Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {2 b (a+b \text {arcsinh}(c+d x))^3}{d e^3}-\frac {2 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 d e^3 (c+d x)^2}+\frac {6 b^2 (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {6 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^4 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3} \]
2*b*(a+b*arcsinh(d*x+c))^3/d/e^3-1/2*(a+b*arcsinh(d*x+c))^4/d/e^3/(d*x+c)^ 2+6*b^2*(a+b*arcsinh(d*x+c))^2*ln(1-1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3 -6*b^3*(a+b*arcsinh(d*x+c))*polylog(2,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e ^3-3*b^4*polylog(3,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e^3-2*b*(a+b*arcsinh (d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d/e^3/(d*x+c)
Time = 1.35 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=-\frac {\frac {a^4}{(c+d x)^2}+\frac {4 a^3 b \sqrt {1+(c+d x)^2}}{c+d x}+\frac {4 a^3 b \text {arcsinh}(c+d x)}{(c+d x)^2}+\frac {b^4 \text {arcsinh}(c+d x)^4}{(c+d x)^2}-12 a^2 b^2 \left (-\frac {\sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}-\frac {\text {arcsinh}(c+d x)^2}{2 (c+d x)^2}+\log (c+d x)\right )-4 a b^3 \left (\text {arcsinh}(c+d x) \left (3 \text {arcsinh}(c+d x)-\frac {3 \sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}-\frac {\text {arcsinh}(c+d x)^2}{(c+d x)^2}+6 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )-3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )\right )-2 b^4 \left (2 \text {arcsinh}(c+d x)^2 \left (\text {arcsinh}(c+d x)-\frac {\sqrt {1+(c+d x)^2} \text {arcsinh}(c+d x)}{c+d x}+3 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )-6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )-3 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )\right )}{2 d e^3} \]
-1/2*(a^4/(c + d*x)^2 + (4*a^3*b*Sqrt[1 + (c + d*x)^2])/(c + d*x) + (4*a^3 *b*ArcSinh[c + d*x])/(c + d*x)^2 + (b^4*ArcSinh[c + d*x]^4)/(c + d*x)^2 - 12*a^2*b^2*(-((Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x)) - ArcSin h[c + d*x]^2/(2*(c + d*x)^2) + Log[c + d*x]) - 4*a*b^3*(ArcSinh[c + d*x]*( 3*ArcSinh[c + d*x] - (3*Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x) - ArcSinh[c + d*x]^2/(c + d*x)^2 + 6*Log[1 - E^(-2*ArcSinh[c + d*x])]) - 3 *PolyLog[2, E^(-2*ArcSinh[c + d*x])]) - 2*b^4*(2*ArcSinh[c + d*x]^2*(ArcSi nh[c + d*x] - (Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x) + 3*Log[1 - E^(-2*ArcSinh[c + d*x])]) - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-2*ArcSinh [c + d*x])] - 3*PolyLog[3, E^(-2*ArcSinh[c + d*x])]))/(d*e^3)
Result contains complex when optimal does not.
Time = 1.08 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6274, 27, 6191, 6215, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c+d x)^3}d(c+d x)}{d e^3}\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {2 b \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x)^2 \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6215 |
\(\displaystyle \frac {2 b \left (3 b \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6190 |
\(\displaystyle \frac {2 b \left (3 \int -(a+b \text {arcsinh}(c+d x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 b \left (-3 \int (a+b \text {arcsinh}(c+d x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}-3 \int -i (a+b \text {arcsinh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c+d x))\right )}{d e^3}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \int (a+b \text {arcsinh}(c+d x))^2 \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )d(a+b \text {arcsinh}(c+d x))\right )}{d e^3}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi } (a+b \text {arcsinh}(c+d x))^2}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }}d(a+b \text {arcsinh}(c+d x))-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d e^3}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \left (2 i \left (b \int (a+b \text {arcsinh}(c+d x)) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d e^3}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d e^3}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \left (2 i \left (b \left (\frac {1}{4} b^2 \int \exp \left (-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+i \pi \right ) \operatorname {PolyLog}(2,-c-d x)de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d e^3}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{2 (c+d x)^2}+2 b \left (-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3}{c+d x}+3 i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}(3,-c-d x)+\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d e^3}\) |
(-1/2*(a + b*ArcSinh[c + d*x])^4/(c + d*x)^2 + 2*b*(-((Sqrt[1 + (c + d*x)^ 2]*(a + b*ArcSinh[c + d*x])^3)/(c + d*x)) + (3*I)*((-1/3*I)*(a + b*ArcSinh [c + d*x])^3 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c + d*x])^2*Log[1 + E^((2*a)/ b - I*Pi - (2*(a + b*ArcSinh[c + d*x]))/b)]) + b*((b*(a + b*ArcSinh[c + d* x])*PolyLog[2, -E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c + d*x]))/b)])/2 + (b^2*PolyLog[3, -c - d*x])/4)))))/(d*e^3)
3.2.53.3.1 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[(((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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(560\) vs. \(2(206)=412\).
