\(\int \frac {\text {arcsinh}(a x)}{(c+a^2 c x^2)^{7/2}} \, dx\) [111]

Optimal result

Integrand size = 19, antiderivative size = 200 \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {4 \sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \]

[Out]

1/5*x*arcsinh(a*x)/c/(a^2*c*x^2+c)^(5/2)+4/15*x*arcsinh(a*x)/c^2/(a^2*c*x^2+c)^(3/2)+1/20/a/c^3/(a^2*x^2+1)^(3
/2)/(a^2*c*x^2+c)^(1/2)+8/15*x*arcsinh(a*x)/c^3/(a^2*c*x^2+c)^(1/2)+2/15/a/c^3/(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)
^(1/2)-4/15*ln(a^2*x^2+1)*(a^2*x^2+1)^(1/2)/a/c^3/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5788, 5787, 266, 267} \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {8 x \text {arcsinh}(a x)}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \text {arcsinh}(a x)}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \text {arcsinh}(a x)}{5 c \left (a^2 c x^2+c\right )^{5/2}}+\frac {2}{15 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {1}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {4 \sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{15 a c^3 \sqrt {a^2 c x^2+c}} \]

[In]

Int[ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2),x]

[Out]

1/(20*a*c^3*(1 + a^2*x^2)^(3/2)*Sqrt[c + a^2*c*x^2]) + 2/(15*a*c^3*Sqrt[1 + a^2*x^2]*Sqrt[c + a^2*c*x^2]) + (x
*ArcSinh[a*x])/(5*c*(c + a^2*c*x^2)^(5/2)) + (4*x*ArcSinh[a*x])/(15*c^2*(c + a^2*c*x^2)^(3/2)) + (8*x*ArcSinh[
a*x])/(15*c^3*Sqrt[c + a^2*c*x^2]) - (4*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2])/(15*a*c^3*Sqrt[c + a^2*c*x^2])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5787

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh
[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Dist[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]], Int[x*((a + b*ArcS
inh[c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]

Rule 5788

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)^(
p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a
+ b*ArcSinh[c*x])^n, x], x] + Dist[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2
)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] &
& LtQ[p, -1] && NeQ[p, -3/2]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arcsinh}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \text {arcsinh}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \text {arcsinh}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \text {arcsinh}(a x)}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {4 \sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{15 a c^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.60 \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (4 a x \sqrt {1+a^2 x^2} \left (15+20 a^2 x^2+8 a^4 x^4\right ) \text {arcsinh}(a x)-\left (1+a^2 x^2\right ) \left (-11-8 a^2 x^2+16 \left (1+a^2 x^2\right )^2 \log \left (1+a^2 x^2\right )\right )\right )}{60 a c^4 \left (1+a^2 x^2\right )^{7/2}} \]

[In]

Integrate[ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2),x]

[Out]

(Sqrt[c + a^2*c*x^2]*(4*a*x*Sqrt[1 + a^2*x^2]*(15 + 20*a^2*x^2 + 8*a^4*x^4)*ArcSinh[a*x] - (1 + a^2*x^2)*(-11
- 8*a^2*x^2 + 16*(1 + a^2*x^2)^2*Log[1 + a^2*x^2])))/(60*a*c^4*(1 + a^2*x^2)^(7/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(170)=340\).

Time = 0.28 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.82

method	result	size
default	\(\frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+20 a^{3} x^{3}-16 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+15 a x -8 \sqrt {a^{2} x^{2}+1}\right ) \left (-64 a^{8} x^{8}-64 \sqrt {a^{2} x^{2}+1}\, a^{7} x^{7}-280 a^{6} x^{6}-248 x^{5} a^{5} \sqrt {a^{2} x^{2}+1}+160 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )-456 a^{4} x^{4}-340 a^{3} x^{3} \sqrt {a^{2} x^{2}+1}+380 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )-328 a^{2} x^{2}-165 a x \sqrt {a^{2} x^{2}+1}+256 \,\operatorname {arcsinh}\left (a x \right )-88\right )}{60 \left (40 a^{10} x^{10}+215 a^{8} x^{8}+469 a^{6} x^{6}+517 a^{4} x^{4}+287 a^{2} x^{2}+64\right ) a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}\)	\(363\)

