Integrand size = 30, antiderivative size = 84 \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=-\frac {\left (1+(a+b x)^2\right )^2}{2 b \text {arcsinh}(a+b x)^2}-\frac {2 (a+b x) \left (1+(a+b x)^2\right )^{3/2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b}+\frac {\text {Chi}(4 \text {arcsinh}(a+b x))}{b} \]
-1/2*(1+(b*x+a)^2)^2/b/arcsinh(b*x+a)^2-2*(b*x+a)*(1+(b*x+a)^2)^(3/2)/b/ar csinh(b*x+a)+Chi(2*arcsinh(b*x+a))/b+Chi(4*arcsinh(b*x+a))/b
Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.29 \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\frac {-\frac {\left (1+a^2+2 a b x+b^2 x^2\right ) \left (1+a^2+2 a b x+b^2 x^2+4 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)\right )}{\text {arcsinh}(a+b x)^2}+2 \text {Chi}(2 \text {arcsinh}(a+b x))+2 \text {Chi}(4 \text {arcsinh}(a+b x))}{2 b} \]
(-(((1 + a^2 + 2*a*b*x + b^2*x^2)*(1 + a^2 + 2*a*b*x + b^2*x^2 + 4*(a + b* x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]))/ArcSinh[a + b*x]^2 ) + 2*CoshIntegral[2*ArcSinh[a + b*x]] + 2*CoshIntegral[4*ArcSinh[a + b*x] ])/(2*b)
Time = 1.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6275, 6205, 6229, 6206, 3042, 3793, 2009, 6234, 5971, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 6275 |
| \(\displaystyle \frac {\int \frac {\left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)^3}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 6205 |
| \(\displaystyle \frac {2 \int \frac {(a+b x) \left ((a+b x)^2+1\right )}{\text {arcsinh}(a+b x)^2}d(a+b x)-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 6229 |
| \(\displaystyle \frac {2 \left (\int \frac {\sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)+4 \int \frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 6206 |
| \(\displaystyle \frac {2 \left (4 \int \frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)+\int \frac {(a+b x)^2+1}{\text {arcsinh}(a+b x)}d\text {arcsinh}(a+b x)-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}+2 \left (4 \int \frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)+\int \frac {\sin \left (i \text {arcsinh}(a+b x)+\frac {\pi }{2}\right )^2}{\text {arcsinh}(a+b x)}d\text {arcsinh}(a+b x)-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}\right )}{b}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 \left (4 \int \frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)+\int \left (\frac {\cosh (2 \text {arcsinh}(a+b x))}{2 \text {arcsinh}(a+b x)}+\frac {1}{2 \text {arcsinh}(a+b x)}\right )d\text {arcsinh}(a+b x)-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (4 \int \frac {(a+b x)^2 \sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)}d(a+b x)+\frac {1}{2} \text {Chi}(2 \text {arcsinh}(a+b x))-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}+\frac {1}{2} \log (\text {arcsinh}(a+b x))\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {2 \left (4 \int \frac {(a+b x)^2 \left ((a+b x)^2+1\right )}{\text {arcsinh}(a+b x)}d\text {arcsinh}(a+b x)+\frac {1}{2} \text {Chi}(2 \text {arcsinh}(a+b x))-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}+\frac {1}{2} \log (\text {arcsinh}(a+b x))\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {2 \left (4 \int \left (\frac {\cosh (4 \text {arcsinh}(a+b x))}{8 \text {arcsinh}(a+b x)}-\frac {1}{8 \text {arcsinh}(a+b x)}\right )d\text {arcsinh}(a+b x)+\frac {1}{2} \text {Chi}(2 \text {arcsinh}(a+b x))-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}+\frac {1}{2} \log (\text {arcsinh}(a+b x))\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \text {Chi}(2 \text {arcsinh}(a+b x))+4 \left (\frac {1}{8} \text {Chi}(4 \text {arcsinh}(a+b x))-\frac {1}{8} \log (\text {arcsinh}(a+b x))\right )-\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{\text {arcsinh}(a+b x)}+\frac {1}{2} \log (\text {arcsinh}(a+b x))\right )-\frac {\left ((a+b x)^2+1\right )^2}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
