Integrand size = 23, antiderivative size = 81 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d (a+b \arcsin (c x))}{3 x^3}+\frac {c^2 d (a+b \arcsin (c x))}{x}+\frac {5}{6} b c^3 d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
-1/3*d*(a+b*arcsin(c*x))/x^3+c^2*d*(a+b*arcsin(c*x))/x+5/6*b*c^3*d*arctanh ((-c^2*x^2+1)^(1/2))-1/6*b*c*d*(-c^2*x^2+1)^(1/2)/x^2
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=-\frac {a d}{3 x^3}+\frac {a c^2 d}{x}-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {b d \arcsin (c x)}{3 x^3}+\frac {b c^2 d \arcsin (c x)}{x}+\frac {5}{6} b c^3 d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
-1/3*(a*d)/x^3 + (a*c^2*d)/x - (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*d*Ar cSin[c*x])/(3*x^3) + (b*c^2*d*ArcSin[c*x])/x + (5*b*c^3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5192, 27, 354, 87, 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.
\(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 5192 |
\(\displaystyle -b c \int -\frac {d \left (1-3 c^2 x^2\right )}{3 x^3 \sqrt {1-c^2 x^2}}dx+\frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c d \int \frac {1-3 c^2 x^2}{x^3 \sqrt {1-c^2 x^2}}dx+\frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{6} b c d \int \frac {1-3 c^2 x^2}{x^4 \sqrt {1-c^2 x^2}}dx^2+\frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{6} b c d \left (-\frac {5}{2} c^2 \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx^2-\frac {\sqrt {1-c^2 x^2}}{x^2}\right )+\frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{6} b c d \left (5 \int \frac {1}{\frac {1}{c^2}-\frac {x^4}{c^2}}d\sqrt {1-c^2 x^2}-\frac {\sqrt {1-c^2 x^2}}{x^2}\right )+\frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {c^2 d (a+b \arcsin (c x))}{x}-\frac {d (a+b \arcsin (c x))}{3 x^3}+\frac {1}{6} b c d \left (5 c^2 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {\sqrt {1-c^2 x^2}}{x^2}\right )\) |
-1/3*(d*(a + b*ArcSin[c*x]))/x^3 + (c^2*d*(a + b*ArcSin[c*x]))/x + (b*c*d* (-(Sqrt[1 - c^2*x^2]/x^2) + 5*c^2*ArcTanh[Sqrt[1 - c^2*x^2]]))/6
3.1.9.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ (a + b*ArcSin[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 - c ^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 ] && IGtQ[p, 0]
Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07
method | result | size |
parts | \(-d a \left (-\frac {c^{2}}{x}+\frac {1}{3 x^{3}}\right )-d b \,c^{3} \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\) | \(87\) |
derivativedivides | \(c^{3} \left (-d a \left (\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )-d b \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(91\) |
default | \(c^{3} \left (-d a \left (\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )-d b \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(91\) |
-d*a*(-c^2/x+1/3/x^3)-d*b*c^3*(1/3/c^3/x^3*arcsin(c*x)-1/c/x*arcsin(c*x)+1 /6/c^2/x^2*(-c^2*x^2+1)^(1/2)-5/6*arctanh(1/(-c^2*x^2+1)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=\frac {5 \, b c^{3} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 12 \, a c^{2} d x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b c d x - 4 \, a d + 4 \, {\left (3 \, b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )}{12 \, x^{3}} \]
1/12*(5*b*c^3*d*x^3*log(sqrt(-c^2*x^2 + 1) + 1) - 5*b*c^3*d*x^3*log(sqrt(- c^2*x^2 + 1) - 1) + 12*a*c^2*d*x^2 - 2*sqrt(-c^2*x^2 + 1)*b*c*d*x - 4*a*d + 4*(3*b*c^2*d*x^2 - b*d)*arcsin(c*x))/x^3
Time = 3.02 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.19 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=\frac {a c^{2} d}{x} - \frac {a d}{3 x^{3}} - b c^{3} d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) + \frac {b c^{2} d \operatorname {asin}{\left (c x \right )}}{x} + \frac {b c d \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} - \frac {b d \operatorname {asin}{\left (c x \right )}}{3 x^{3}} \]
a*c**2*d/x - a*d/(3*x**3) - b*c**3*d*Piecewise((-acosh(1/(c*x)), 1/Abs(c** 2*x**2) > 1), (I*asin(1/(c*x)), True)) + b*c**2*d*asin(c*x)/x + b*c*d*Piec ewise((-c**2*acosh(1/(c*x))/2 + c/(2*x*sqrt(-1 + 1/(c**2*x**2))) - 1/(2*c* x**3*sqrt(-1 + 1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (I*c**2*asin(1/(c*x ))/2 - I*c*sqrt(1 - 1/(c**2*x**2))/(2*x), True))/3 - b*d*asin(c*x)/(3*x**3 )
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.52 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx={\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b c^{2} d - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d + \frac {a c^{2} d}{x} - \frac {a d}{3 \, x^{3}} \]
(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*c^2*d - 1/6*((c^2*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1) /x^2)*c + 2*arcsin(c*x)/x^3)*b*d + a*c^2*d/x - 1/3*a*d/x^3
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (71) = 142\).
Time = 4.56 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.65 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=-\frac {b c^{6} d x^{3} \arcsin \left (c x\right )}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {a c^{6} d x^{3}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {b c^{5} d x^{2}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {3 \, b c^{4} d x \arcsin \left (c x\right )}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} + \frac {3 \, a c^{4} d x}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {5}{6} \, b c^{3} d \log \left ({\left | c \right |} {\left | x \right |}\right ) + \frac {5}{6} \, b c^{3} d \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + \frac {3 \, b c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{8 \, x} + \frac {3 \, a c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{8 \, x} - \frac {b c d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}}{24 \, x^{2}} - \frac {b d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3} \arcsin \left (c x\right )}{24 \, x^{3}} - \frac {a d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}{24 \, x^{3}} \]
-1/24*b*c^6*d*x^3*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1)^3 - 1/24*a*c^6*d*x^ 3/(sqrt(-c^2*x^2 + 1) + 1)^3 + 1/24*b*c^5*d*x^2/(sqrt(-c^2*x^2 + 1) + 1)^2 + 3/8*b*c^4*d*x*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1) + 3/8*a*c^4*d*x/(sqr t(-c^2*x^2 + 1) + 1) - 5/6*b*c^3*d*log(abs(c)*abs(x)) + 5/6*b*c^3*d*log(sq rt(-c^2*x^2 + 1) + 1) + 3/8*b*c^2*d*(sqrt(-c^2*x^2 + 1) + 1)*arcsin(c*x)/x + 3/8*a*c^2*d*(sqrt(-c^2*x^2 + 1) + 1)/x - 1/24*b*c*d*(sqrt(-c^2*x^2 + 1) + 1)^2/x^2 - 1/24*b*d*(sqrt(-c^2*x^2 + 1) + 1)^3*arcsin(c*x)/x^3 - 1/24*a *d*(sqrt(-c^2*x^2 + 1) + 1)^3/x^3
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^4} \,d x \]