3.779 \(\int \frac {1}{(-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{5/2}} \, dx\)

Optimal. Leaf size=211 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt {2} \left (b^2-c^2\right )^{5/4}}+\frac {3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {b \sinh (x)+c \cosh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]

[Out]

-3/32*arctanh(1/2*(b^2-c^2)^(1/4)*sinh(x+I*arctan(b,-I*c))*2^(1/2)/(-(b^2-c^2)^(1/2)+cosh(x+I*arctan(b,-I*c))*
(b^2-c^2)^(1/2))^(1/2))/(b^2-c^2)^(5/4)*2^(1/2)+3/16*(c*cosh(x)+b*sinh(x))/(b^2-c^2)/(b*cosh(x)+c*sinh(x)-(b^2
-c^2)^(1/2))^(3/2)+1/4*(-c*cosh(x)-b*sinh(x))/(b*cosh(x)+c*sinh(x)-(b^2-c^2)^(1/2))^(5/2)/(b^2-c^2)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3116, 3115, 2649, 204} \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt {2} \left (b^2-c^2\right )^{5/4}}+\frac {3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {b \sinh (x)+c \cosh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-5/2),x]

[Out]

(-3*ArcTanh[((b^2 - c^2)^(1/4)*Sinh[x + I*ArcTan[b, (-I)*c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 - c^2] + Sqrt[b^2 - c^2]
*Cosh[x + I*ArcTan[b, (-I)*c]]])])/(16*Sqrt[2]*(b^2 - c^2)^(5/4)) - (c*Cosh[x] + b*Sinh[x])/(4*Sqrt[b^2 - c^2]
*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2)) + (3*(c*Cosh[x] + b*Sinh[x]))/(16*(b^2 - c^2)*(-Sqrt[b^2 -
c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))

Rule 204

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3115

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Int[1/Sqrt[a +
Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0]

Rule 3116

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

Rubi steps

\begin {align*} \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \, dx &=-\frac {c \cosh (x)+b \sinh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}-\frac {3 \int \frac {1}{\left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}} \, dx}{8 \sqrt {b^2-c^2}}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac {3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {3 \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx}{32 \left (b^2-c^2\right )}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac {3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {3 \int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}} \, dx}{32 \left (b^2-c^2\right )}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac {3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{-2 \sqrt {b^2-c^2}-x^2} \, dx,x,-\frac {i \sqrt {b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \left (b^2-c^2\right )}\\ &=-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt {2} \left (b^2-c^2\right )^{5/4}}-\frac {c \cosh (x)+b \sinh (x)}{4 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}}+\frac {3 (c \cosh (x)+b \sinh (x))}{16 \left (b^2-c^2\right ) \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}\\ \end {align*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-5/2),x]

[Out]

$Aborted

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fricas [B]  time = 0.69, size = 5675, normalized size = 26.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cosh(x)+c*sinh(x)-(b^2-c^2)^(1/2))^(5/2),x, algorithm="fricas")

[Out]

-1/16*(3*sqrt(1/2)*((b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^10 + 10*(b^5 + 5*b^4*c +
10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)*sinh(x)^9 + (b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c
^4 + c^5)*sinh(x)^10 - 5*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^8 - 5*(b^5 + 3*b^4*c
+ 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5 - 9*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^
2)*sinh(x)^8 + 40*(3*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^3 - (b^5 + 3*b^4*c + 2*
b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x))*sinh(x)^7 + 10*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^
5)*cosh(x)^6 + 10*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5 + 21*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2
*c^3 + 5*b*c^4 + c^5)*cosh(x)^4 - 14*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^2)*sinh(x
)^6 + 4*(63*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^5 - 70*(b^5 + 3*b^4*c + 2*b^3*c^
2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^3 + 15*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x))*sin
h(x)^5 - b^5 + 5*b^4*c - 10*b^3*c^2 + 10*b^2*c^3 - 5*b*c^4 + c^5 - 10*(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b
*c^4 - c^5)*cosh(x)^4 + 10*(21*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^6 - b^5 + b^4
*c + 2*b^3*c^2 - 2*b^2*c^3 - b*c^4 + c^5 - 35*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^
4 + 15*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^2)*sinh(x)^4 + 40*(3*(b^5 + 5*b^4*c + 10*b^
3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^7 - 7*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh
(x)^5 + 5*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^3 - (b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3
+ b*c^4 - c^5)*cosh(x))*sinh(x)^3 + 5*(b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^2*c^3 - 3*b*c^4 + c^5)*cosh(x)^2 + 5*(
9*(b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^8 - 28*(b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*
c^3 - 3*b*c^4 - c^5)*cosh(x)^6 + b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^2*c^3 - 3*b*c^4 + c^5 + 30*(b^5 + b^4*c - 2*b
^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^4 - 12*(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*cosh(x)^2
)*sinh(x)^2 + 10*((b^5 + 5*b^4*c + 10*b^3*c^2 + 10*b^2*c^3 + 5*b*c^4 + c^5)*cosh(x)^9 - 4*(b^5 + 3*b^4*c + 2*b
^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^7 + 6*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cosh(x)^
5 - 4*(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5)*cosh(x)^3 + (b^5 - 3*b^4*c + 2*b^3*c^2 + 2*b^2*c^3 -
3*b*c^4 + c^5)*cosh(x))*sinh(x))*log(-((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)^3*sinh(x
) + 6*(b^2 + 2*b*c + c^2)*cosh(x)^2*sinh(x)^2 + 4*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^3 + (b^2 + 2*b*c + c^2)*
sinh(x)^4 - b^2 + 2*b*c - c^2 + 2*((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh(x)^3 - (b -
c)*cosh(x) + (3*(b + c)*cosh(x)^2 - b + c)*sinh(x))*sqrt(b^2 - c^2) + 2*(2*(b^2 - c^2)*cosh(x)^2 + 4*(b^2 - c^
2)*cosh(x)*sinh(x) + 2*(b^2 - c^2)*sinh(x)^2 + ((b + c)*cosh(x)^3 + 3*(b + c)*cosh(x)*sinh(x)^2 + (b + c)*sinh
(x)^3 + (b - c)*cosh(x) + (3*(b + c)*cosh(x)^2 + b - c)*sinh(x))*sqrt(b^2 - c^2))*sqrt(((b + c)*cosh(x)^2 + 2*
(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 2*sqrt(b^2 - c^2)*(cosh(x) + sinh(x)) + b - c)/(cosh(x) + sinh(x
)))/(b^2 - c^2)^(1/4))/((b^2 + 2*b*c + c^2)*cosh(x)^4 + 4*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^3 + (b^2 + 2*b*c
+ c^2)*sinh(x)^4 - 2*(b^2 - c^2)*cosh(x)^2 + 2*(3*(b^2 + 2*b*c + c^2)*cosh(x)^2 - b^2 + c^2)*sinh(x)^2 + b^2
- 2*b*c + c^2 + 4*((b^2 + 2*b*c + c^2)*cosh(x)^3 - (b^2 - c^2)*cosh(x))*sinh(x)))/(b^2 - c^2)^(1/4) - 2*sqrt(1
/2)*(3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^9 + 27*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*
cosh(x)*sinh(x)^8 + 3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*sinh(x)^9 - 36*(b^4 + 2*b^3*c - 2*b*c^3 - c^
4)*cosh(x)^7 - 36*(b^4 + 2*b^3*c - 2*b*c^3 - c^4 - 3*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^2)*si
nh(x)^7 + 252*((b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^3 - (b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)
)*sinh(x)^6 - 190*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^5 + 2*(189*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(
x)^4 - 95*b^4 + 190*b^2*c^2 - 95*c^4 - 378*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^2)*sinh(x)^5 + 2*(189*(b^4
+ 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^5 - 630*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^3 - 475*(b^4 -
2*b^2*c^2 + c^4)*cosh(x))*sinh(x)^4 - 36*(b^4 - 2*b^3*c + 2*b*c^3 - c^4)*cosh(x)^3 + 4*(63*(b^4 + 4*b^3*c + 6*
b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^6 - 315*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^4 - 9*b^4 + 18*b^3*c - 18*b*c
^3 + 9*c^4 - 475*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^2)*sinh(x)^3 + 4*(27*(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c
^4)*cosh(x)^7 - 189*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^5 - 475*(b^4 - 2*b^2*c^2 + c^4)*cosh(x)^3 - 27*(b^
4 - 2*b^3*c + 2*b*c^3 - c^4)*cosh(x))*sinh(x)^2 + 3*(b^4 - 4*b^3*c + 6*b^2*c^2 - 4*b*c^3 + c^4)*cosh(x) + (27*
(b^4 + 4*b^3*c + 6*b^2*c^2 + 4*b*c^3 + c^4)*cosh(x)^8 - 252*(b^4 + 2*b^3*c - 2*b*c^3 - c^4)*cosh(x)^6 - 950*(b
^4 - 2*b^2*c^2 + c^4)*cosh(x)^4 + 3*b^4 - 12*b^3*c + 18*b^2*c^2 - 12*b*c^3 + 3*c^4 - 108*(b^4 - 2*b^3*c + 2*b*
c^3 - c^4)*cosh(x)^2)*sinh(x) + 4*((b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^8 + 8*(b^3 + 3*b^2*c + 3*b*c^2 + c^
3)*cosh(x)*sinh(x)^7 + (b^3 + 3*b^2*c + 3*b*c^2 + c^3)*sinh(x)^8 - 33*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^6 -
(33*b^3 + 33*b^2*c - 33*b*c^2 - 33*c^3 - 28*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2)*sinh(x)^6 + 2*(28*(b^3
+ 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 - 99*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x))*sinh(x)^5 - 33*(b^3 - b^2*c - b
*c^2 + c^3)*cosh(x)^4 + (70*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4 - 33*b^3 + 33*b^2*c + 33*b*c^2 - 33*c^3
- 495*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^5 - 165
*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 - 33*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x))*sinh(x)^3 + (b^3 - 3*b^2*c +
3*b*c^2 - c^3)*cosh(x)^2 + (28*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^6 - 495*(b^3 + b^2*c - b*c^2 - c^3)*cos
h(x)^4 + b^3 - 3*b^2*c + 3*b*c^2 - c^3 - 198*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^2)*sinh(x)^2 + 2*(4*(b^3 + 3*
b^2*c + 3*b*c^2 + c^3)*cosh(x)^7 - 99*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^5 - 66*(b^3 - b^2*c - b*c^2 + c^3)*c
osh(x)^3 + (b^3 - 3*b^2*c + 3*b*c^2 - c^3)*cosh(x))*sinh(x))*sqrt(b^2 - c^2))*sqrt(((b + c)*cosh(x)^2 + 2*(b +
c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - 2*sqrt(b^2 - c^2)*(cosh(x) + sinh(x)) + b - c)/(cosh(x) + sinh(x))))
/((b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^10 + 10*(b^7 + 5*b^6
*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)*sinh(x)^9 + (b^7 + 5*b^6*c + 9*b^5
*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*sinh(x)^10 - 5*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3
- 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^8 - 5*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^
5 + 3*b*c^6 + c^7 - 9*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^
2)*sinh(x)^8 + 40*(3*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^3
- (b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x))*sinh(x)^7 - b^7 + 5*b^
6*c - 9*b^5*c^2 + 5*b^4*c^3 + 5*b^3*c^4 - 9*b^2*c^5 + 5*b*c^6 - c^7 + 10*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3
+ 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^6 + 10*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2
*c^5 - b*c^6 - c^7 + 21*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x
)^4 - 14*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^2)*sinh(x)^6 + 4*
(63*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^5 - 70*(b^7 + 3*b^
6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^3 + 15*(b^7 + b^6*c - 3*b^5*c^2 - 3*b
^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x))*sinh(x)^5 - 10*(b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3
*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7)*cosh(x)^4 - 10*(b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2*c^
5 - b*c^6 + c^7 - 21*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^6
+ 35*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^4 - 15*(b^7 + b^6*c
- 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^2)*sinh(x)^4 + 40*(3*(b^7 + 5*b^6*c + 9
*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^7 - 7*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c
^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^5 + 5*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b
^2*c^5 - b*c^6 - c^7)*cosh(x)^3 - (b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7)*
cosh(x))*sinh(x)^3 + 5*(b^7 - 3*b^6*c + b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - b^2*c^5 + 3*b*c^6 - c^7)*cosh(x)^2 +
5*(9*(b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^8 + b^7 - 3*b^6*
c + b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - b^2*c^5 + 3*b*c^6 - c^7 - 28*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^
3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^6 + 30*(b^7 + b^6*c - 3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 -
b*c^6 - c^7)*cosh(x)^4 - 12*(b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c^3 + 3*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7)*cosh(
x)^2)*sinh(x)^2 + 10*((b^7 + 5*b^6*c + 9*b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - 9*b^2*c^5 - 5*b*c^6 - c^7)*cosh(x)^
9 - 4*(b^7 + 3*b^6*c + b^5*c^2 - 5*b^4*c^3 - 5*b^3*c^4 + b^2*c^5 + 3*b*c^6 + c^7)*cosh(x)^7 + 6*(b^7 + b^6*c -
3*b^5*c^2 - 3*b^4*c^3 + 3*b^3*c^4 + 3*b^2*c^5 - b*c^6 - c^7)*cosh(x)^5 - 4*(b^7 - b^6*c - 3*b^5*c^2 + 3*b^4*c
^3 + 3*b^3*c^4 - 3*b^2*c^5 - b*c^6 + c^7)*cosh(x)^3 + (b^7 - 3*b^6*c + b^5*c^2 + 5*b^4*c^3 - 5*b^3*c^4 - b^2*c
^5 + 3*b*c^6 - c^7)*cosh(x))*sinh(x))

