Integrand size = 21, antiderivative size = 121 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=-\frac {b \sqrt {1-(c+d x)^2}}{20 d e^6 (c+d x)^4}-\frac {3 b \sqrt {1-(c+d x)^2}}{40 d e^6 (c+d x)^2}-\frac {a+b \arcsin (c+d x)}{5 d e^6 (c+d x)^5}-\frac {3 b \text {arctanh}\left (\sqrt {1-(c+d x)^2}\right )}{40 d e^6} \]
1/5*(-a-b*arcsin(d*x+c))/d/e^6/(d*x+c)^5-3/40*b*arctanh((1-(d*x+c)^2)^(1/2 ))/d/e^6-1/20*b*(1-(d*x+c)^2)^(1/2)/d/e^6/(d*x+c)^4-3/40*b*(1-(d*x+c)^2)^( 1/2)/d/e^6/(d*x+c)^2
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.54 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=-\frac {\frac {a+b \arcsin (c+d x)}{(c+d x)^5}+b \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},1-(c+d x)^2\right )}{5 d e^6} \]
-1/5*((a + b*ArcSin[c + d*x])/(c + d*x)^5 + b*Sqrt[1 - (c + d*x)^2]*Hyperg eometric2F1[1/2, 3, 3/2, 1 - (c + d*x)^2])/(d*e^6)
Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {5304, 27, 5138, 243, 52, 52, 73, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx\) |
| \(\Big \downarrow \) 5304 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c+d x)}{e^6 (c+d x)^6}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c+d x)}{(c+d x)^6}d(c+d x)}{d e^6}\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {\frac {1}{5} b \int \frac {1}{(c+d x)^5 \sqrt {1-(c+d x)^2}}d(c+d x)-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{10} b \int \frac {1}{\sqrt {-c-d x+1} (c+d x)^6}d(c+d x)^2-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\frac {1}{10} b \left (\frac {3}{4} \int \frac {1}{\sqrt {-c-d x+1} (c+d x)^4}d(c+d x)^2-\frac {\sqrt {-c-d x+1}}{2 (c+d x)^4}\right )-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {\frac {1}{10} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {-c-d x+1} (c+d x)^2}d(c+d x)^2-\frac {\sqrt {-c-d x+1}}{(c+d x)^2}\right )-\frac {\sqrt {-c-d x+1}}{2 (c+d x)^4}\right )-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{10} b \left (\frac {3}{4} \left (-\int \frac {1}{1-(c+d x)^4}d\sqrt {-c-d x+1}-\frac {\sqrt {-c-d x+1}}{(c+d x)^2}\right )-\frac {\sqrt {-c-d x+1}}{2 (c+d x)^4}\right )-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{10} b \left (\frac {3}{4} \left (-\text {arctanh}\left (\sqrt {-c-d x+1}\right )-\frac {\sqrt {-c-d x+1}}{(c+d x)^2}\right )-\frac {\sqrt {-c-d x+1}}{2 (c+d x)^4}\right )-\frac {a+b \arcsin (c+d x)}{5 (c+d x)^5}}{d e^6}\) |
(-1/5*(a + b*ArcSin[c + d*x])/(c + d*x)^5 + (b*(-1/2*Sqrt[1 - c - d*x]/(c + d*x)^4 + (3*(-(Sqrt[1 - c - d*x]/(c + d*x)^2) - ArcTanh[Sqrt[1 - c - d*x ]]))/4))/10)/(d*e^6)
3.2.87.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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[(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[(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[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A rcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}-\frac {3 \sqrt {1-\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d}\) | \(100\) |
| default | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}-\frac {3 \sqrt {1-\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6}}}{d}\) | \(100\) |
| parts | \(-\frac {a}{5 e^{6} \left (d x +c \right )^{5} d}+\frac {b \left (-\frac {\arcsin \left (d x +c \right )}{5 \left (d x +c \right )^{5}}-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{20 \left (d x +c \right )^{4}}-\frac {3 \sqrt {1-\left (d x +c \right )^{2}}}{40 \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (d x +c \right )^{2}}}\right )}{40}\right )}{e^{6} d}\) | \(102\) |
1/d*(-1/5*a/e^6/(d*x+c)^5+b/e^6*(-1/5/(d*x+c)^5*arcsin(d*x+c)-1/20/(d*x+c) ^4*(1-(d*x+c)^2)^(1/2)-3/40/(d*x+c)^2*(1-(d*x+c)^2)^(1/2)-3/40*arctanh(1/( 1-(d*x+c)^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (107) = 214\).
