Integrand size = 28, antiderivative size = 167 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\frac {c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {b^2 c \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{\sqrt {d+c^2 d x^2}} \]
c*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+2*b*c*(a+b*ar csinh(c*x))*ln(1-1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/(c^2*d*x^2 +d)^(1/2)-b^2*c*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)*(c^2*x^2+1)^(1/2)/( c^2*d*x^2+d)^(1/2)-(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/d/x
Time = 0.69 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\frac {b^2 \left (-1-c^2 x^2+c x \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-2 b \text {arcsinh}(c x) \left (a+a c^2 x^2-b c x \sqrt {1+c^2 x^2} \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )-a \left (a+a c^2 x^2-2 b c x \sqrt {1+c^2 x^2} \log (c x)\right )-b^2 c x \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )}{x \sqrt {d+c^2 d x^2}} \]
(b^2*(-1 - c^2*x^2 + c*x*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - 2*b*ArcSinh[c *x]*(a + a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2]*Log[1 - E^(-2*ArcSinh[c*x])]) - a*(a + a*c^2*x^2 - 2*b*c*x*Sqrt[1 + c^2*x^2]*Log[c*x]) - b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(x*Sqrt[d + c^2*d*x^2])
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {6215, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {c^2 d x^2+d}} \, dx\) |
| \(\Big \downarrow \) 6215 |
| \(\displaystyle \frac {2 b c \sqrt {c^2 x^2+1} \int \frac {a+b \text {arcsinh}(c x)}{x}dx}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle \frac {2 c \sqrt {c^2 x^2+1} \int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 c \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c \sqrt {c^2 x^2+1} \int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{d x}+\frac {2 i c \sqrt {c^2 x^2+1} \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 d x^2+d}}\) |
-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(d*x)) + ((2*I)*c*Sqrt[1 + c^2*x^2]*((-1/2*I)*(a + b*ArcSinh[c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[ c*x])*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c*x]))/b)]) + (b^2*Pol yLog[2, -a - b*ArcSinh[c*x]])/4)))/Sqrt[d + c^2*d*x^2]
3.3.97.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
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. \(477\) vs. \(2(171)=342\).
Time = 0.26 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.86
| method | result | size |
| default | \(-\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\left (c^{2} x^{2}+1\right ) d x}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\left (c^{2} x^{2}+1\right ) d x}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\right )\) | \(478\) |
| parts | \(-\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{d x}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )^{2}}{\left (c^{2} x^{2}+1\right ) d x}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\right )+2 a b \left (-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\left (c^{2} x^{2}+1\right ) d x}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\right )\) | \(478\) |
-a^2/d/x*(c^2*d*x^2+d)^(1/2)+b^2*(-(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2 *x^2+1)^(1/2)+1)*arcsinh(c*x)^2/(c^2*x^2+1)/d/x-2*(d*(c^2*x^2+1))^(1/2)/(c ^2*x^2+1)^(1/2)/d*arcsinh(c*x)^2*c+2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/ 2)/d*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*c+2*(d*(c^2*x^2+1))^(1/2)/(c ^2*x^2+1)^(1/2)/d*polylog(2,-c*x-(c^2*x^2+1)^(1/2))*c+2*(d*(c^2*x^2+1))^(1 /2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))*c+2*(d*(c ^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*polylog(2,c*x+(c^2*x^2+1)^(1/2))*c)+2 *a*b*(-2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*arcsinh(c*x)*c-(d*(c^2* x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*arcsinh(c*x)/(c^2*x^2+1)/d /x+(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d*ln((c*x+(c^2*x^2+1)^(1/2))^2- 1)*c)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{2}} \,d x } \]
integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^ 2)/(c^2*d*x^4 + d*x^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{2}} \,d x } \]
-((-1)^(2*c^2*d*x^2 + 2*d)*sqrt(d)*log(2*c^2*d + 2*d/x^2) - sqrt(d)*log(x^ 2 + 1/c^2))*a*b*c/d + b^2*integrate(log(c*x + sqrt(c^2*x^2 + 1))^2/(sqrt(c ^2*d*x^2 + d)*x^2), x) - 2*sqrt(c^2*d*x^2 + d)*a*b*arcsinh(c*x)/(d*x) - sq rt(c^2*d*x^2 + d)*a^2/(d*x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {c^{2} d x^{2} + d} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \]