Integrand size = 26, antiderivative size = 78 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {b c \pi ^{3/2}}{20 x^4}-\frac {b c^3 \pi ^{3/2}}{5 x^2}-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}+\frac {1}{5} b c^5 \pi ^{3/2} \log (x) \] Output:
-1/20*b*c*Pi^(3/2)/x^4-1/5*b*c^3*Pi^(3/2)/x^2-1/5*(Pi*c^2*x^2+Pi)^(5/2)*(a +b*arcsinh(c*x))/Pi/x^5+1/5*b*c^5*Pi^(3/2)*ln(x)
Time = 0.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.63 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {\pi ^{3/2} \left (-3 b c x-12 b c^3 x^3-25 b c^5 x^5-12 a \sqrt {1+c^2 x^2}-24 a c^2 x^2 \sqrt {1+c^2 x^2}-12 a c^4 x^4 \sqrt {1+c^2 x^2}-12 b \left (1+c^2 x^2\right )^{5/2} \text {arcsinh}(c x)+12 b c^5 x^5 \log (x)\right )}{60 x^5} \] Input:
Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^6,x]
Output:
(Pi^(3/2)*(-3*b*c*x - 12*b*c^3*x^3 - 25*b*c^5*x^5 - 12*a*Sqrt[1 + c^2*x^2] - 24*a*c^2*x^2*Sqrt[1 + c^2*x^2] - 12*a*c^4*x^4*Sqrt[1 + c^2*x^2] - 12*b* (1 + c^2*x^2)^(5/2)*ArcSinh[c*x] + 12*b*c^5*x^5*Log[x]))/(60*x^5)
Time = 0.41 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6215, 243, 49, 2009}
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.
\(\displaystyle \int \frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx\) |
\(\Big \downarrow \) 6215 |
\(\displaystyle \frac {1}{5} \pi ^{3/2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^5}dx-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{10} \pi ^{3/2} b c \int \frac {\left (c^2 x^2+1\right )^2}{x^6}dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{10} \pi ^{3/2} b c \int \left (\frac {c^4}{x^2}+\frac {2 c^2}{x^4}+\frac {1}{x^6}\right )dx^2-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{10} \pi ^{3/2} b c \left (c^4 \log \left (x^2\right )-\frac {2 c^2}{x^2}-\frac {1}{2 x^4}\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi x^5}\) |
Input:
Int[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^6,x]
Output:
-1/5*((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(Pi*x^5) + (b*c*Pi^(3/ 2)*(-1/2*1/x^4 - (2*c^2)/x^2 + c^4*Log[x^2]))/10
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e *x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b *ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ [e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1188\) vs. \(2(62)=124\).
Time = 1.19 (sec) , antiderivative size = 1189, normalized size of antiderivative = 15.24
method | result | size |
default | \(\text {Expression too large to display}\) | \(1189\) |
parts | \(\text {Expression too large to display}\) | \(1189\) |
Input:
int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(x*c))/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*a/Pi/x^5*(Pi*c^2*x^2+Pi)^(5/2)-2/5*b*Pi^(3/2)*c^5*arcsinh(x*c)+b*Pi^( 3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^8*arcsinh(x*c)*c^13-b *Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^7*(c^2*x^2+1)^(1 /2)*arcsinh(x*c)*c^12+1/5*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^ 2*x^2+1)*x^8*c^13-1/5*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^ 2+1)*x^6*(c^2*x^2+1)*c^11+2*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5* c^2*x^2+1)*x^6*arcsinh(x*c)*c^11-4*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4 *x^4+5*c^2*x^2+1)*x^5*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^10+1/20*b*Pi^(3/2)/ (5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^6*c^11-17/20*b*Pi^(3/2)/(5 *c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^4*(c^2*x^2+1)*c^9+2*b*Pi^(3/ 2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^4*arcsinh(x*c)*c^9-7*b* Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*x^3*(c^2*x^2+1)^(1/ 2)*arcsinh(x*c)*c^8-7/5*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2* x^2+1)*x^2*(c^2*x^2+1)*c^7+b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c ^2*x^2+1)*x^2*arcsinh(x*c)*c^7-7*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x ^4+5*c^2*x^2+1)*x*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^6-11/10*b*Pi^(3/2)/(5*c ^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*(c^2*x^2+1)*c^5+1/5*b*Pi^(3/2)/( 5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)*arcsinh(x*c)*c^5-21/5*b*Pi^(3 /2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+1)/x*(c^2*x^2+1)^(1/2)*arcs inh(x*c)*c^4-2/5*b*Pi^(3/2)/(5*c^8*x^8+10*c^6*x^6+10*c^4*x^4+5*c^2*x^2+...
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (62) = 124\).