Time = 0.52 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.02
method | result | size |
derivativedivides | \(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {6 a^{2} b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(561\) |
default | \(\frac {-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {6 a^{2} b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\) | \(561\) |
parts | \(-\frac {a^{4}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (-4 \left (d x +c \right )^{2}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-4 \operatorname {arcsinh}\left (d x +c \right )^{3}+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3} d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3} d}+\frac {6 a^{2} b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3} d}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3} d}\) | \(572\) |
1/d*(-1/2*a^4/e^3/(d*x+c)^2+b^4/e^3*(-1/2*arcsinh(d*x+c)^3*(-4*(d*x+c)^2+4 *(d*x+c)*(1+(d*x+c)^2)^(1/2)+arcsinh(d*x+c))/(d*x+c)^2-4*arcsinh(d*x+c)^3+ 6*arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+12*arcsinh(d*x+c)*polyl og(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-12*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+ 6*arcsinh(d*x+c)^2*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+12*arcsinh(d*x+c)*polyl og(2,d*x+c+(1+(d*x+c)^2)^(1/2))-12*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2)))+4 *a*b^3/e^3*(-1/2*arcsinh(d*x+c)^2*(3*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3*(d*x+c) ^2+arcsinh(d*x+c))/(d*x+c)^2-3*arcsinh(d*x+c)^2+3*arcsinh(d*x+c)*ln(1+d*x+ c+(1+(d*x+c)^2)^(1/2))+3*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+3*arcsinh(d *x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+3*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2 )))+6*a^2*b^2/e^3*(-2*arcsinh(d*x+c)-1/2*arcsinh(d*x+c)*(-2*(d*x+c)^2+2*(d *x+c)*(1+(d*x+c)^2)^(1/2)+arcsinh(d*x+c))/(d*x+c)^2+ln((d*x+c+(1+(d*x+c)^2 )^(1/2))^2-1))+4*b*a^3/e^3*(-1/2/(d*x+c)^2*arcsinh(d*x+c)-1/2/(d*x+c)*(1+( d*x+c)^2)^(1/2)))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2* arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh(d*x + c) + a^4)/(d^3*e^3*x^3 + 3*c*d^ 2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{4}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
(Integral(a**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integ ral(b**4*asinh(c + d*x)**4/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3) , x) + Integral(4*a*b**3*asinh(c + d*x)**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x **2 + d**3*x**3), x) + Integral(6*a**2*b**2*asinh(c + d*x)**2/(c**3 + 3*c* *2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(4*a**3*b*asinh(c + d*x) /(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e**3
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
-1/2*b^4*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 6*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d*arcsi nh(d*x + c)/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c)/(d*e^3))*a^2*b^2 - 2*a^ 3*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d/(d^3*e^3*x + c*d^2*e^3) + arcsinh (d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 3*a^2*b^2*arcsinh(d *x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^4/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + integrate(2*(2*(c^3 + c)*a*b^3 + (c^3 + c) *b^4 + (2*a*b^3*d^3 + b^4*d^3)*x^3 + 3*(2*a*b^3*c*d^2 + b^4*c*d^2)*x^2 + ( 2*(3*c^2*d + d)*a*b^3 + (3*c^2*d + d)*b^4)*x + (b^4*c^2 + 2*(c^2 + 1)*a*b^ 3 + (2*a*b^3*d^2 + b^4*d^2)*x^2 + 2*(2*a*b^3*c*d + b^4*c*d)*x)*sqrt(d^2*x^ 2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^3 /(d^6*e^3*x^6 + 6*c*d^5*e^3*x^5 + c^6*e^3 + c^4*e^3 + (15*c^2*d^4*e^3 + d^ 4*e^3)*x^4 + 4*(5*c^3*d^3*e^3 + c*d^3*e^3)*x^3 + 3*(5*c^4*d^2*e^3 + 2*c^2* d^2*e^3)*x^2 + 2*(3*c^5*d*e^3 + 2*c^3*d*e^3)*x + (d^5*e^3*x^5 + 5*c*d^4*e^ 3*x^4 + c^5*e^3 + c^3*e^3 + (10*c^2*d^3*e^3 + d^3*e^3)*x^3 + (10*c^3*d^2*e ^3 + 3*c*d^2*e^3)*x^2 + (5*c^4*d*e^3 + 3*c^2*d*e^3)*x)*sqrt(d^2*x^2 + 2*c* d*x + c^2 + 1)), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]