Time = 0.34 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.65 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=-\frac {16 \, b \arcsin \left (d x + c\right ) + 3 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 1\right ) - 3 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} - 1\right ) + 2 \, {\left (3 \, b d^{3} x^{3} + 9 \, b c d^{2} x^{2} + 3 \, b c^{3} + {\left (9 \, b c^{2} + 2 \, b\right )} d x + 2 \, b c\right )} \sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} + 16 \, a}{80 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} \]
-1/80*(16*b*arcsin(d*x + c) + 3*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3* x^3 + 10*b*c^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*log(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) + 1) - 3*(b*d^5*x^5 + 5*b*c*d^4*x^4 + 10*b*c^2*d^3*x^3 + 10*b*c ^3*d^2*x^2 + 5*b*c^4*d*x + b*c^5)*log(sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) - 1) + 2*(3*b*d^3*x^3 + 9*b*c*d^2*x^2 + 3*b*c^3 + (9*b*c^2 + 2*b)*d*x + 2*b *c)*sqrt(-d^2*x^2 - 2*c*d*x - c^2 + 1) + 16*a)/(d^6*e^6*x^5 + 5*c*d^5*e^6* x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^ 6)
\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=\frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {asin}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \]
(Integral(a/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 1 5*c**2*d**4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x) + Integral(b*asin(c + d* x)/(c**6 + 6*c**5*d*x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*d* *4*x**4 + 6*c*d**5*x**5 + d**6*x**6), x))/e**6
\[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=\int { \frac {b \arcsin \left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}} \,d x } \]
-1/5*(5*(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e ^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)*integrate(1/5*e^(1/2*log(d*x + c + 1 ) + 1/2*log(-d*x - c + 1))/(d^9*e^6*x^9 + 9*c*d^8*e^6*x^8 + (36*c^2 - 1)*d ^7*e^6*x^7 + 7*(12*c^3 - c)*d^6*e^6*x^6 + 21*(6*c^4 - c^2)*d^5*e^6*x^5 + 7 *(18*c^5 - 5*c^3)*d^4*e^6*x^4 + 7*(12*c^6 - 5*c^4)*d^3*e^6*x^3 + 3*(12*c^7 - 7*c^5)*d^2*e^6*x^2 + (9*c^8 - 7*c^6)*d*e^6*x + (c^9 - c^7)*e^6 + (d^7*e ^6*x^7 + 7*c*d^6*e^6*x^6 + (21*c^2 - 1)*d^5*e^6*x^5 + 5*(7*c^3 - c)*d^4*e^ 6*x^4 + 5*(7*c^4 - 2*c^2)*d^3*e^6*x^3 + (21*c^5 - 10*c^3)*d^2*e^6*x^2 + (7 *c^6 - 5*c^4)*d*e^6*x + (c^7 - c^5)*e^6)*e^(log(d*x + c + 1) + log(-d*x - c + 1))), x) + arctan2(d*x + c, sqrt(d*x + c + 1)*sqrt(-d*x - c + 1)))*b/( d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6) - 1/5*a/(d^6*e^6*x^5 + 5*c*d^5*e^6*x^4 + 10*c ^2*d^4*e^6*x^3 + 10*c^3*d^3*e^6*x^2 + 5*c^4*d^2*e^6*x + c^5*d*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 598 vs. \(2 (107) = 214\).