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cosh(x)+c*sinh(x)-(b^2-c^2)^(1/2))^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep),abs((-sqrt(b^2-c^2))*t_nostep+b-c)]Evaluation time: 1.45Unable to divide, perhaps due to rounding
error%%%{%%%{32,[5,0]%%%}+%%%{96,[4,1]%%%}+%%%{-32,[4,0]%%%}+%%%{64,[3,2]%%%}+%%%{-64,[3,1]%%%}+%%%{-64,[2,3]%
%%}+%%%{-96,[1,4]%%%}+%%%{64,[1,3]%%%}+%%%{-32,[0,5]%%%}+%%%{32,[0,4]%%%},[10,1]%%%}+%%%{%%{[%%%{80,[2,0]%%%}+
%%%{160,[1,1]%%%}+%%%{80,[0,2]%%%},0,%%%{-240,[4,0]%%%}+%%%{-480,[3,1]%%%}+%%%{-80,[3,0]%%%}+%%%{-80,[2,1]%%%}
+%%%{480,[1,3]%%%}+%%%{80,[1,2]%%%}+%%%{240,[0,4]%%%}+%%%{80,[0,3]%%%},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2,[1,0]%%
%}+%%%{2,[0,2]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%%%}+%%%{-2,[3,0]%%%}+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[
2,0]%%%}+%%%{2,[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,4]%%%}+%%%{-2,[0,3]%%%}+%%%{1,[0,2]%%%}]%%},[8,1]%%%}+%%%{%
%%{320,[5,0]%%%}+%%%{320,[4,1]%%%}+%%%{-320,[4,0]%%%}+%%%{-640,[3,2]%%%}+%%%{-640,[2,3]%%%}+%%%{640,[2,2]%%%}+
%%%{320,[1,4]%%%}+%%%{320,[0,5]%%%}+%%%{-320,[0,4]%%%},[6,1]%%%}+%%%{%%{[%%%{160,[2,0]%%%}+%%%{-160,[0,2]%%%},
0,%%%{-480,[4,0]%%%}+%%%{-160,[3,0]%%%}+%%%{960,[2,2]%%%}+%%%{160,[2,1]%%%}+%%%{160,[1,2]%%%}+%%%{-480,[0,4]%%
%}+%%%{-160,[0,3]%%%},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2,[1,0]%%%}+%%%{2,[0,2]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%
%%}+%%%{-2,[3,0]%%%}+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[2,0]%%%}+%%%{2,[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[
0,4]%%%}+%%%{-2,[0,3]%%%}+%%%{1,[0,2]%%%}]%%},[4,1]%%%}+%%%{%%%{160,[5,0]%%%}+%%%{-160,[4,1]%%%}+%%%{-160,[4,0
]%%%}+%%%{-320,[3,2]%%%}+%%%{320,[3,1]%%%}+%%%{320,[2,3]%%%}+%%%{160,[1,4]%%%}+%%%{-320,[1,3]%%%}+%%%{-160,[0,
5]%%%}+%%%{160,[0,4]%%%},[2,1]%%%}+%%%{%%{[%%%{16,[2,0]%%%}+%%%{-32,[1,1]%%%}+%%%{16,[0,2]%%%},0,%%%{-48,[4,0]
%%%}+%%%{96,[3,1]%%%}+%%%{-16,[3,0]%%%}+%%%{48,[2,1]%%%}+%%%{-96,[1,3]%%%}+%%%{-48,[1,2]%%%}+%%%{48,[0,4]%%%}+
%%%{16,[0,3]%%%},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2,[1,0]%%%}+%%%{2,[0,2]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%%%}+%
%%{-2,[3,0]%%%}+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[2,0]%%%}+%%%{2,[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,4]%
%%}+%%%{-2,[0,3]%%%}+%%%{1,[0,2]%%%}]%%},[0,1]%%%} / %%%{%%%{32,[15,0]%%%}+%%%{160,[14,1]%%%}+%%%{-160,[14,0]%
%%}+%%%{160,[13,2]%%%}+%%%{-640,[13,1]%%%}+%%%{320,[13,0]%%%}+%%%{-480,[12,3]%%%}+%%%{-160,[12,2]%%%}+%%%{960,
[12,1]%%%}+%%%{-320,[12,0]%%%}+%%%{-1120,[11,4]%%%}+%%%{2560,[11,3]%%%}+%%%{-640,[11,2]%%%}+%%%{-640,[11,1]%%%
}+%%%{160,[11,0]%%%}+%%%{32,[10,5]%%%}+%%%{3040,[10,4]%%%}+%%%{-4480,[10,3]%%%}+%%%{1280,[10,2]%%%}+%%%{160,[1
0,1]%%%}+%%%{-32,[10,0]%%%}+%%%{2080,[9,6]%%%}+%%%{-3200,[9,5]%%%}+%%%{-1600,[9,4]%%%}+%%%{3200,[9,3]%%%}+%%%{
