3.518 \(\int \frac{a+b \cosh ^{-1}(c x)}{(d+e x^2)^{7/2}} \, dx\)

Optimal. Leaf size=284 \[ \frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{2 b c \left (1-c^2 x^2\right ) \left (3 c^2 d+2 e\right )}{15 d^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^2 \sqrt{d+e x^2}}-\frac{8 b \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \left (1-c^2 x^2\right )}{15 d \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}} \]

[Out]

(b*c*(1 - c^2*x^2))/(15*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)^(3/2)) + (2*b*c*(3*c^2*d + 2*e)
*(1 - c^2*x^2))/(15*d^2*(c^2*d + e)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]) + (x*(a + b*ArcCosh[c*x]))
/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcCosh[c*x]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcCosh[c*x]))/
(15*d^3*Sqrt[d + e*x^2]) - (8*b*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/
(15*d^3*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.801408, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {192, 191, 5705, 12, 519, 6715, 949, 78, 63, 217, 206} \[ \frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{2 b c \left (1-c^2 x^2\right ) \left (3 c^2 d+2 e\right )}{15 d^2 \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right )^2 \sqrt{d+e x^2}}-\frac{8 b \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \left (1-c^2 x^2\right )}{15 d \sqrt{c x-1} \sqrt{c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2),x]

[Out]

(b*c*(1 - c^2*x^2))/(15*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)^(3/2)) + (2*b*c*(3*c^2*d + 2*e)
*(1 - c^2*x^2))/(15*d^2*(c^2*d + e)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[d + e*x^2]) + (x*(a + b*ArcCosh[c*x]))
/(5*d*(d + e*x^2)^(5/2)) + (4*x*(a + b*ArcCosh[c*x]))/(15*d^2*(d + e*x^2)^(3/2)) + (8*x*(a + b*ArcCosh[c*x]))/
(15*d^3*Sqrt[d + e*x^2]) - (8*b*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/(c*Sqrt[d + e*x^2])])/
(15*d^3*Sqrt[e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 192

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 949

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[(R*(d + e*x)^(m + 1)*(f + g*x)^(n + 1))/((m + 1)*(e*f - d*g)), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 78

