3.6.18 \(\int \frac {a+b \cosh ^{-1}(c x)}{(d+e x^2)^{7/2}} \, dx\) [518]

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

[Out]

1/5*x*(a+b*arccosh(c*x))/d/(e*x^2+d)^(5/2)+4/15*x*(a+b*arccosh(c*x))/d^2/(e*x^2+d)^(3/2)+1/15*b*c*(-c^2*x^2+1)
/d/(c^2*d+e)/(e*x^2+d)^(3/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-8/15*b*arctanh(e^(1/2)*(c^2*x^2-1)^(1/2)/c/(e*x^2+d)^
(1/2))*(c^2*x^2-1)^(1/2)/d^3/e^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/15*x*(a+b*arccosh(c*x))/d^3/(e*x^2+d)^(1/2)
+2/15*b*c*(3*c^2*d+2*e)*(-c^2*x^2+1)/d^2/(c^2*d+e)^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(e*x^2+d)^(1/2)

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Rubi [A]
time = 0.57, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {198, 197, 5908, 12, 533, 6847, 963, 79, 65, 223, 212} \begin {gather*} \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 {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 {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 {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}} \end {gather*}

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 12

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

Rule 65

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 79

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] || L
tQ[p, n]))))

Rule 197

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 198

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 212

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

Rule 223

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 533

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)^FracPa
rt[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 963

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 5908

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 6847

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 2.95, size = 685, normalized size = 2.41 \begin {gather*} \frac {\frac {a x \left (15 d^2+20 d e x^2+8 e^2 x^4\right )}{d^3}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right ) \left (e \left (5 d+4 e x^2\right )+c^2 d \left (7 d+6 e x^2\right )\right )}{d^2 \left (c^2 d+e\right )^2}+\frac {b x \left (15 d^2+20 d e x^2+8 e^2 x^4\right ) \cosh ^{-1}(c x)}{d^3}+\frac {16 b (-1+c x)^{3/2} \sqrt {\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (-1+c x)}} \left (d+e x^2\right )^2 \left (\frac {c \left (-i c \sqrt {d}+\sqrt {e}\right ) \left (i \sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} F\left (\text {ArcSin}\left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )}{-1+c x}+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 (-1+c x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \Pi \left (\frac {2 c \sqrt {d}}{c \sqrt {d}+i \sqrt {e}};\text {ArcSin}\left (\sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{2-2 c x}}\right )|\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}+i \sqrt {e}\right )^2}\right )\right )}{c d^3 \left (c^2 d+e\right ) \sqrt {1+c x} \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}}}}{15 \left (d+e x^2\right )^{5/2}} \end {gather*}

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))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arccosh}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {7}{2}}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2519 vs. \(2 (245) = 490\).
time = 0.49, size = 2519, normalized size = 8.87 \begin {gather*} \text {Too large to display} \end {gather*}