-800,[9,2]%%%}+%%%{1440,[8,7]%%%}+%%%{-7200,[8,6]%%%}+%%%{8000,[8,5]%%%}+%%%{-1600,[8,4]%%%}+%%%{-800,[8,3]%%%
}+%%%{160,[8,2]%%%}+%%%{-1440,[7,8]%%%}+%%%{6400,[7,6]%%%}+%%%{-6400,[7,5]%%%}+%%%{1600,[7,4]%%%}+%%%{-2080,[6
,9]%%%}+%%%{7200,[6,8]%%%}+%%%{-6400,[6,7]%%%}+%%%{1600,[6,5]%%%}+%%%{-320,[6,4]%%%}+%%%{-32,[5,10]%%%}+%%%{32
00,[5,9]%%%}+%%%{-8000,[5,8]%%%}+%%%{6400,[5,7]%%%}+%%%{-1600,[5,6]%%%}+%%%{1120,[4,11]%%%}+%%%{-3040,[4,10]%%
%}+%%%{1600,[4,9]%%%}+%%%{1600,[4,8]%%%}+%%%{-1600,[4,7]%%%}+%%%{320,[4,6]%%%}+%%%{480,[3,12]%%%}+%%%{-2560,[3
,11]%%%}+%%%{4480,[3,10]%%%}+%%%{-3200,[3,9]%%%}+%%%{800,[3,8]%%%}+%%%{-160,[2,13]%%%}+%%%{160,[2,12]%%%}+%%%{
640,[2,11]%%%}+%%%{-1280,[2,10]%%%}+%%%{800,[2,9]%%%}+%%%{-160,[2,8]%%%}+%%%{-160,[1,14]%%%}+%%%{640,[1,13]%%%
}+%%%{-960,[1,12]%%%}+%%%{640,[1,11]%%%}+%%%{-160,[1,10]%%%}+%%%{-32,[0,15]%%%}+%%%{160,[0,14]%%%}+%%%{-320,[0
,13]%%%}+%%%{320,[0,12]%%%}+%%%{-160,[0,11]%%%}+%%%{32,[0,10]%%%},[10,0]%%%}+%%%{%%{[%%%{80,[12,0]%%%}+%%%{320
,[11,1]%%%}+%%%{-320,[11,0]%%%}+%%%{160,[10,2]%%%}+%%%{-960,[10,1]%%%}+%%%{480,[10,0]%%%}+%%%{-960,[9,3]%%%}+%
%%{320,[9,2]%%%}+%%%{960,[9,1]%%%}+%%%{-320,[9,0]%%%}+%%%{-1360,[8,4]%%%}+%%%{3520,[8,3]%%%}+%%%{-1440,[8,2]%%
%}+%%%{-320,[8,1]%%%}+%%%{80,[8,0]%%%}+%%%{640,[7,5]%%%}+%%%{1920,[7,4]%%%}+%%%{-3840,[7,3]%%%}+%%%{1280,[7,2]
%%%}+%%%{2240,[6,6]%%%}+%%%{-4480,[6,5]%%%}+%%%{960,[6,4]%%%}+%%%{1280,[6,3]%%%}+%%%{-320,[6,2]%%%}+%%%{640,[5
,7]%%%}+%%%{-4480,[5,6]%%%}+%%%{5760,[5,5]%%%}+%%%{-1920,[5,4]%%%}+%%%{-1360,[4,8]%%%}+%%%{1920,[4,7]%%%}+%%%{
960,[4,6]%%%}+%%%{-1920,[4,5]%%%}+%%%{480,[4,4]%%%}+%%%{-960,[3,9]%%%}+%%%{3520,[3,8]%%%}+%%%{-3840,[3,7]%%%}+
%%%{1280,[3,6]%%%}+%%%{160,[2,10]%%%}+%%%{320,[2,9]%%%}+%%%{-1440,[2,8]%%%}+%%%{1280,[2,7]%%%}+%%%{-320,[2,6]%
%%}+%%%{320,[1,11]%%%}+%%%{-960,[1,10]%%%}+%%%{960,[1,9]%%%}+%%%{-320,[1,8]%%%}+%%%{80,[0,12]%%%}+%%%{-320,[0,
11]%%%}+%%%{480,[0,10]%%%}+%%%{-320,[0,9]%%%}+%%%{80,[0,8]%%%},0,%%%{-240,[14,0]%%%}+%%%{-960,[13,1]%%%}+%%%{8
80,[13,0]%%%}+%%%{-240,[12,2]%%%}+%%%{2640,[12,1]%%%}+%%%{-1120,[12,0]%%%}+%%%{3840,[11,3]%%%}+%%%{-1760,[11,2
]%%%}+%%%{-2240,[11,1]%%%}+%%%{480,[11,0]%%%}+%%%{4560,[10,4]%%%}+%%%{-12320,[10,3]%%%}+%%%{4480,[10,2]%%%}+%%
%{480,[10,1]%%%}+%%%{80,[10,0]%%%}+%%%{-4800,[9,5]%%%}+%%%{-4400,[9,4]%%%}+%%%{11200,[9,3]%%%}+%%%{-2400,[9,2]
%%%}+%%%{-80,[9,0]%%%}+%%%{-10800,[8,6]%%%}+%%%{22000,[8,5]%%%}+%%%{-5600,[8,4]%%%}+%%%{-2400,[8,3]%%%}+%%%{-4
00,[8,2]%%%}+%%%{80,[8,1]%%%}+%%%{17600,[7,6]%%%}+%%%{-22400,[7,5]%%%}+%%%{4800,[7,4]%%%}+%%%{320,[7,2]%%%}+%%
%{10800,[6,8]%%%}+%%%{-17600,[6,7]%%%}+%%%{4800,[6,5]%%%}+%%%{800,[6,4]%%%}+%%%{-320,[6,3]%%%}+%%%{4800,[5,9]%
%%}+%%%{-22000,[5,8]%%%}+%%%{22400,[5,7]%%%}+%%%{-4800,[5,6]%%%}+%%%{-480,[5,4]%%%}+%%%{-4560,[4,10]%%%}+%%%{4
400,[4,9]%%%}+%%%{5600,[4,8]%%%}+%%%{-4800,[4,7]%%%}+%%%{-800,[4,6]%%%}+%%%{480,[4,5]%%%}+%%%{-3840,[3,11]%%%}
