*x])/2 + (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcT anh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]
Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5750, 1892, 1730, 540, 25, 27, 538, 223, 243, 73, 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.
| \(\displaystyle \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5750 |
| \(\displaystyle \frac {b \int \frac {(d+e x)^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{2 c e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}\) |
| \(\Big \downarrow \) 1892 |
| \(\displaystyle \frac {b \int \frac {\left (\frac {d}{x}+e\right )^2}{\sqrt {1-\frac {1}{c^2 x^2}}}dx}{2 c e}+\frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}\) |
| \(\Big \downarrow \) 1730 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \int \frac {\left (\frac {d}{x}+e\right )^2 x^2}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}}{2 c e}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (e^2 x \left (-\sqrt {1-\frac {1}{c^2 x^2}}\right )-\int -\frac {d \left (\frac {d}{x}+2 e\right ) x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}\right )}{2 c e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (\int \frac {d \left (\frac {d}{x}+2 e\right ) x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \int \frac {\left (\frac {d}{x}+2 e\right ) x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \left (d \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}+2 e \int \frac {x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}\right )-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \left (2 e \int \frac {x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}+c d \arcsin \left (\frac {1}{c x}\right )\right )-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \left (e \int \frac {x}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x^2}+c d \arcsin \left (\frac {1}{c x}\right )\right )-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \left (c d \arcsin \left (\frac {1}{c x}\right )-2 c^2 e \int \frac {1}{c^2-c^2 \sqrt {1-\frac {1}{c^2 x^2}}}d\sqrt {1-\frac {1}{c^2 x^2}}\right )-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(d+e x)^2 \left (a+b \csc ^{-1}(c x)\right )}{2 e}-\frac {b \left (d \left (c d \arcsin \left (\frac {1}{c x}\right )-2 e \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )-e^2 x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c e}\) |
((d + e*x)^2*(a + b*ArcCsc[c*x]))/(2*e) - (b*(-(e^2*Sqrt[1 - 1/(c^2*x^2)]* x) + d*(c*d*ArcSin[1/(c*x)] - 2*e*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]])))/(2*c*e )
3.1.46.3.1 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_))^(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[((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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol ] :> -Subst[Int[(d + e/x^n)^q*((a + c/x^(2*n))^p/x^2), x], x, 1/x] /; FreeQ [{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ (p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 2] || !IntegerQ[p])
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 0.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33
| method | result | size |
| parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\frac {c \,\operatorname {arccsc}\left (c x \right ) x^{2} e}{2}+\operatorname {arccsc}\left (c x \right ) x c d +\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 c^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}\right )}{c}\) | \(110\) |
| derivativedivides | \(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\) | \(127\) |
| default | \(\frac {\frac {a \left (d x \,c^{2}+\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccsc}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccsc}\left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 d c \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )+e \sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c}}{c}\) | \(127\) |
a*(1/2*e*x^2+d*x)+b/c*(1/2*c*arccsc(c*x)*x^2*e+arccsc(c*x)*x*c*d+1/2/c^2/( (c^2*x^2-1)/c^2/x^2)^(1/2)/x*(c^2*x^2-1)^(1/2)*(2*d*c*ln(c*x+(c^2*x^2-1)^( 1/2))+e*(c^2*x^2-1)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.55 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {a c^{2} e x^{2} + 2 \, a c^{2} d x - 2 \, b c d \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + \sqrt {c^{2} x^{2} - 1} b e + {\left (b c^{2} e x^{2} + 2 \, b c^{2} d x - 2 \, b c^{2} d - b c^{2} e\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, {\left (2 \, b c^{2} d + b c^{2} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, c^{2}} \]
1/2*(a*c^2*e*x^2 + 2*a*c^2*d*x - 2*b*c*d*log(-c*x + sqrt(c^2*x^2 - 1)) + s qrt(c^2*x^2 - 1)*b*e + (b*c^2*e*x^2 + 2*b*c^2*d*x - 2*b*c^2*d - b*c^2*e)*a rccsc(c*x) - 2*(2*b*c^2*d + b*c^2*e)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/c^2
Time = 2.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=a d x + \frac {a e x^{2}}{2} + b d x \operatorname {acsc}{\left (c x \right )} + \frac {b e x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b d \left (\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}\right )}{c} + \frac {b e \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} \]
a*d*x + a*e*x**2/2 + b*d*x*acsc(c*x) + b*e*x**2*acsc(c*x)/2 + b*d*Piecewis e((acosh(c*x), Abs(c**2*x**2) > 1), (-I*asin(c*x), True))/c + b*e*Piecewis e((sqrt(c**2*x**2 - 1)/c, Abs(c**2*x**2) > 1), (I*sqrt(-c**2*x**2 + 1)/c, True))/(2*c)
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b d}{2 \, c} \]
1/2*a*e*x^2 + 1/2*(x^2*arccsc(c*x) + x*sqrt(-1/(c^2*x^2) + 1)/c)*b*e + a*d *x + 1/2*(2*c*x*arccsc(c*x) + log(sqrt(-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. 341 vs. \(2 (73) = 146\).
Time = 0.44 (sec) , antiderivative size = 341, normalized size of antiderivative = 4.11 \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{8} \, {\left (\frac {b e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a e x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {4 \, b d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {4 \, a d x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c} + \frac {2 \, b e x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {8 \, b d \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {8 \, b d \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{2}} + \frac {2 \, b e \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2 \, a e}{c^{3}} + \frac {4 \, b d \arcsin \left (\frac {1}{c x}\right )}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {4 \, a d}{c^{3} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {2 \, b e}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b e \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a e}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \]
1/8*(b*e*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c + a*e*x^2*(s qrt(-1/(c^2*x^2) + 1) + 1)^2/c + 4*b*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)*arcs in(1/(c*x))/c + 4*a*d*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c + 2*b*e*x*(sqrt(-1/ (c^2*x^2) + 1) + 1)/c^2 + 8*b*d*log(sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 - 8*b* d*log(1/(abs(c)*abs(x)))/c^2 + 2*b*e*arcsin(1/(c*x))/c^3 + 2*a*e/c^3 + 4*b *d*arcsin(1/(c*x))/(c^3*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 4*a*d/(c^3*x*(sq rt(-1/(c^2*x^2) + 1) + 1)) - 2*b*e/(c^4*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + b*e*arcsin(1/(c*x))/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + a*e/(c^5*x^ 2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2))*c
Timed out. \[ \int (d+e x) \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\left (d+e\,x\right ) \,d x \]