(-1/2*(1 + (a + b*x)^2)^2/ArcSinh[a + b*x]^2 + 2*(-(((a + b*x)*(1 + (a + b *x)^2)^(3/2))/ArcSinh[a + b*x]) + CoshIntegral[2*ArcSinh[a + b*x]]/2 + 4*( CoshIntegral[4*ArcSinh[a + b*x]]/8 - Log[ArcSinh[a + b*x]]/8) + Log[ArcSin h[a + b*x]]/2))/b
3.3.71.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[Simp[Sqrt[1 + c^2*x^2]*(d + e*x^2)^p]*((a + b*ArcSinh[c*x] )^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x ^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x]) ^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Subst[Int [x^n*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Sqrt[1 + c^2*x^2]*(d + e*x^2)^p *((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (-Simp[f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f* (n + 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}, x] && EqQ[e, c^2*d] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1 , 0] && IGtQ[m, -3]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(C/d^2 + (C/d^2)*x^2 )^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B, C , n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Time = 0.87 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {16 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}+16 \,\operatorname {Chi}\left (4 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-8 \sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-4 \sinh \left (4 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-4 \cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-\cosh \left (4 \,\operatorname {arcsinh}\left (b x +a \right )\right )-3}{16 b \operatorname {arcsinh}\left (b x +a \right )^{2}}\) | \(110\) |
1/16/b*(16*Chi(2*arcsinh(b*x+a))*arcsinh(b*x+a)^2+16*Chi(4*arcsinh(b*x+a)) *arcsinh(b*x+a)^2-8*sinh(2*arcsinh(b*x+a))*arcsinh(b*x+a)-4*sinh(4*arcsinh (b*x+a))*arcsinh(b*x+a)-4*cosh(2*arcsinh(b*x+a))-cosh(4*arcsinh(b*x+a))-3) /arcsinh(b*x+a)^2
\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}}}{\operatorname {asinh}^{3}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
-1/2*((b^6*x^6 + 6*a*b^5*x^5 + a^6 + (15*a^2*b^4 + 2*b^4)*x^4 + 2*a^4 + 4* (5*a^3*b^3 + 2*a*b^3)*x^3 + (15*a^4*b^2 + 12*a^2*b^2 + b^2)*x^2 + a^2 + 2* (3*a^5*b + 4*a^3*b + a*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + (3*b^7*x^7 + 21*a*b^6*x^6 + 3*a^7 + (63*a^2*b^5 + 8*b^5)*x^5 + 8*a^5 + 5*(21*a^3*b^4 + 8*a*b^4)*x^4 + (105*a^4*b^3 + 80*a^2*b^3 + 7*b^3)*x^3 + 7*a^3 + (63*a^5* b^2 + 80*a^3*b^2 + 21*a*b^2)*x^2 + (21*a^6*b + 40*a^4*b + 21*a^2*b + 2*b)* x + 2*a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*b^8*x^8 + 24*a*b^7*x^7 + 3*a^8 + 2*(42*a^2*b^6 + 5*b^6)*x^6 + 10*a^6 + 12*(14*a^3*b^5 + 5*a*b^5)*x ^5 + 6*(35*a^4*b^4 + 25*a^2*b^4 + 2*b^4)*x^4 + 12*a^4 + 8*(21*a^5*b^3 + 25 *a^3*b^3 + 6*a*b^3)*x^3 + 6*(14*a^6*b^2 + 25*a^4*b^2 + 12*a^2*b^2 + b^2)*x ^2 + 6*a^2 + 12*(2*a^7*b + 5*a^5*b + 4*a^3*b + a*b)*x + 1)*(b^2*x^2 + 2*a* b*x + a^2 + 1) + ((4*b^6*x^6 + 24*a*b^5*x^5 + 4*a^6 + (60*a^2*b^4 + 7*b^4) *x^4 + 7*a^4 + 4*(20*a^3*b^3 + 7*a*b^3)*x^3 + 2*(30*a^4*b^2 + 21*a^2*b^2 + b^2)*x^2 + 2*a^2 + 4*(6*a^5*b + 7*a^3*b + a*b)*x - 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + 3*(4*b^7*x^7 + 28*a*b^6*x^6 + 4*a^7 + 3*(28*a^2*b^5 + 3*b^5 )*x^5 + 9*a^5 + 5*(28*a^3*b^4 + 9*a*b^4)*x^4 + 2*(70*a^4*b^3 + 45*a^2*b^3 + 3*b^3)*x^3 + 6*a^3 + 6*(14*a^5*b^2 + 15*a^3*b^2 + 3*a*b^2)*x^2 + (28*a^6 *b + 45*a^4*b + 18*a^2*b + b)*x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (12*b^8*x^8 + 96*a*b^7*x^7 + 12*a^8 + 3*(112*a^2*b^6 + 11*b^6)*x^6 + 33*a ^6 + 6*(112*a^3*b^5 + 33*a*b^5)*x^5 + (840*a^4*b^4 + 495*a^2*b^4 + 31*b...
\[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2}}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}}{{\mathrm {asinh}\left (a+b\,x\right )}^3} \,d x \]