+%%%{12320,[3,10]%%%}+%%%{-11200,[3,9]%%%}+%%%{2400,[3,8]%%%}+%%%{320,[3,6]%%%}+%%%{240,[2,12]%%%}+%%%{1760,[2
,11]%%%}+%%%{-4480,[2,10]%%%}+%%%{2400,[2,9]%%%}+%%%{400,[2,8]%%%}+%%%{-320,[2,7]%%%}+%%%{960,[1,13]%%%}+%%%{-
2640,[1,12]%%%}+%%%{2240,[1,11]%%%}+%%%{-480,[1,10]%%%}+%%%{-80,[1,8]%%%}+%%%{240,[0,14]%%%}+%%%{-880,[0,13]%%
%}+%%%{1120,[0,12]%%%}+%%%{-480,[0,11]%%%}+%%%{-80,[0,10]%%%}+%%%{80,[0,9]%%%},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2
,[1,0]%%%}+%%%{2,[0,2]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%%%}+%%%{-2,[3,0]%%%}+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}
+%%%{1,[2,0]%%%}+%%%{2,[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,4]%%%}+%%%{-2,[0,3]%%%}+%%%{1,[0,2]%%%}]%%},[8,0]%%
%}+%%%{%%%{320,[15,0]%%%}+%%%{960,[14,1]%%%}+%%%{-1600,[14,0]%%%}+%%%{-960,[13,2]%%%}+%%%{-3200,[13,1]%%%}+%%%
{3200,[13,0]%%%}+%%%{-5440,[12,3]%%%}+%%%{8000,[12,2]%%%}+%%%{3200,[12,1]%%%}+%%%{-3200,[12,0]%%%}+%%%{-960,[1
1,4]%%%}+%%%{19200,[11,3]%%%}+%%%{-19200,[11,2]%%%}+%%%{1600,[11,0]%%%}+%%%{12480,[10,5]%%%}+%%%{-14400,[10,4]
%%%}+%%%{-19200,[10,3]%%%}+%%%{19200,[10,2]%%%}+%%%{-1600,[10,1]%%%}+%%%{-320,[10,0]%%%}+%%%{8000,[9,6]%%%}+%%
%{-48000,[9,5]%%%}+%%%{48000,[9,4]%%%}+%%%{-8000,[9,2]%%%}+%%%{640,[9,1]%%%}+%%%{-14400,[8,7]%%%}+%%%{8000,[8,
6]%%%}+%%%{48000,[8,5]%%%}+%%%{-48000,[8,4]%%%}+%%%{8000,[8,3]%%%}+%%%{960,[8,2]%%%}+%%%{-14400,[7,8]%%%}+%%%{
64000,[7,7]%%%}+%%%{-64000,[7,6]%%%}+%%%{16000,[7,4]%%%}+%%%{-2560,[7,3]%%%}+%%%{8000,[6,9]%%%}+%%%{8000,[6,8]
%%%}+%%%{-64000,[6,7]%%%}+%%%{64000,[6,6]%%%}+%%%{-16000,[6,5]%%%}+%%%{-640,[6,4]%%%}+%%%{12480,[5,10]%%%}+%%%
{-48000,[5,9]%%%}+%%%{48000,[5,8]%%%}+%%%{-16000,[5,6]%%%}+%%%{3840,[5,5]%%%}+%%%{-960,[4,11]%%%}+%%%{-14400,[
4,10]%%%}+%%%{48000,[4,9]%%%}+%%%{-48000,[4,8]%%%}+%%%{16000,[4,7]%%%}+%%%{-640,[4,6]%%%}+%%%{-5440,[3,12]%%%}
+%%%{19200,[3,11]%%%}+%%%{-19200,[3,10]%%%}+%%%{8000,[3,8]%%%}+%%%{-2560,[3,7]%%%}+%%%{-960,[2,13]%%%}+%%%{800
0,[2,12]%%%}+%%%{-19200,[2,11]%%%}+%%%{19200,[2,10]%%%}+%%%{-8000,[2,9]%%%}+%%%{960,[2,8]%%%}+%%%{960,[1,14]%%
%}+%%%{-3200,[1,13]%%%}+%%%{3200,[1,12]%%%}+%%%{-1600,[1,10]%%%}+%%%{640,[1,9]%%%}+%%%{320,[0,15]%%%}+%%%{-160
0,[0,14]%%%}+%%%{3200,[0,13]%%%}+%%%{-3200,[0,12]%%%}+%%%{1600,[0,11]%%%}+%%%{-320,[0,10]%%%},[6,0]%%%}+%%%{%%
{[%%%{160,[12,0]%%%}+%%%{320,[11,1]%%%}+%%%{-640,[11,0]%%%}+%%%{-640,[10,2]%%%}+%%%{-640,[10,1]%%%}+%%%{960,[1
0,0]%%%}+%%%{-1600,[9,3]%%%}+%%%{3200,[9,2]%%%}+%%%{-640,[9,0]%%%}+%%%{800,[8,4]%%%}+%%%{3200,[8,3]%%%}+%%%{-4
800,[8,2]%%%}+%%%{640,[8,1]%%%}+%%%{160,[8,0]%%%}+%%%{3200,[7,5]%%%}+%%%{-6400,[7,4]%%%}+%%%{2560,[7,2]%%%}+%%
%{-320,[7,1]%%%}+%%%{-6400,[6,5]%%%}+%%%{9600,[6,4]%%%}+%%%{-2560,[6,3]%%%}+%%%{-320,[6,2]%%%}+%%%{-3200,[5,7]
%%%}+%%%{6400,[5,6]%%%}+%%%{-3840,[5,4]%%%}+%%%{960,[5,3]%%%}+%%%{-800,[4,8]%%%}+%%%{6400,[4,7]%%%}+%%%{-9600,
[4,6]%%%}+%%%{3840,[4,5]%%%}+%%%{1600,[3,9]%%%}+%%%{-3200,[3,8]%%%}+%%%{2560,[3,6]%%%}+%%%{-960,[3,5]%%%}+%%%{