Time = 0.72 (sec) , antiderivative size = 598, normalized size of antiderivative = 4.94 \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=-\frac {{\left (d x + c\right )}^{5} b \arcsin \left (d x + c\right )}{160 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5}} - \frac {{\left (d x + c\right )}^{3} b \arcsin \left (d x + c\right )}{32 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} - \frac {{\left (d x + c\right )} b \arcsin \left (d x + c\right )}{16 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )} \arcsin \left (d x + c\right )}{16 \, {\left (d x + c\right )} d e^{6}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3} \arcsin \left (d x + c\right )}{32 \, {\left (d x + c\right )}^{3} d e^{6}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5} \arcsin \left (d x + c\right )}{160 \, {\left (d x + c\right )}^{5} d e^{6}} - \frac {3 \, b \log \left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}{40 \, d e^{6}} + \frac {3 \, b \log \left ({\left | d x + c \right |}\right )}{40 \, d e^{6}} - \frac {{\left (d x + c\right )}^{5} a}{160 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5}} + \frac {{\left (d x + c\right )}^{4} b}{320 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4}} - \frac {{\left (d x + c\right )}^{3} a}{32 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}} + \frac {{\left (d x + c\right )}^{2} b}{40 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}} - \frac {{\left (d x + c\right )} a}{16 \, d e^{6} {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}}{16 \, {\left (d x + c\right )} d e^{6}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{2}}{40 \, {\left (d x + c\right )}^{2} d e^{6}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{3}}{32 \, {\left (d x + c\right )}^{3} d e^{6}} - \frac {b {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{4}}{320 \, {\left (d x + c\right )}^{4} d e^{6}} - \frac {a {\left (\sqrt {-{\left (d x + c\right )}^{2} + 1} + 1\right )}^{5}}{160 \, {\left (d x + c\right )}^{5} d e^{6}} \]
-1/160*(d*x + c)^5*b*arcsin(d*x + c)/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1)^5 ) - 1/32*(d*x + c)^3*b*arcsin(d*x + c)/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1) ^3) - 1/16*(d*x + c)*b*arcsin(d*x + c)/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1) ) - 1/16*b*(sqrt(-(d*x + c)^2 + 1) + 1)*arcsin(d*x + c)/((d*x + c)*d*e^6) - 1/32*b*(sqrt(-(d*x + c)^2 + 1) + 1)^3*arcsin(d*x + c)/((d*x + c)^3*d*e^6 ) - 1/160*b*(sqrt(-(d*x + c)^2 + 1) + 1)^5*arcsin(d*x + c)/((d*x + c)^5*d* e^6) - 3/40*b*log(sqrt(-(d*x + c)^2 + 1) + 1)/(d*e^6) + 3/40*b*log(abs(d*x + c))/(d*e^6) - 1/160*(d*x + c)^5*a/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1)^5 ) + 1/320*(d*x + c)^4*b/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1)^4) - 1/32*(d*x + c)^3*a/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1)^3) + 1/40*(d*x + c)^2*b/(d*e ^6*(sqrt(-(d*x + c)^2 + 1) + 1)^2) - 1/16*(d*x + c)*a/(d*e^6*(sqrt(-(d*x + c)^2 + 1) + 1)) - 1/16*a*(sqrt(-(d*x + c)^2 + 1) + 1)/((d*x + c)*d*e^6) - 1/40*b*(sqrt(-(d*x + c)^2 + 1) + 1)^2/((d*x + c)^2*d*e^6) - 1/32*a*(sqrt( -(d*x + c)^2 + 1) + 1)^3/((d*x + c)^3*d*e^6) - 1/320*b*(sqrt(-(d*x + c)^2 + 1) + 1)^4/((d*x + c)^4*d*e^6) - 1/160*a*(sqrt(-(d*x + c)^2 + 1) + 1)^5/( (d*x + c)^5*d*e^6)
Timed out. \[ \int \frac {a+b \arcsin (c+d x)}{(c e+d e x)^6} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \]