Integrand size = 24, antiderivative size = 187 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{4} b c^3 d^2 x \sqrt {1+c^2 x^2}-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2}}{2 x}+\frac {1}{4} b c^2 d^2 \text {arcsinh}(c x)+c^2 d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))-\frac {d^2 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {c^2 d^2 (a+b \text {arcsinh}(c x))^2}{b}+2 c^2 d^2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )-b c^2 d^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right ) \]
-1/2*b*c*d^2*(c^2*x^2+1)^(3/2)/x+1/4*b*c^2*d^2*arcsinh(c*x)+c^2*d^2*(c^2*x ^2+1)*(a+b*arcsinh(c*x))-1/2*d^2*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))/x^2+c^2* d^2*(a+b*arcsinh(c*x))^2/b+2*c^2*d^2*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x ^2+1)^(1/2))^2)-b*c^2*d^2*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)+1/4*b*c^3 *d^2*x*(c^2*x^2+1)^(1/2)
Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{4} d^2 \left (2 a c^4 x^2-\frac {2 b c \sqrt {1+c^2 x^2}}{x}-b c^3 x \sqrt {1+c^2 x^2}+b c^2 \text {arcsinh}(c x)+2 b c^4 x^2 \text {arcsinh}(c x)-\frac {2 (a+b \text {arcsinh}(c x))}{x^2}-\frac {4 c^2 (a+b \text {arcsinh}(c x))^2}{b}+4 c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )\right ) \]
(d^2*(2*a*c^4*x^2 - (2*b*c*Sqrt[1 + c^2*x^2])/x - b*c^3*x*Sqrt[1 + c^2*x^2 ] + b*c^2*ArcSinh[c*x] + 2*b*c^4*x^2*ArcSinh[c*x] - (2*(a + b*ArcSinh[c*x] ))/x^2 - (4*c^2*(a + b*ArcSinh[c*x])^2)/b + 4*c^2*(2*(a + b*ArcSinh[c*x])* Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2*ArcSinh[c*x])])))/4
Result contains complex when optimal does not.
Time = 1.03 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.34, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6217, 27, 247, 211, 222, 6216, 211, 222, 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.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx\) |
| \(\Big \downarrow \) 6217 |
| \(\displaystyle 2 c^2 d \int \frac {d \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c d^2 \int \frac {\left (c^2 x^2+1\right )^{3/2}}{x^2}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c d^2 \int \frac {\left (c^2 x^2+1\right )^{3/2}}{x^2}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 247 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c d^2 \left (3 c^2 \int \sqrt {c^2 x^2+1}dx-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle 2 c^2 d^2 \int \frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{x}dx-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 6216 |
| \(\displaystyle 2 c^2 d^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \int \sqrt {c^2 x^2+1}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle 2 c^2 d^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx-\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle 2 c^2 d^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x}dx+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 6190 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {\int -\left ((a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 c^2 d^2 \left (-\frac {\int (a+b \text {arcsinh}(c x)) \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 2 c^2 d^2 \left (-\frac {\int -i (a+b \text {arcsinh}(c x)) \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \int (a+b \text {arcsinh}(c x)) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )d(a+b \text {arcsinh}(c x))}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi } (a+b \text {arcsinh}(c x))}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 2 c^2 d^2 \left (\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}+\frac {1}{2} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )\right )-\frac {d^2 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c d^2 \left (3 c^2 \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )-\frac {\left (c^2 x^2+1\right )^{3/2}}{x}\right )\) |
-1/2*(d^2*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/x^2 + (b*c*d^2*(-((1 + c^2 *x^2)^(3/2)/x) + 3*c^2*((x*Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c))))/2 + 2*c^2*d^2*(((1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^ 2*x^2])/2 + ArcSinh[c*x]/(2*c)))/2 + (I*((-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*PolyLog[2, -a - b*ArcSinh[c*x]])/4)))/b)
3.1.17.3.1 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[(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_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.))*((d_) + (e_.)*(x_)^2)^(p_.))/(x_), x_Symbol] :> Simp[(d + e*x^2)^p*((a + b*ArcSinh[c*x])/(2*p)), x] + (Simp[d Int[(d + e*x^2)^(p - 1)*((a + b*ArcSinh[c*x])/x), x], x] - Simp[b*c*(d^p /(2*p)) Int[(1 + c^2*x^2)^(p - 1/2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcSinh[c *x])/(f*(m + 1))), x] + (-Simp[b*c*(d^p/(f*(m + 1))) Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2), x], x] - Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]
Time = 0.24 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.28
| method | result | size |
| parts | \(d^{2} a \left (\frac {c^{4} x^{2}}{2}-\frac {1}{2 x^{2}}+2 c^{2} \ln \left (x \right )\right )+d^{2} b \,c^{2} \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\) | \(240\) |
| derivativedivides | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(243\) |
| default | \(c^{2} \left (d^{2} a \left (\frac {c^{2} x^{2}}{2}+2 \ln \left (c x \right )-\frac {1}{2 c^{2} x^{2}}\right )+d^{2} b \left (-\operatorname {arcsinh}\left (c x \right )^{2}+\frac {\left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right ) \left (2 c^{2} x^{2}+1+2 c x \sqrt {c^{2} x^{2}+1}\right )}{16}+\frac {\left (-2 c x \sqrt {c^{2} x^{2}+1}+2 c^{2} x^{2}+1\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16}-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )\right )\) | \(243\) |
d^2*a*(1/2*c^4*x^2-1/2/x^2+2*c^2*ln(x))+d^2*b*c^2*(-arcsinh(c*x)^2+1/16*(- 1+2*arcsinh(c*x))*(2*c^2*x^2+1+2*c*x*(c^2*x^2+1)^(1/2))+1/16*(-2*c*x*(c^2* x^2+1)^(1/2)+2*c^2*x^2+1)*(1+2*arcsinh(c*x))-1/2*(c*x*(c^2*x^2+1)^(1/2)-c^ 2*x^2+arcsinh(c*x))/c^2/x^2+2*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))+2*p olylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2) )+2*polylog(2,c*x+(c^2*x^2+1)^(1/2)))
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
integral((a*c^4*d^2*x^4 + 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 + 2*b*c ^2*d^2*x^2 + b*d^2)*arcsinh(c*x))/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \frac {2 a c^{2}}{x}\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b c^{2} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
d**2*(Integral(a/x**3, x) + Integral(2*a*c**2/x, x) + Integral(a*c**4*x, x ) + Integral(b*asinh(c*x)/x**3, x) + Integral(2*b*c**2*asinh(c*x)/x, x) + Integral(b*c**4*x*asinh(c*x), x))
\[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/2*a*c^4*d^2*x^2 + 2*a*c^2*d^2*log(x) - 1/2*b*d^2*(sqrt(c^2*x^2 + 1)*c/x + arcsinh(c*x)/x^2) - 1/2*a*d^2/x^2 + integrate(b*c^4*d^2*x*log(c*x + sqrt (c^2*x^2 + 1)) + 2*b*c^2*d^2*log(c*x + sqrt(c^2*x^2 + 1))/x, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2}{x^3} \,d x \]