[In]

int(arcsinh(a*x)/(a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

16/15*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)/a/c^4*arcsinh(a*x)+1/60*(c*(a^2*x^2+1))^(1/2)*(8*a^5*x^5-8*a^4*x
^4*(a^2*x^2+1)^(1/2)+20*a^3*x^3-16*a^2*x^2*(a^2*x^2+1)^(1/2)+15*a*x-8*(a^2*x^2+1)^(1/2))*(-64*a^8*x^8-64*(a^2*
x^2+1)^(1/2)*a^7*x^7-280*a^6*x^6-248*x^5*a^5*(a^2*x^2+1)^(1/2)+160*a^4*x^4*arcsinh(a*x)-456*a^4*x^4-340*a^3*x^
3*(a^2*x^2+1)^(1/2)+380*a^2*x^2*arcsinh(a*x)-328*a^2*x^2-165*a*x*(a^2*x^2+1)^(1/2)+256*arcsinh(a*x)-88)/(40*a^
10*x^10+215*a^8*x^8+469*a^6*x^6+517*a^4*x^4+287*a^2*x^2+64)/a/c^4-8/15*(c*(a^2*x^2+1))^(1/2)/(a^2*x^2+1)^(1/2)
/a/c^4*ln(1+(a*x+(a^2*x^2+1)^(1/2))^2)

Fricas [F]

\[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(arcsinh(a*x)/(a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)*arcsinh(a*x)/(a^8*c^4*x^8 + 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 + 4*a^2*c^4*x^2 + c^4),
x)

Sympy [F]

\[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\operatorname {asinh}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(asinh(a*x)/(a**2*c*x**2+c)**(7/2),x)

[Out]

Integral(asinh(a*x)/(c*(a**2*x**2 + 1))**(7/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.72 \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\frac {1}{60} \, a {\left (\frac {3}{{\left (a^{6} c^{\frac {5}{2}} x^{4} + 2 \, a^{4} c^{\frac {5}{2}} x^{2} + a^{2} c^{\frac {5}{2}}\right )} c} + \frac {8}{{\left (a^{4} c^{\frac {3}{2}} x^{2} + a^{2} c^{\frac {3}{2}}\right )} c^{2}} - \frac {16 \, \log \left (x^{2} + \frac {1}{a^{2}}\right )}{a^{2} c^{\frac {7}{2}}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arsinh}\left (a x\right ) \]

[In]

integrate(arcsinh(a*x)/(a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

1/60*a*(3/((a^6*c^(5/2)*x^4 + 2*a^4*c^(5/2)*x^2 + a^2*c^(5/2))*c) + 8/((a^4*c^(3/2)*x^2 + a^2*c^(3/2))*c^2) -
16*log(x^2 + 1/a^2)/(a^2*c^(7/2))) + 1/15*(8*x/(sqrt(a^2*c*x^2 + c)*c^3) + 4*x/((a^2*c*x^2 + c)^(3/2)*c^2) + 3
*x/((a^2*c*x^2 + c)^(5/2)*c))*arcsinh(a*x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.62 \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=-\frac {1}{60} \, \sqrt {c} {\left (\frac {16 \, \log \left (a^{2} x^{2} + 1\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} + 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} + 1\right )}^{2} a c^{4}}\right )} + \frac {{\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} + \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{15 \, {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \]

[In]

integrate(arcsinh(a*x)/(a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

-1/60*sqrt(c)*(16*log(a^2*x^2 + 1)/(a*c^4) - (24*a^4*x^4 + 56*a^2*x^2 + 35)/((a^2*x^2 + 1)^2*a*c^4)) + 1/15*(4
*(2*a^4*x^2/c + 5*a^2/c)*x^2 + 15/c)*x*log(a*x + sqrt(a^2*x^2 + 1))/(a^2*c*x^2 + c)^(5/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\mathrm {asinh}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{7/2}} \,d x \]

[In]

int(asinh(a*x)/(c + a^2*c*x^2)^(7/2),x)

[Out]

int(asinh(a*x)/(c + a^2*c*x^2)^(7/2), x)