640,[2,10]%%%}+%%%{-3200,[2,9]%%%}+%%%{4800,[2,8]%%%}+%%%{-2560,[2,7]%%%}+%%%{320,[2,6]%%%}+%%%{-320,[1,11]%%%
}+%%%{640,[1,10]%%%}+%%%{-640,[1,8]%%%}+%%%{320,[1,7]%%%}+%%%{-160,[0,12]%%%}+%%%{640,[0,11]%%%}+%%%{-960,[0,1
0]%%%}+%%%{640,[0,9]%%%}+%%%{-160,[0,8]%%%},0,%%%{-480,[14,0]%%%}+%%%{-960,[13,1]%%%}+%%%{1760,[13,0]%%%}+%%%{
2400,[12,2]%%%}+%%%{1760,[12,1]%%%}+%%%{-2240,[12,0]%%%}+%%%{5760,[11,3]%%%}+%%%{-10560,[11,2]%%%}+%%%{960,[11
,0]%%%}+%%%{-4320,[10,4]%%%}+%%%{-10560,[10,3]%%%}+%%%{13440,[10,2]%%%}+%%%{-960,[10,1]%%%}+%%%{160,[10,0]%%%}
+%%%{-14400,[9,5]%%%}+%%%{26400,[9,4]%%%}+%%%{-4800,[9,2]%%%}+%%%{-320,[9,1]%%%}+%%%{-160,[9,0]%%%}+%%%{2400,[
8,6]%%%}+%%%{26400,[8,5]%%%}+%%%{-33600,[8,4]%%%}+%%%{4800,[8,3]%%%}+%%%{-480,[8,2]%%%}+%%%{480,[8,1]%%%}+%%%{
19200,[7,7]%%%}+%%%{-35200,[7,6]%%%}+%%%{9600,[7,4]%%%}+%%%{1280,[7,3]%%%}+%%%{2400,[6,8]%%%}+%%%{-35200,[6,7]
%%%}+%%%{44800,[6,6]%%%}+%%%{-9600,[6,5]%%%}+%%%{320,[6,4]%%%}+%%%{-1280,[6,3]%%%}+%%%{-14400,[5,9]%%%}+%%%{26
400,[5,8]%%%}+%%%{-9600,[5,6]%%%}+%%%{-1920,[5,5]%%%}+%%%{960,[5,4]%%%}+%%%{-4320,[4,10]%%%}+%%%{26400,[4,9]%%
%}+%%%{-33600,[4,8]%%%}+%%%{9600,[4,7]%%%}+%%%{320,[4,6]%%%}+%%%{960,[4,5]%%%}+%%%{5760,[3,11]%%%}+%%%{-10560,
[3,10]%%%}+%%%{4800,[3,8]%%%}+%%%{1280,[3,7]%%%}+%%%{-1280,[3,6]%%%}+%%%{2400,[2,12]%%%}+%%%{-10560,[2,11]%%%}
+%%%{13440,[2,10]%%%}+%%%{-4800,[2,9]%%%}+%%%{-480,[2,8]%%%}+%%%{-960,[1,13]%%%}+%%%{1760,[1,12]%%%}+%%%{-960,
[1,10]%%%}+%%%{-320,[1,9]%%%}+%%%{480,[1,8]%%%}+%%%{-480,[0,14]%%%}+%%%{1760,[0,13]%%%}+%%%{-2240,[0,12]%%%}+%
%%{960,[0,11]%%%}+%%%{160,[0,10]%%%}+%%%{-160,[0,9]%%%},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2,[1,0]%%%}+%%%{2,[0,2]%
%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%%%}+%%%{-2,[3,0]%%%}+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[2,0]%%%}+%%%{2,
[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,4]%%%}+%%%{-2,[0,3]%%%}+%%%{1,[0,2]%%%}]%%},[4,0]%%%}+%%%{%%%{160,[15,0]%%
%}+%%%{160,[14,1]%%%}+%%%{-800,[14,0]%%%}+%%%{-1120,[13,2]%%%}+%%%{1600,[13,0]%%%}+%%%{-1120,[12,3]%%%}+%%%{56
00,[12,2]%%%}+%%%{-1600,[12,1]%%%}+%%%{-1600,[12,0]%%%}+%%%{3360,[11,4]%%%}+%%%{-9600,[11,2]%%%}+%%%{3200,[11,
1]%%%}+%%%{800,[11,0]%%%}+%%%{3360,[10,5]%%%}+%%%{-16800,[10,4]%%%}+%%%{9600,[10,3]%%%}+%%%{6400,[10,2]%%%}+%%
%{-2400,[10,1]%%%}+%%%{-160,[10,0]%%%}+%%%{-5600,[9,6]%%%}+%%%{24000,[9,4]%%%}+%%%{-16000,[9,3]%%%}+%%%{-800,[
9,2]%%%}+%%%{640,[9,1]%%%}+%%%{-5600,[8,7]%%%}+%%%{28000,[8,6]%%%}+%%%{-24000,[8,5]%%%}+%%%{-8000,[8,4]%%%}+%%
%{8800,[8,3]%%%}+%%%{-480,[8,2]%%%}+%%%{5600,[7,8]%%%}+%%%{-32000,[7,6]%%%}+%%%{32000,[7,5]%%%}+%%%{-4800,[7,4
]%%%}+%%%{-1280,[7,3]%%%}+%%%{5600,[6,9]%%%}+%%%{-28000,[6,8]%%%}+%%%{32000,[6,7]%%%}+%%%{-11200,[6,5]%%%}+%%%
{2240,[6,4]%%%}+%%%{-3360,[5,10]%%%}+%%%{24000,[5,8]%%%}+%%%{-32000,[5,7]%%%}+%%%{11200,[5,6]%%%}+%%%{-3360,[4