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*x^6*cosh(1)^5 + b*x^6*sinh(1)^5 + b*c^4*d^5 + (2*b*c^2*d*x^6 + 3*b*d*x^4)*cosh(1)^4 + (2*b*c^2*d*x^
6 + 5*b*x^6*cosh(1) + 3*b*d*x^4)*sinh(1)^4 + (b*c^4*d^2*x^6 + 6*b*c^2*d^2*x^4 + 3*b*d^2*x^2)*cosh(1)^3 + (b*c^
4*d^2*x^6 + 6*b*c^2*d^2*x^4 + 10*b*x^6*cosh(1)^2 + 3*b*d^2*x^2 + 4*(2*b*c^2*d*x^6 + 3*b*d*x^4)*cosh(1))*sinh(1
)^3 + (3*b*c^4*d^3*x^4 + 6*b*c^2*d^3*x^2 + b*d^3)*cosh(1)^2 + (3*b*c^4*d^3*x^4 + 10*b*x^6*cosh(1)^3 + 6*b*c^2*
d^3*x^2 + b*d^3 + 6*(2*b*c^2*d*x^6 + 3*b*d*x^4)*cosh(1)^2 + 3*(b*c^4*d^2*x^6 + 6*b*c^2*d^2*x^4 + 3*b*d^2*x^2)*
cosh(1))*sinh(1)^2 + (3*b*c^4*d^4*x^2 + 2*b*c^2*d^4)*cosh(1) + (3*b*c^4*d^4*x^2 + 5*b*x^6*cosh(1)^4 + 2*b*c^2*
d^4 + 4*(2*b*c^2*d*x^6 + 3*b*d*x^4)*cosh(1)^3 + 3*(b*c^4*d^2*x^6 + 6*b*c^2*d^2*x^4 + 3*b*d^2*x^2)*cosh(1)^2 +
2*(3*b*c^4*d^3*x^4 + 6*b*c^2*d^3*x^2 + b*d^3)*cosh(1))*sinh(1))*sqrt(cosh(1) + sinh(1))*log(c^4*d^2 + (8*c^4*x
^4 - 8*c^2*x^2 + 1)*cosh(1)^2 + (8*c^4*x^4 - 8*c^2*x^2 + 1)*sinh(1)^2 - 4*(c^3*d + (2*c^3*x^2 - c)*cosh(1) + (
2*c^3*x^2 - c)*sinh(1))*sqrt(c^2*x^2 - 1)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(cosh(1) + sinh(1)) + 2*(4*c
^4*d*x^2 - 3*c^2*d)*cosh(1) + 2*(4*c^4*d*x^2 - 3*c^2*d + (8*c^4*x^4 - 8*c^2*x^2 + 1)*cosh(1))*sinh(1)) + (15*b
*c^4*d^4*x*cosh(1) + 8*b*x^5*cosh(1)^5 + 8*b*x^5*sinh(1)^5 + 4*(4*b*c^2*d*x^5 + 5*b*d*x^3)*cosh(1)^4 + 4*(4*b*
c^2*d*x^5 + 10*b*x^5*cosh(1) + 5*b*d*x^3)*sinh(1)^4 + (8*b*c^4*d^2*x^5 + 40*b*c^2*d^2*x^3 + 15*b*d^2*x)*cosh(1
)^3 + (8*b*c^4*d^2*x^5 + 40*b*c^2*d^2*x^3 + 80*b*x^5*cosh(1)^2 + 15*b*d^2*x + 16*(4*b*c^2*d*x^5 + 5*b*d*x^3)*c
osh(1))*sinh(1)^3 + 10*(2*b*c^4*d^3*x^3 + 3*b*c^2*d^3*x)*cosh(1)^2 + (20*b*c^4*d^3*x^3 + 80*b*x^5*cosh(1)^3 +
30*b*c^2*d^3*x + 24*(4*b*c^2*d*x^5 + 5*b*d*x^3)*cosh(1)^2 + 3*(8*b*c^4*d^2*x^5 + 40*b*c^2*d^2*x^3 + 15*b*d^2*x
)*cosh(1))*sinh(1)^2 + (15*b*c^4*d^4*x + 40*b*x^5*cosh(1)^4 + 16*(4*b*c^2*d*x^5 + 5*b*d*x^3)*cosh(1)^3 + 3*(8*
b*c^4*d^2*x^5 + 40*b*c^2*d^2*x^3 + 15*b*d^2*x)*cosh(1)^2 + 20*(2*b*c^4*d^3*x^3 + 3*b*c^2*d^3*x)*cosh(1))*sinh(
1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (15*a*c^4*d^4*x*cosh(1) + 8*a*x^5*cosh(
1)^5 + 8*a*x^5*sinh(1)^5 + 4*(4*a*c^2*d*x^5 + 5*a*d*x^3)*cosh(1)^4 + 4*(4*a*c^2*d*x^5 + 10*a*x^5*cosh(1) + 5*a