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b \cosh ^{-1}(c x)}{\left (d+e x^2\right )^{7/2}} \, dx &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-(b c) \int \frac{x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{15 d^3 \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{5/2}} \, dx\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{5/2}} \, dx}{15 d^3}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \int \frac{x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{\sqrt{-1+c^2 x^2} \left (d+e x^2\right )^{5/2}} \, dx}{15 d^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15 d^2+20 d e x+8 e^2 x^2}{\sqrt{-1+c^2 x} (d+e x)^{5/2}} \, dx,x,x^2\right )}{30 d^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{15 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{3 d \left (7 c^2 d+6 e\right )+12 e \left (c^2 d+e\right ) x}{\sqrt{-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{45 d^3 \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{15 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}}+\frac{2 b c \left (3 c^2 d+2 e\right ) \left (1-c^2 x^2\right )}{15 d^2 \left (c^2 d+e\right )^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (4 b c \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{15 d^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{15 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}}+\frac{2 b c \left (3 c^2 d+2 e\right ) \left (1-c^2 x^2\right )}{15 d^2 \left (c^2 d+e\right )^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (8 b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1+c^2 x^2}\right )}{15 c d^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{15 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}}+\frac{2 b c \left (3 c^2 d+2 e\right ) \left (1-c^2 x^2\right )}{15 d^2 \left (c^2 d+e\right )^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{\left (8 b \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{15 c d^3 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c \left (1-c^2 x^2\right )}{15 d \left (c^2 d+e\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (d+e x^2\right )^{3/2}}+\frac{2 b c \left (3 c^2 d+2 e\right ) \left (1-c^2 x^2\right )}{15 d^2 \left (c^2 d+e\right )^2 \sqrt{-1+c x} \sqrt{1+c x} \sqrt{d+e x^2}}+\frac{x \left (a+b \cosh ^{-1}(c x)\right )}{5 d \left (d+e x^2\right )^{5/2}}+\frac{4 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^2 \left (d+e x^2\right )^{3/2}}+\frac{8 x \left (a+b \cosh ^{-1}(c x)\right )}{15 d^3 \sqrt{d+e x^2}}-\frac{8 b \sqrt{-1+c^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{-1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{15 d^3 \sqrt{e} \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [C]  time = 3.79011, size = 685, normalized size = 2.41 \[ \frac{\frac{16 b (c x-1)^{3/2} \left (d+e x^2\right )^2 \sqrt{\frac{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x-1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (\frac{c \left (\sqrt{e}-i c \sqrt{d}\right ) \left (\sqrt{e} x+i \sqrt{d}\right ) \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{2-2 c x}}\right ),\frac{4 i c \sqrt{d} \sqrt{e}}{\left (c \sqrt{d}+i \sqrt{e}\right )^2}\right )}{c x-1}+c \sqrt{d} \left (-c \sqrt{d}+i \sqrt{e}\right ) \sqrt{\frac{\left (c^2 d+e\right ) \left (d+e x^2\right )}{d e (c x-1)^2}} \sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \Pi \left (\frac{2 c \sqrt{d}}{\sqrt{d} c+i \sqrt{e}};\sin ^{-1}\left (\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )-1}{2-2 c x}}\right )|\frac{4 i c \sqrt{d} \sqrt{e}}{\left (\sqrt{d} c+i \sqrt{e}\right )^2}\right )\right )}{c d^3 \sqrt{c x+1} \left (c^2 d+e\right ) \sqrt{-\frac{c \left (x+\frac{i \sqrt{d}}{\sqrt{e}}\right )+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}}}+\frac{a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{d^3}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (d+e x^2\right ) \left (c^2 d \left (7 d+6 e x^2\right )+e \left (5 d+4 e x^2\right )\right )}{d^2 \left (c^2 d+e\right )^2}+\frac{b x \cosh ^{-1}(c x) \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{d^3}}{15 \left (d+e x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2),x]

[Out]

((a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4))/d^3 - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)*(e*(5*d + 4*e*x^2
) + c^2*d*(7*d + 6*e*x^2)))/(d^2*(c^2*d + e)^2) + (b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcCosh[c*x])/d^3 + (
16*b*(-1 + c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(-1 + c*x))]*(d + e*x^
2)^2*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*Sqrt[d] + Sqrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]
*x)/Sqrt[d])/(1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2
- 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d]) + I*Sq
rt[e])*Sqrt[((c^2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/
Sqrt[e] + x))/(1 - c*x))]*EllipticPi[(2*c*Sqrt[d])/(c*Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/
Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2]))/
(c*d^3*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]
))/(15*(d + e*x^2)^(5/2))

________________________________________________________________________________________

Maple [F]  time = 1.055, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arccosh} \left (cx\right )) \left ( e{x}^{2}+d \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x)

[Out]

int((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{15} \, a{\left (\frac{8 \, x}{\sqrt{e x^{2} + d} d^{3}} + \frac{4 \, x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d^{2}} + \frac{3 \, x}{{\left (e x^{2} + d\right )}^{\frac{5}{2}} d}\right )} + b \int \frac{\log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{{\left (e x^{2} + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="maxima")

[Out]

1/15*a*(8*x/(sqrt(e*x^2 + d)*d^3) + 4*x/((e*x^2 + d)^(3/2)*d^2) + 3*x/((e*x^2 + d)^(5/2)*d)) + b*integrate(log
(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e*x^2 + d)^(7/2), x)

________________________________________________________________________________________

Fricas [B]  time = 3.40401, size = 2782, normalized size = 9.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="fricas")

[Out]

