Integrand size = 16, antiderivative size = 190 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{12 \left (1-c^2\right ) x^4}-\frac {b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right )^2 x^2}-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}-\frac {b \left (1+2 c^2\right ) d^3 \text {arctanh}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{12 \left (1-c^2\right )^{5/2}} \]
1/6*(-a-b*arcsin(d*x^2+c))/x^6-1/12*b*(2*c^2+1)*d^3*arctanh((-c*d*x^2-c^2+ 1)/(-c^2+1)^(1/2)/(-d^2*x^4-2*c*d*x^2-c^2+1)^(1/2))/(-c^2+1)^(5/2)-1/12*b* d*(-d^2*x^4-2*c*d*x^2-c^2+1)^(1/2)/(-c^2+1)/x^4-1/4*b*c*d^2*(-d^2*x^4-2*c* d*x^2-c^2+1)^(1/2)/(-c^2+1)^2/x^2
Time = 0.33 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=-\frac {a}{6 x^6}+b \left (\frac {d}{12 \left (-1+c^2\right ) x^4}-\frac {c d^2}{4 \left (-1+c^2\right )^2 x^2}\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}-\frac {b \arcsin \left (c+d x^2\right )}{6 x^6}-\frac {b \left (1+2 c^2\right ) d^3 \arctan \left (\frac {\sqrt {-d^2} x^2-\sqrt {1-c^2-2 c d x^2-d^2 x^4}}{\sqrt {-1+c^2}}\right )}{6 (-1+c)^2 (1+c)^2 \sqrt {-1+c^2}} \]
-1/6*a/x^6 + b*(d/(12*(-1 + c^2)*x^4) - (c*d^2)/(4*(-1 + c^2)^2*x^2))*Sqrt [1 - c^2 - 2*c*d*x^2 - d^2*x^4] - (b*ArcSin[c + d*x^2])/(6*x^6) - (b*(1 + 2*c^2)*d^3*ArcTan[(Sqrt[-d^2]*x^2 - Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/S qrt[-1 + c^2]])/(6*(-1 + c)^2*(1 + c)^2*Sqrt[-1 + c^2])
Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5341, 27, 1434, 1167, 25, 27, 1228, 1154, 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.
| \(\displaystyle \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx\) |
| \(\Big \downarrow \) 5341 |
| \(\displaystyle \frac {1}{6} b \int \frac {2 d}{x^5 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} b d \int \frac {1}{x^5 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 1434 |
| \(\displaystyle \frac {1}{6} b d \int \frac {1}{x^6 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx^2-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle \frac {1}{6} b d \left (-\frac {\int -\frac {d \left (d x^2+3 c\right )}{x^4 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx^2}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} b d \left (\frac {\int \frac {d \left (d x^2+3 c\right )}{x^4 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx^2}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} b d \left (\frac {d \int \frac {d x^2+3 c}{x^4 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx^2}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{6} b d \left (\frac {d \left (\frac {\left (2 c^2+1\right ) d \int \frac {1}{x^2 \sqrt {-d^2 x^4-2 c d x^2-c^2+1}}dx^2}{1-c^2}-\frac {3 c \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{\left (1-c^2\right ) x^2}\right )}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{6} b d \left (\frac {d \left (-\frac {2 \left (2 c^2+1\right ) d \int \frac {1}{4 \left (1-c^2\right )-x^4}d\frac {2 \left (-c^2-d x^2 c+1\right )}{\sqrt {-d^2 x^4-2 c d x^2-c^2+1}}}{1-c^2}-\frac {3 c \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{\left (1-c^2\right ) x^2}\right )}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} b d \left (\frac {d \left (-\frac {\left (2 c^2+1\right ) d \text {arctanh}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{\left (1-c^2\right )^{3/2}}-\frac {3 c \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{\left (1-c^2\right ) x^2}\right )}{2 \left (1-c^2\right )}-\frac {\sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{2 \left (1-c^2\right ) x^4}\right )-\frac {a+b \arcsin \left (c+d x^2\right )}{6 x^6}\) |
-1/6*(a + b*ArcSin[c + d*x^2])/x^6 + (b*d*(-1/2*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]/((1 - c^2)*x^4) + (d*((-3*c*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4]) /((1 - c^2)*x^2) - ((1 + 2*c^2)*d*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^ 2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(1 - c^2)^(3/2)))/(2*(1 - c^2))) )/6
3.4.92.3.1 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/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d ^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[ (d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcSin[u])/(d*(m + 1))), x] - Simp[b/(d*(m + 1) ) Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 - u^2]), x], x] , x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