*d*x^3)*sinh(1)^4 + (8*a*c^4*d^2*x^5 + 40*a*c^2*d^2*x^3 + 15*a*d^2*x)*cosh(1)^3 + (8*a*c^4*d^2*x^5 + 40*a*c^2*
d^2*x^3 + 80*a*x^5*cosh(1)^2 + 15*a*d^2*x + 16*(4*a*c^2*d*x^5 + 5*a*d*x^3)*cosh(1))*sinh(1)^3 + 10*(2*a*c^4*d^
3*x^3 + 3*a*c^2*d^3*x)*cosh(1)^2 + (20*a*c^4*d^3*x^3 + 80*a*x^5*cosh(1)^3 + 30*a*c^2*d^3*x + 24*(4*a*c^2*d*x^5
+ 5*a*d*x^3)*cosh(1)^2 + 3*(8*a*c^4*d^2*x^5 + 40*a*c^2*d^2*x^3 + 15*a*d^2*x)*cosh(1))*sinh(1)^2 + (15*a*c^4*d
^4*x + 40*a*x^5*cosh(1)^4 + 16*(4*a*c^2*d*x^5 + 5*a*d*x^3)*cosh(1)^3 + 3*(8*a*c^4*d^2*x^5 + 40*a*c^2*d^2*x^3 +
15*a*d^2*x)*cosh(1)^2 + 20*(2*a*c^4*d^3*x^3 + 3*a*c^2*d^3*x)*cosh(1))*sinh(1) - (4*b*c*d*x^4*cosh(1)^4 + 4*b*
c*d*x^4*sinh(1)^4 + 7*b*c^3*d^4*cosh(1) + 3*(2*b*c^3*d^2*x^4 + 3*b*c*d^2*x^2)*cosh(1)^3 + (6*b*c^3*d^2*x^4 + 1
6*b*c*d*x^4*cosh(1) + 9*b*c*d^2*x^2)*sinh(1)^3 + (13*b*c^3*d^3*x^2 + 5*b*c*d^3)*cosh(1)^2 + (13*b*c^3*d^3*x^2
+ 24*b*c*d*x^4*cosh(1)^2 + 5*b*c*d^3 + 9*(2*b*c^3*d^2*x^4 + 3*b*c*d^2*x^2)*cosh(1))*sinh(1)^2 + (16*b*c*d*x^4*
cosh(1)^3 + 7*b*c^3*d^4 + 9*(2*b*c^3*d^2*x^4 + 3*b*c*d^2*x^2)*cosh(1)^2 + 2*(13*b*c^3*d^3*x^2 + 5*b*c*d^3)*cos
h(1))*sinh(1))*sqrt(c^2*x^2 - 1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d))/(d^3*x^6*cosh(1)^6 + d^3*x^6*sinh(1)^6
+ c^4*d^8*cosh(1) + (2*c^2*d^4*x^6 + 3*d^4*x^4)*cosh(1)^5 + (2*c^2*d^4*x^6 + 6*d^3*x^6*cosh(1) + 3*d^4*x^4)*si
nh(1)^5 + (c^4*d^5*x^6 + 6*c^2*d^5*x^4 + 3*d^5*x^2)*cosh(1)^4 + (c^4*d^5*x^6 + 6*c^2*d^5*x^4 + 15*d^3*x^6*cosh
(1)^2 + 3*d^5*x^2 + 5*(2*c^2*d^4*x^6 + 3*d^4*x^4)*cosh(1))*sinh(1)^4 + (3*c^4*d^6*x^4 + 6*c^2*d^6*x^2 + d^6)*c
osh(1)^3 + (3*c^4*d^6*x^4 + 20*d^3*x^6*cosh(1)^3 + 6*c^2*d^6*x^2 + d^6 + 10*(2*c^2*d^4*x^6 + 3*d^4*x^4)*cosh(1
)^2 + 4*(c^4*d^5*x^6 + 6*c^2*d^5*x^4 + 3*d^5*x^2)*cosh(1))*sinh(1)^3 + (3*c^4*d^7*x^2 + 2*c^2*d^7)*cosh(1)^2 +
(3*c^4*d^7*x^2 + 15*d^3*x^6*cosh(1)^4 + 2*c^2*d^7 + 10*(2*c^2*d^4*x^6 + 3*d^4*x^4)*cosh(1)^3 + 6*(c^4*d^5*x^6
+ 6*c^2*d^5*x^4 + 3*d^5*x^2)*cosh(1)^2 + 3*(3*c^4*d^6*x^4 + 6*c^2*d^6*x^2 + d^6)*cosh(1))*sinh(1)^2 + (6*d^3*
x^6*cosh(1)^5 + c^4*d^8 + 5*(2*c^2*d^4*x^6 + 3*d^4*x^4)*cosh(1)^4 + 4*(c^4*d^5*x^6 + 6*c^2*d^5*x^4 + 3*d^5*x^2
)*cosh(1)^3 + 3*(3*c^4*d^6*x^4 + 6*c^2*d^6*x^2 + d^6)*cosh(1)^2 + 2*(3*c^4*d^7*x^2 + 2*c^2*d^7)*cosh(1))*sinh(
1))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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