[1/15*(2*(b*c^4*d^5 + 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e
^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 3*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(e)*log(8*c^4*e^2
*x^4 + c^4*d^2 - 6*c^2*d*e + 8*(c^4*d*e - c^2*e^2)*x^2 - 4*(2*c^3*e*x^2 + c^3*d - c*e)*sqrt(c^2*x^2 - 1)*sqrt(
e*x^2 + d)*sqrt(e) + e^2) + (8*(b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^5 + 20*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e
^3 + b*d*e^4)*x^3 + 15*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x)*sqrt(e*x^2 + d)*log(c*x + sqrt(c^2*x^2 -
1)) + (8*(a*c^4*d^2*e^3 + 2*a*c^2*d*e^4 + a*e^5)*x^5 + 20*(a*c^4*d^3*e^2 + 2*a*c^2*d^2*e^3 + a*d*e^4)*x^3 + 1
5*(a*c^4*d^4*e + 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x - (7*b*c^3*d^4*e + 5*b*c*d^3*e^2 + 2*(3*b*c^3*d^2*e^3 + 2*b*c*
d*e^4)*x^4 + (13*b*c^3*d^3*e^2 + 9*b*c*d^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2 + d))/(c^4*d^8*e + 2*c^2*d^
7*e^2 + d^6*e^3 + (c^4*d^5*e^4 + 2*c^2*d^4*e^5 + d^3*e^6)*x^6 + 3*(c^4*d^6*e^3 + 2*c^2*d^5*e^4 + d^4*e^5)*x^4
+ 3*(c^4*d^7*e^2 + 2*c^2*d^6*e^3 + d^5*e^4)*x^2), 1/15*(4*(b*c^4*d^5 + 2*b*c^2*d^4*e + (b*c^4*d^2*e^3 + 2*b*c^
2*d*e^4 + b*e^5)*x^6 + b*d^3*e^2 + 3*(b*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^4 + 3*(b*c^4*d^4*e + 2*b*c^
2*d^3*e^2 + b*d^2*e^3)*x^2)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^2 + c^2*d - e)*sqrt(c^2*x^2 - 1)*sqrt(e*x^2 + d)*sq
rt(-e)/(c^3*e^2*x^4 - c*d*e + (c^3*d*e - c*e^2)*x^2)) + (8*(b*c^4*d^2*e^3 + 2*b*c^2*d*e^4 + b*e^5)*x^5 + 20*(b
*c^4*d^3*e^2 + 2*b*c^2*d^2*e^3 + b*d*e^4)*x^3 + 15*(b*c^4*d^4*e + 2*b*c^2*d^3*e^2 + b*d^2*e^3)*x)*sqrt(e*x^2 +
d)*log(c*x + sqrt(c^2*x^2 - 1)) + (8*(a*c^4*d^2*e^3 + 2*a*c^2*d*e^4 + a*e^5)*x^5 + 20*(a*c^4*d^3*e^2 + 2*a*c^
2*d^2*e^3 + a*d*e^4)*x^3 + 15*(a*c^4*d^4*e + 2*a*c^2*d^3*e^2 + a*d^2*e^3)*x - (7*b*c^3*d^4*e + 5*b*c*d^3*e^2 +
2*(3*b*c^3*d^2*e^3 + 2*b*c*d*e^4)*x^4 + (13*b*c^3*d^3*e^2 + 9*b*c*d^2*e^3)*x^2)*sqrt(c^2*x^2 - 1))*sqrt(e*x^2
+ d))/(c^4*d^8*e + 2*c^2*d^7*e^2 + d^6*e^3 + (c^4*d^5*e^4 + 2*c^2*d^4*e^5 + d^3*e^6)*x^6 + 3*(c^4*d^6*e^3 + 2
*c^2*d^5*e^4 + d^4*e^5)*x^4 + 3*(c^4*d^7*e^2 + 2*c^2*d^6*e^3 + d^5*e^4)*x^2)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/(e*x**2+d)**(7/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcosh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/(e*x^2+d)^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/(e*x^2 + d)^(7/2), x)