Integrand size = 23, antiderivative size = 115 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=-\frac {(a+b \text {arcsinh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )}{d e}+\frac {b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {b^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )}{2 d e} \] Output:
-1/3*(a+b*arcsinh(d*x+c))^3/b/d/e+(a+b*arcsinh(d*x+c))^2*ln(1-(d*x+c+(1+(d *x+c)^2)^(1/2))^2)/d/e+b*(a+b*arcsinh(d*x+c))*polylog(2,(d*x+c+(1+(d*x+c)^ 2)^(1/2))^2)/d/e-1/2*b^2*polylog(3,(d*x+c+(1+(d*x+c)^2)^(1/2))^2)/d/e
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {-2 (a+b \text {arcsinh}(c+d x))^2 \left (a+b \text {arcsinh}(c+d x)-3 b \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )\right )+6 b^2 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )-3 b^3 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )}{6 b d e} \] Input:
Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x),x]
Output:
(-2*(a + b*ArcSinh[c + d*x])^2*(a + b*ArcSinh[c + d*x] - 3*b*Log[1 - E^(2* ArcSinh[c + d*x])]) + 6*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^(2*ArcSi nh[c + d*x])] - 3*b^3*PolyLog[3, E^(2*ArcSinh[c + d*x])])/(6*b*d*e)
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.34, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6274, 27, 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.
\(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{e (c+d x)}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}d(c+d x)}{d e}\) |
\(\Big \downarrow \) 6190 |
\(\displaystyle \frac {\int -(a+b \text {arcsinh}(c+d x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int (a+b \text {arcsinh}(c+d x))^2 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -i (a+b \text {arcsinh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int (a+b \text {arcsinh}(c+d x))^2 \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi } (a+b \text {arcsinh}(c+d x))^2}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }}d(a+b \text {arcsinh}(c+d x))-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )}{b d e}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {i \left (2 i \left (b \int (a+b \text {arcsinh}(c+d x)) \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )}{b d e}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )}{b d e}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int \exp \left (-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+i \pi \right ) \operatorname {PolyLog}(2,-c-d x)de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }+\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )}{b d e}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \operatorname {PolyLog}(3,-c-d x)+\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{3} i (a+b \text {arcsinh}(c+d x))^3\right )}{b d e}\) |
Input:
Int[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x),x]
Output:
(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*A rcSinh[c + d*x]))/b)])/2 + (b^2*PolyLog[3, -c - d*x])/4))))/(b*d*e)
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_) + (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. \(311\) vs. \(2(135)=270\).
Time = 0.65 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.71
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(312\) |
default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e}}{d}\) | \(312\) |
parts | \(\frac {a^{2} \ln \left (d x +c \right )}{e d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{2}+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e d}\) | \(317\) |
Input:
int((a+b*arcsinh(d*x+c))^2/(d*e*x+c*e),x,method=_RETURNVERBOSE)
Output:
1/d*(a^2/e*ln(d*x+c)+b^2/e*(-1/3*arcsinh(d*x+c)^3+arcsinh(d*x+c)^2*ln(1-d* x-c-(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*polylog(2,d*x+c+(1+(d*x+c)^2)^(1 /2))-2*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))+arcsinh(d*x+c)^2*ln(1+d*x+c+(1 +(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))- 2*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2)))+2*a*b/e*(-1/2*arcsinh(d*x+c)^2+ar csinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+polylog(2,d*x+c+(1+(d*x+c)^2) ^(1/2))+arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+polylog(2,-d*x-c-(1 +(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \] Input:
integrate((a+b*arcsinh(d*x+c))^2/(d*e*x+c*e),x, algorithm="fricas")
Output:
integral((b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2)/(d*e*x + c*e), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \] Input:
integrate((a+b*asinh(d*x+c))**2/(d*e*x+c*e),x)
Output:
(Integral(a**2/(c + d*x), x) + Integral(b**2*asinh(c + d*x)**2/(c + d*x), x) + Integral(2*a*b*asinh(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \] Input:
integrate((a+b*arcsinh(d*x+c))^2/(d*e*x+c*e),x, algorithm="maxima")
Output:
a^2*log(d*e*x + c*e)/(d*e) + integrate(b^2*log(d*x + c + sqrt((d*x + c)^2 + 1))^2/(d*e*x + c*e) + 2*a*b*log(d*x + c + sqrt((d*x + c)^2 + 1))/(d*e*x + c*e), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \] Input:
integrate((a+b*arcsinh(d*x+c))^2/(d*e*x+c*e),x, algorithm="giac")
Output:
integrate((b*arcsinh(d*x + c) + a)^2/(d*e*x + c*e), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \] Input:
int((a + b*asinh(c + d*x))^2/(c*e + d*e*x),x)
Output:
int((a + b*asinh(c + d*x))^2/(c*e + d*e*x), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{c e+d e x} \, dx=\frac {2 \left (\int \frac {\mathit {asinh} \left (d x +c \right )}{d x +c}d x \right ) a b d +\left (\int \frac {\mathit {asinh} \left (d x +c \right )^{2}}{d x +c}d x \right ) b^{2} d +\mathrm {log}\left (d x +c \right ) a^{2}}{d e} \] Input:
int((a+b*asinh(d*x+c))^2/(d*e*x+c*e),x)
Output:
(2*int(asinh(c + d*x)/(c + d*x),x)*a*b*d + int(asinh(c + d*x)**2/(c + d*x) ,x)*b**2*d + log(c + d*x)*a**2)/(d*e)