Time = 0.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.29
| method | result | size |
| default | \(-\frac {a}{6 x^{6}}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{6 x^{6}}+\frac {d \left (-\frac {\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right ) x^{4}}-\frac {3 d c \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right )^{2} x^{2}}-\frac {3 d^{2} c^{2} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {5}{2}}}-\frac {d^{2} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {3}{2}}}\right )}{3}\right )\) | \(245\) |
| parts | \(-\frac {a}{6 x^{6}}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{6 x^{6}}+\frac {d \left (-\frac {\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right ) x^{4}}-\frac {3 d c \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right )^{2} x^{2}}-\frac {3 d^{2} c^{2} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {5}{2}}}-\frac {d^{2} \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {3}{2}}}\right )}{3}\right )\) | \(245\) |
-1/6*a/x^6+b*(-1/6/x^6*arcsin(d*x^2+c)+1/3*d*(-1/4/(-c^2+1)/x^4*(-d^2*x^4- 2*c*d*x^2-c^2+1)^(1/2)-3/4*d*c/(-c^2+1)^2/x^2*(-d^2*x^4-2*c*d*x^2-c^2+1)^( 1/2)-3/4*d^2*c^2/(-c^2+1)^(5/2)*ln((-2*c^2+2-2*c*d*x^2+2*(-c^2+1)^(1/2)*(- d^2*x^4-2*c*d*x^2-c^2+1)^(1/2))/x^2)-1/4*d^2/(-c^2+1)^(3/2)*ln((-2*c^2+2-2 *c*d*x^2+2*(-c^2+1)^(1/2)*(-d^2*x^4-2*c*d*x^2-c^2+1)^(1/2))/x^2)))
Time = 0.39 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.61 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=\left [-\frac {{\left (2 \, b c^{2} + b\right )} \sqrt {-c^{2} + 1} d^{3} x^{6} \log \left (\frac {{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \, {\left (c^{3} - c\right )} d x^{2} + 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 4 \, a c^{6} - 12 \, a c^{4} + 12 \, a c^{2} + 4 \, {\left (b c^{6} - 3 \, b c^{4} + 3 \, b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) + 2 \, {\left (3 \, {\left (b c^{3} - b c\right )} d^{2} x^{4} - {\left (b c^{4} - 2 \, b c^{2} + b\right )} d x^{2}\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} - 4 \, a}{24 \, {\left (c^{6} - 3 \, c^{4} + 3 \, c^{2} - 1\right )} x^{6}}, \frac {{\left (2 \, b c^{2} + b\right )} \sqrt {c^{2} - 1} d^{3} x^{6} \arctan \left (\frac {\sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \, {\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) - 2 \, a c^{6} + 6 \, a c^{4} - 6 \, a c^{2} - 2 \, {\left (b c^{6} - 3 \, b c^{4} + 3 \, b c^{2} - b\right )} \arcsin \left (d x^{2} + c\right ) - {\left (3 \, {\left (b c^{3} - b c\right )} d^{2} x^{4} - {\left (b c^{4} - 2 \, b c^{2} + b\right )} d x^{2}\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} + 2 \, a}{12 \, {\left (c^{6} - 3 \, c^{4} + 3 \, c^{2} - 1\right )} x^{6}}\right ] \]
[-1/24*((2*b*c^2 + b)*sqrt(-c^2 + 1)*d^3*x^6*log(((2*c^2 - 1)*d^2*x^4 + 2* c^4 + 4*(c^3 - c)*d*x^2 + 2*sqrt(-d^2*x^4 - 2*c*d*x^2 - c^2 + 1)*(c*d*x^2 + c^2 - 1)*sqrt(-c^2 + 1) - 4*c^2 + 2)/x^4) + 4*a*c^6 - 12*a*c^4 + 12*a*c^ 2 + 4*(b*c^6 - 3*b*c^4 + 3*b*c^2 - b)*arcsin(d*x^2 + c) + 2*(3*(b*c^3 - b* c)*d^2*x^4 - (b*c^4 - 2*b*c^2 + b)*d*x^2)*sqrt(-d^2*x^4 - 2*c*d*x^2 - c^2 + 1) - 4*a)/((c^6 - 3*c^4 + 3*c^2 - 1)*x^6), 1/12*((2*b*c^2 + b)*sqrt(c^2 - 1)*d^3*x^6*arctan(sqrt(-d^2*x^4 - 2*c*d*x^2 - c^2 + 1)*(c*d*x^2 + c^2 - 1)*sqrt(c^2 - 1)/((c^2 - 1)*d^2*x^4 + c^4 + 2*(c^3 - c)*d*x^2 - 2*c^2 + 1) ) - 2*a*c^6 + 6*a*c^4 - 6*a*c^2 - 2*(b*c^6 - 3*b*c^4 + 3*b*c^2 - b)*arcsin (d*x^2 + c) - (3*(b*c^3 - b*c)*d^2*x^4 - (b*c^4 - 2*b*c^2 + b)*d*x^2)*sqrt (-d^2*x^4 - 2*c*d*x^2 - c^2 + 1) + 2*a)/((c^6 - 3*c^4 + 3*c^2 - 1)*x^6)]
\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{7}}\, dx \]
Exception generated. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for more details)Is
\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=\int { \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{7}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^7} \, dx=\int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^7} \,d x \]