Time = 0.14 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.40 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {4 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (\pi b c^{6} x^{6} + 3 \, \pi b c^{4} x^{4} + 3 \, \pi b c^{2} x^{2} + \pi b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi } {\left (\pi b c^{7} x^{7} + \pi b c^{5} x^{5}\right )} \log \left (\frac {\pi + \pi c^{2} x^{6} + \pi c^{2} x^{2} + \pi x^{4} + \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} {\left (x^{4} - 1\right )}}{c^{2} x^{4} + x^{2}}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (4 \, \pi a c^{6} x^{6} + 12 \, \pi a c^{4} x^{4} + 12 \, \pi a c^{2} x^{2} + 4 \, \pi a + {\left (4 \, \pi b c^{3} x^{3} - \pi {\left (4 \, b c^{3} + b c\right )} x^{5} + \pi b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{20 \, {\left (c^{2} x^{7} + x^{5}\right )}} \] Input:
integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="frica s")
Output:
-1/20*(4*sqrt(pi + pi*c^2*x^2)*(pi*b*c^6*x^6 + 3*pi*b*c^4*x^4 + 3*pi*b*c^2 *x^2 + pi*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 2*sqrt(pi)*(pi*b*c^7*x^7 + pi* b*c^5*x^5)*log((pi + pi*c^2*x^6 + pi*c^2*x^2 + pi*x^4 + sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 + 1)*(x^4 - 1))/(c^2*x^4 + x^2)) + sqrt(pi + pi* c^2*x^2)*(4*pi*a*c^6*x^6 + 12*pi*a*c^4*x^4 + 12*pi*a*c^2*x^2 + 4*pi*a + (4 *pi*b*c^3*x^3 - pi*(4*b*c^3 + b*c)*x^5 + pi*b*c*x)*sqrt(c^2*x^2 + 1)))/(c^ 2*x^7 + x^5)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\pi ^{\frac {3}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{6}}\, dx + \int \frac {a c^{2} \sqrt {c^{2} x^{2} + 1}}{x^{4}}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{6}}\, dx + \int \frac {b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{4}}\, dx\right ) \] Input:
integrate((pi*c**2*x**2+pi)**(3/2)*(a+b*asinh(c*x))/x**6,x)
Output:
pi**(3/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**6, x) + Integral(a*c**2*sqrt( c**2*x**2 + 1)/x**4, x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**6, x) + Integral(b*c**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**4, x))
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (62) = 124\).
Time = 0.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.15 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=-\frac {{\left (2 \, \pi ^{\frac {5}{2}} \left (-1\right )^{2 \, \pi + 2 \, \pi c^{2} x^{2}} c^{4} \log \left (2 \, \pi c^{2} + \frac {2 \, \pi }{x^{2}}\right ) - 2 \, \pi ^{\frac {5}{2}} c^{4} \log \left (x^{2} + \frac {1}{c^{2}}\right ) + \frac {3 \, \pi ^{2} \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}} c^{2}}{x^{2}} + \frac {\pi ^{2} \sqrt {\pi + \pi c^{4} x^{4} + 2 \, \pi c^{2} x^{2}}}{x^{4}}\right )} b c}{20 \, \pi } - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} b \operatorname {arsinh}\left (c x\right )}{5 \, \pi x^{5}} - \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} a}{5 \, \pi x^{5}} \] Input:
integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="maxim a")
Output:
-1/20*(2*pi^(5/2)*(-1)^(2*pi + 2*pi*c^2*x^2)*c^4*log(2*pi*c^2 + 2*pi/x^2) - 2*pi^(5/2)*c^4*log(x^2 + 1/c^2) + 3*pi^2*sqrt(pi + pi*c^4*x^4 + 2*pi*c^2 *x^2)*c^2/x^2 + pi^2*sqrt(pi + pi*c^4*x^4 + 2*pi*c^2*x^2)/x^4)*b*c/pi - 1/ 5*(pi + pi*c^2*x^2)^(5/2)*b*arcsinh(c*x)/(pi*x^5) - 1/5*(pi + pi*c^2*x^2)^ (5/2)*a/(pi*x^5)
Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((pi*c^2*x^2+pi)^(3/2)*(a+b*arcsinh(c*x))/x^6,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^6} \,d x \] Input:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^6,x)
Output:
int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(3/2))/x^6, x)
\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x^6} \, dx=\frac {\sqrt {\pi }\, \pi \left (-\sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-2 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a +5 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{6}}d x \right ) b \,x^{5}+5 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) b \,c^{2} x^{5}-a \,c^{5} x^{5}\right )}{5 x^{5}} \] Input:
int((Pi*c^2*x^2+Pi)^(3/2)*(a+b*asinh(c*x))/x^6,x)
Output:
(sqrt(pi)*pi*( - sqrt(c**2*x**2 + 1)*a*c**4*x**4 - 2*sqrt(c**2*x**2 + 1)*a *c**2*x**2 - sqrt(c**2*x**2 + 1)*a + 5*int((sqrt(c**2*x**2 + 1)*asinh(c*x) )/x**6,x)*b*x**5 + 5*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4,x)*b*c**2*x **5 - a*c**5*x**5))/(5*x**5)