,11]%%%}+%%%{16800,[4,10]%%%}+%%%{-24000,[4,9]%%%}+%%%{8000,[4,8]%%%}+%%%{4800,[4,7]%%%}+%%%{-2240,[4,6]%%%}+%
%%{1120,[3,12]%%%}+%%%{-9600,[3,10]%%%}+%%%{16000,[3,9]%%%}+%%%{-8800,[3,8]%%%}+%%%{1280,[3,7]%%%}+%%%{1120,[2
,13]%%%}+%%%{-5600,[2,12]%%%}+%%%{9600,[2,11]%%%}+%%%{-6400,[2,10]%%%}+%%%{800,[2,9]%%%}+%%%{480,[2,8]%%%}+%%%
{-160,[1,14]%%%}+%%%{1600,[1,12]%%%}+%%%{-3200,[1,11]%%%}+%%%{2400,[1,10]%%%}+%%%{-640,[1,9]%%%}+%%%{-160,[0,1
5]%%%}+%%%{800,[0,14]%%%}+%%%{-1600,[0,13]%%%}+%%%{1600,[0,12]%%%}+%%%{-800,[0,11]%%%}+%%%{160,[0,10]%%%},[2,0
]%%%}+%%%{%%{[%%%{16,[12,0]%%%}+%%%{-64,[11,0]%%%}+%%%{-96,[10,2]%%%}+%%%{64,[10,1]%%%}+%%%{96,[10,0]%%%}+%%%{
320,[9,2]%%%}+%%%{-192,[9,1]%%%}+%%%{-64,[9,0]%%%}+%%%{240,[8,4]%%%}+%%%{-320,[8,3]%%%}+%%%{-288,[8,2]%%%}+%%%
{192,[8,1]%%%}+%%%{16,[8,0]%%%}+%%%{-640,[7,4]%%%}+%%%{768,[7,3]%%%}+%%%{-64,[7,1]%%%}+%%%{-320,[6,6]%%%}+%%%{
640,[6,5]%%%}+%%%{192,[6,4]%%%}+%%%{-512,[6,3]%%%}+%%%{64,[6,2]%%%}+%%%{640,[5,6]%%%}+%%%{-1152,[5,5]%%%}+%%%{
384,[5,4]%%%}+%%%{64,[5,3]%%%}+%%%{240,[4,8]%%%}+%%%{-640,[4,7]%%%}+%%%{192,[4,6]%%%}+%%%{384,[4,5]%%%}+%%%{-1
60,[4,4]%%%}+%%%{-320,[3,8]%%%}+%%%{768,[3,7]%%%}+%%%{-512,[3,6]%%%}+%%%{64,[3,5]%%%}+%%%{-96,[2,10]%%%}+%%%{3
20,[2,9]%%%}+%%%{-288,[2,8]%%%}+%%%{64,[2,6]%%%}+%%%{64,[1,10]%%%}+%%%{-192,[1,9]%%%}+%%%{192,[1,8]%%%}+%%%{-6
4,[1,7]%%%}+%%%{16,[0,12]%%%}+%%%{-64,[0,11]%%%}+%%%{96,[0,10]%%%}+%%%{-64,[0,9]%%%}+%%%{16,[0,8]%%%},0,%%%{-4
8,[14,0]%%%}+%%%{176,[13,0]%%%}+%%%{336,[12,2]%%%}+%%%{-176,[12,1]%%%}+%%%{-224,[12,0]%%%}+%%%{-1056,[11,2]%%%
}+%%%{448,[11,1]%%%}+%%%{96,[11,0]%%%}+%%%{-1008,[10,4]%%%}+%%%{1056,[10,3]%%%}+%%%{896,[10,2]%%%}+%%%{-288,[1
0,1]%%%}+%%%{16,[10,0]%%%}+%%%{2640,[9,4]%%%}+%%%{-2240,[9,3]%%%}+%%%{-96,[9,2]%%%}+%%%{-64,[9,1]%%%}+%%%{-16,
[9,0]%%%}+%%%{1680,[8,6]%%%}+%%%{-2640,[8,5]%%%}+%%%{-1120,[8,4]%%%}+%%%{1056,[8,3]%%%}+%%%{48,[8,2]%%%}+%%%{8
0,[8,1]%%%}+%%%{-3520,[7,6]%%%}+%%%{4480,[7,5]%%%}+%%%{-576,[7,4]%%%}+%%%{128,[7,3]%%%}+%%%{-128,[7,2]%%%}+%%%
{-1680,[6,8]%%%}+%%%{3520,[6,7]%%%}+%%%{-1344,[6,5]%%%}+%%%{-224,[6,4]%%%}+%%%{2640,[5,8]%%%}+%%%{-4480,[5,7]%
%%}+%%%{1344,[5,6]%%%}+%%%{224,[5,4]%%%}+%%%{1008,[4,10]%%%}+%%%{-2640,[4,9]%%%}+%%%{1120,[4,8]%%%}+%%%{576,[4
,7]%%%}+%%%{224,[4,6]%%%}+%%%{-224,[4,5]%%%}+%%%{-1056,[3,10]%%%}+%%%{2240,[3,9]%%%}+%%%{-1056,[3,8]%%%}+%%%{-
128,[3,7]%%%}+%%%{-336,[2,12]%%%}+%%%{1056,[2,11]%%%}+%%%{-896,[2,10]%%%}+%%%{96,[2,9]%%%}+%%%{-48,[2,8]%%%}+%
%%{128,[2,7]%%%}+%%%{176,[1,12]%%%}+%%%{-448,[1,11]%%%}+%%%{288,[1,10]%%%}+%%%{64,[1,9]%%%}+%%%{-80,[1,8]%%%}+
%%%{48,[0,14]%%%}+%%%{-176,[0,13]%%%}+%%%{224,[0,12]%%%}+%%%{-96,[0,11]%%%}+%%%{-16,[0,10]%%%}+%%%{16,[0,9]%%%
},0]:[1,0,%%%{-2,[2,0]%%%}+%%%{-2,[1,0]%%%}+%%%{2,[0,2]%%%}+%%%{2,[0,1]%%%},0,%%%{1,[4,0]%%%}+%%%{-2,[3,0]%%%}
+%%%{-2,[2,2]%%%}+%%%{2,[2,1]%%%}+%%%{1,[2,0]%%%}+%%%{2,[1,2]%%%}+%%%{-2,[1,1]%%%}+%%%{1,[0,4]%%%}+%%%{-2,[0,3
]%%%}+%%%{1,[0,2]%%%}]%%},[0,0]%%%} Error: Bad Argument Value

