Integrand size = 30, antiderivative size = 71 \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\frac {-1-(a+b x)^2}{2 b \text {arcsinh}(a+b x)^2}-\frac {(a+b x) \sqrt {1+(a+b x)^2}}{b \text {arcsinh}(a+b x)}+\frac {\text {Chi}(2 \text {arcsinh}(a+b x))}{b} \]
1/2*(-1-(b*x+a)^2)/b/arcsinh(b*x+a)^2+Chi(2*arcsinh(b*x+a))/b-(b*x+a)*(1+( b*x+a)^2)^(1/2)/b/arcsinh(b*x+a)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=-\frac {1+a^2+2 a b x+b^2 x^2+2 (a+b x) \sqrt {1+a^2+2 a b x+b^2 x^2} \text {arcsinh}(a+b x)-2 \text {arcsinh}(a+b x)^2 \text {Chi}(2 \text {arcsinh}(a+b x))}{2 b \text {arcsinh}(a+b x)^2} \]
-1/2*(1 + a^2 + 2*a*b*x + b^2*x^2 + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b ^2*x^2]*ArcSinh[a + b*x] - 2*ArcSinh[a + b*x]^2*CoshIntegral[2*ArcSinh[a + b*x]])/(b*ArcSinh[a + b*x]^2)
Time = 0.43 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6275, 6205, 6193, 3042, 3782}
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.
| \(\displaystyle \int \frac {\sqrt {a^2+2 a b x+b^2 x^2+1}}{\text {arcsinh}(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 6275 |
| \(\displaystyle \frac {\int \frac {\sqrt {(a+b x)^2+1}}{\text {arcsinh}(a+b x)^3}d(a+b x)}{b}\) |
| \(\Big \downarrow \) 6205 |
| \(\displaystyle \frac {\int \frac {a+b x}{\text {arcsinh}(a+b x)^2}d(a+b x)-\frac {(a+b x)^2+1}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {\int \frac {\cosh (2 \text {arcsinh}(a+b x))}{\text {arcsinh}(a+b x)}d\text {arcsinh}(a+b x)-\frac {\sqrt {(a+b x)^2+1} (a+b x)}{\text {arcsinh}(a+b x)}-\frac {(a+b x)^2+1}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (2 i \text {arcsinh}(a+b x)+\frac {\pi }{2}\right )}{\text {arcsinh}(a+b x)}d\text {arcsinh}(a+b x)-\frac {\sqrt {(a+b x)^2+1} (a+b x)}{\text {arcsinh}(a+b x)}-\frac {(a+b x)^2+1}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle \frac {\text {Chi}(2 \text {arcsinh}(a+b x))-\frac {\sqrt {(a+b x)^2+1} (a+b x)}{\text {arcsinh}(a+b x)}-\frac {(a+b x)^2+1}{2 \text {arcsinh}(a+b x)^2}}{b}\) |
(-1/2*(1 + (a + b*x)^2)/ArcSinh[a + b*x]^2 - ((a + b*x)*Sqrt[1 + (a + b*x) ^2])/ArcSinh[a + b*x] + CoshIntegral[2*ArcSinh[a + b*x]])/b
3.3.65.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
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_) + (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 = 0.59 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {4 \,\operatorname {Chi}\left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )^{2}-2 \sinh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right ) \operatorname {arcsinh}\left (b x +a \right )-\cosh \left (2 \,\operatorname {arcsinh}\left (b x +a \right )\right )-1}{4 b \operatorname {arcsinh}\left (b x +a \right )^{2}}\) | \(63\) |
1/4/b*(4*Chi(2*arcsinh(b*x+a))*arcsinh(b*x+a)^2-2*sinh(2*arcsinh(b*x+a))*a rcsinh(b*x+a)-cosh(2*arcsinh(b*x+a))-1)/arcsinh(b*x+a)^2
\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{\operatorname {asinh}^{3}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
-1/2*((b^4*x^4 + 4*a*b^3*x^3 + a^4 + (6*a^2*b^2 + b^2)*x^2 + a^2 + 2*(2*a^ 3*b + a*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + (3*b^5*x^5 + 15*a*b^4*x^4 + 3*a^5 + 5*(6*a^2*b^3 + b^3)*x^3 + 5*a^3 + 15*(2*a^3*b^2 + a*b^2)*x^2 + ( 15*a^4*b + 15*a^2*b + 2*b)*x + 2*a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (3*b^6*x^6 + 18*a*b^5*x^5 + 3*a^6 + (45*a^2*b^4 + 7*b^4)*x^4 + 7*a^4 + 4*( 15*a^3*b^3 + 7*a*b^3)*x^3 + (45*a^4*b^2 + 42*a^2*b^2 + 5*b^2)*x^2 + 5*a^2 + 2*(9*a^5*b + 14*a^3*b + 5*a*b)*x + 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + (( 2*b^4*x^4 + 8*a*b^3*x^3 + 2*a^4 + (12*a^2*b^2 + b^2)*x^2 + a^2 + 2*(4*a^3* b + a*b)*x - 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^2 + (6*b^5*x^5 + 30*a*b^4*x^ 4 + 6*a^5 + (60*a^2*b^3 + 7*b^3)*x^3 + 7*a^3 + 3*(20*a^3*b^2 + 7*a*b^2)*x^ 2 + (30*a^4*b + 21*a^2*b + b)*x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + (6*b^6*x^6 + 36*a*b^5*x^5 + 6*a^6 + (90*a^2*b^4 + 11*b^4)*x^4 + 11*a^4 + 4*(30*a^3*b^3 + 11*a*b^3)*x^3 + 6*(15*a^4*b^2 + 11*a^2*b^2 + b^2)*x^2 + 6* a^2 + 4*(9*a^5*b + 11*a^3*b + 3*a*b)*x + 1)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + (2*b^7*x^7 + 14*a*b^6*x^6 + 2*a^7 + (42*a^2*b^5 + 5*b^5)*x^5 + 5*a^5 + 5 *(14*a^3*b^4 + 5*a*b^4)*x^4 + 2*(35*a^4*b^3 + 25*a^2*b^3 + 2*b^3)*x^3 + 4* a^3 + 2*(21*a^5*b^2 + 25*a^3*b^2 + 6*a*b^2)*x^2 + (14*a^6*b + 25*a^4*b + 1 2*a^2*b + b)*x + a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt( b^2*x^2 + 2*a*b*x + a^2 + 1)) + (b^7*x^7 + 7*a*b^6*x^6 + a^7 + 3*(7*a^2*b^ 5 + b^5)*x^5 + 3*a^5 + 5*(7*a^3*b^4 + 3*a*b^4)*x^4 + (35*a^4*b^3 + 30*a...
\[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\int { \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{\operatorname {arsinh}\left (b x + a\right )^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {1+a^2+2 a b x+b^2 x^2}}{\text {arcsinh}(a+b x)^3} \, dx=\int \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}}{{\mathrm {asinh}\left (a+b\,x\right )}^3} \,d x \]