Integrand size = 14, antiderivative size = 195 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {4 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {8 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d} \]
-4/15*(d*x+c)/b^2/d/(a+b*arcsinh(d*x+c))^(3/2)-4/15*exp(a/b)*erf((a+b*arcs inh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d+4/15*erfi((a+b*arcsinh(d*x+c ))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d/exp(a/b)-2/5*(1+(d*x+c)^2)^(1/2)/b/d/ (a+b*arcsinh(d*x+c))^(5/2)-8/15*(1+(d*x+c)^2)^(1/2)/b^3/d/(a+b*arcsinh(d*x +c))^(1/2)
Time = 0.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {-6 b^2 e^{\text {arcsinh}(c+d x)}-2 e^{-\text {arcsinh}(c+d x)} \left (4 a^2+2 a b (-1+4 \text {arcsinh}(c+d x))+b^2 \left (3-2 \text {arcsinh}(c+d x)+4 \text {arcsinh}(c+d x)^2\right )\right )+8 e^{a/b} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x))^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )-4 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{\frac {a}{b}+\text {arcsinh}(c+d x)} (2 a+b+2 b \text {arcsinh}(c+d x))+2 b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )}{30 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]
(-6*b^2*E^ArcSinh[c + d*x] - (2*(4*a^2 + 2*a*b*(-1 + 4*ArcSinh[c + d*x]) + b^2*(3 - 2*ArcSinh[c + d*x] + 4*ArcSinh[c + d*x]^2)))/E^ArcSinh[c + d*x] + 8*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[ 1/2, a/b + ArcSinh[c + d*x]] - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSi nh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b))/(30*b ^3*d*(a + b*ArcSinh[c + d*x])^(5/2))
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6273, 6188, 6233, 6188, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634}
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.
| \(\displaystyle \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6273 |
| \(\displaystyle \frac {\int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)}{b}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}\right )}{5 b}-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-i \int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {-\frac {2 \sqrt {(c+d x)^2+1}}{5 b (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {2 \left (-\frac {2 (c+d x)}{3 b (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {(c+d x)^2+1}}{b \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}\right )}{3 b}\right )}{5 b}}{d}\) |
((-2*Sqrt[1 + (c + d*x)^2])/(5*b*(a + b*ArcSinh[c + d*x])^(5/2)) + (2*((-2 *(c + d*x))/(3*b*(a + b*ArcSinh[c + d*x])^(3/2)) + (2*((-2*Sqrt[1 + (c + d *x)^2])/(b*Sqrt[a + b*ArcSinh[c + d*x]]) + ((2*I)*((I/2)*Sqrt[b]*E^(a/b)*S qrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi ]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b)))/b^2))/(3*b)))/(5*b ))/d
3.3.26.3.1 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_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \frac {1}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]