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maple [B]  time = 1.92, size = 984, normalized size = 4.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cosh(x)+c*sinh(x)-(b^2-c^2)^(1/2))^(5/2),x)

[Out]

-1/8/(sinh(x)+1)/sinh(x)/((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+(-b^2+c^2)/(b^2-c^2)^(1/2))^(1/2)/(-sinh(x)*(b^2-
c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)/(b^2-c^2)*(2*2^(1/2)*arctanh(1/2*cosh(x)*2^(1/2))*(-sinh(x)*(b^2-c^2)^(1/2)-
(b^2-c^2)^(1/2))^(1/2)*sinh(x)^2+2^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*ln(2/(co
sh(x)+2^(1/2))*(cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)+sinh(x)*(b^2-c^2)^(1/2
)+(-sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2
)+(b^2-c^2)^(1/2)))*sinh(x)-2^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*ln(2/(-cosh(x
)+2^(1/2))*(cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)-sinh(x)*(b^2-c^2)^(1/2)-(-
sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)-(b
^2-c^2)^(1/2)))*sinh(x)+2*2^(1/2)*arctanh(1/2*cosh(x)*2^(1/2))*(-sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2
)*sinh(x)+2^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*ln(2/(cosh(x)+2^(1/2))*(cosh(x)
*(b^2-c^2)^(1/2)*2^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)+sinh(x)*(b^2-c^2)^(1/2)+(-sinh(x)*(b^2-c^2)^(
1/2)-(b^2-c^2)^(1/2))^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)+(b^2-c^2)^(1/2)))-2^(
1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*ln(2/(-cosh(x)+2^(1/2))*(cosh(x)*(b^2-c^2)^(
1/2)*2^(1/2)*sinh(x)+cosh(x)*(b^2-c^2)^(1/2)*2^(1/2)-sinh(x)*(b^2-c^2)^(1/2)-(-sinh(x)*(b^2-c^2)^(1/2)-(b^2-c^
2)^(1/2))^(1/2)*(-(b^2-c^2)^(1/2)*sinh(x)^3-(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)-(b^2-c^2)^(1/2)))+4*(-sinh(x)*(b^
2-c^2)^(1/2)-(b^2-c^2)^(1/2))^(1/2)*cosh(x)*sinh(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + c \sinh \relax (x) - \sqrt {b^{2} - c^{2}}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cosh(x)+c*sinh(x)-(b^2-c^2)^(1/2))^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + c*sinh(x) - sqrt(b^2 - c^2))^(-5/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (b\,\mathrm {cosh}\relax (x)-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(5/2),x)

[Out]

int(1/(b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \cosh {\relax (x )} + c \sinh {\relax (x )} - \sqrt {b^{2} - c^{2}}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*cosh(x)+c*sinh(x)-(b**2-c**2)**(1/2))**(5/2),x)

[Out]

Integral((b*cosh(x) + c*sinh(x) - sqrt(b**2 - c**2))**(-5/2), x)

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