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

Optimal. Leaf size=205 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt {2} \left (b^2-c^2\right )^{5/4}}+\frac {3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {b \sinh (x)+c \cosh (x)}{4 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3116, 3115, 2649, 206} \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{16 \sqrt {2} \left (b^2-c^2\right )^{5/4}}+\frac {3 (b \sinh (x)+c \cosh (x))}{16 \left (b^2-c^2\right ) \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {b \sinh (x)+c \cosh (x)}{4 \sqrt {b^2-c^2} \left (\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*ArcTan[((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]*Co
sh[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]*(S
qrt[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 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])

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 \tan ^{-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 = 180.01, 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 = 5297, normalized size = 25.84 \[ \text {result too large to display} \]

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]

-1/8*(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))*sinh
(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))*arctan((sqrt(b^2 - c^2)*(cosh(x) + sinh(x)) - b + c)*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) + si
nh(x)))/(((b + c)*cosh(x)^2 + 2*(b + c)*cosh(x)*sinh(x) + (b + c)*sinh(x)^2 - b + c)*(b^2 - c^2)^(1/4)))/(b^2
- c^2)^(1/4) - 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)*sinh(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)*sin
h(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)*co
sh(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*(6
3*(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)*si
nh(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)*cosh(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)*cosh(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))*s
inh(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)*cos
h(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: 1.36Unable 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]%%%}+%%%{9
60,[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
,[10,1]%%%}+%%%{-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]%%%}+%%%
{3200,[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]%%%}+%%%{3
20,[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]%%
%}+%%%{-880,[13,0]%%%}+%%%{240,[12,2]%%%}+%%%{-2640,[12,1]%%%}+%%%{1120,[12,0]%%%}+%%%{-3840,[11,3]%%%}+%%%{17
60,[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]%%%}+%%%{2
400,[9,2]%%%}+%%%{80,[9,0]%%%}+%%%{10800,[8,6]%%%}+%%%{-22000,[8,5]%%%}+%%%{5600,[8,4]%%%}+%%%{2400,[8,3]%%%}+
%%%{400,[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]%%%}+%%%{-4
800,[5,9]%%%}+%%%{22000,[5,8]%%%}+%%%{-22400,[5,7]%%%}+%%%{4800,[5,6]%%%}+%%%{480,[5,4]%%%}+%%%{4560,[4,10]%%%
}+%%%{-4400,[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,[1,1
3]%%%}+%%%{2640,[1,12]%%%}+%%%{-2240,[1,11]%%%}+%%%{480,[1,10]%%%}+%%%{80,[1,8]%%%}+%%%{-240,[0,14]%%%}+%%%{88
0,[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,[11,4]%%%}+%%%{19200,[11,3]%%%}+%%%{-19200,[11,2]%%%}+%%%{1600,[11,0]%%%}+%%%{12480,[10,5]%%%}+%%%{-14
400,[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,[5,1
0]%%%}+%%%{-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]%%
%}+%%%{8000,[2,12]%%%}+%%%{-19200,[2,11]%%%}+%%%{19200,[2,10]%%%}+%%%{-8000,[2,9]%%%}+%%%{960,[2,8]%%%}+%%%{96
0,[1,14]%%%}+%%%{-3200,[1,13]%%%}+%%%{3200,[1,12]%%%}+%%%{-1600,[1,10]%%%}+%%%{640,[1,9]%%%}+%%%{320,[0,15]%%%
}+%%%{-1600,[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,[10,0]%%%}+%%%{1600,[9,3]%%%}+%%%{-3200,[9,2]%%%}+%%%{640,[9,0]%%%}+%%%{-800,[8,4]%%%}+%%%{-3200,[8,3]
%%%}+%%%{4800,[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]%%%}+%%%{3
200,[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,[0,10]%%%}+%%%{-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]%
%%}+%%%{-26400,[5,8]%%%}+%%%{9600,[5,6]%%%}+%%%{1920,[5,5]%%%}+%%%{-960,[5,4]%%%}+%%%{4320,[4,10]%%%}+%%%{-264
00,[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]%%%}+%%%{105
60,[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]%%%}+%%%{%%%{1
60,[15,0]%%%}+%%%{160,[14,1]%%%}+%%%{-800,[14,0]%%%}+%%%{-1120,[13,2]%%%}+%%%{1600,[13,0]%%%}+%%%{-1120,[12,3]
%%%}+%%%{5600,[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,[1
0,2]%%%}+%%%{-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,[0,15]%%%}+%%%{800,[0,14]%%%}+%%%{-1600,[0,13]%%%}+%%%{1600,[0,12]%%%}+%%%{-800,[0,11]%%%}+%%%{160,[0,1
0]%%%},[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]%%%}+%%%{160,[4,4]%%%}+%%%{320,[3,8]%%%}+%%%{-768,[3,7]%%%}+%%%{512,[3,6]%%%}+%%%{-64,[3,5]%%%}+%%%{96,[2
,10]%%%}+%%%{-320,[2,9]%%%}+%%%{288,[2,8]%%%}+%%%{-64,[2,6]%%%}+%%%{-64,[1,10]%%%}+%%%{192,[1,9]%%%}+%%%{-192,
[1,8]%%%}+%%%{64,[1,7]%%%}+%%%{-16,[0,12]%%%}+%%%{64,[0,11]%%%}+%%%{-96,[0,10]%%%}+%%%{64,[0,9]%%%}+%%%{-16,[0
,8]%%%},0,%%%{48,[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,[10,1]%%%}+%%%{-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]%%%}+%%%{-80,[8,1]%%%}+%%%{3520,[7,6]%%%}+%%%{-4480,[7,5]%%%}+%%%{576,[7,4]%%%}+%%%{-128,[7,3]%%%}+%%%{12
8,[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 = 1.90, size = 954, normalized size = 4.65 \[ \text {result too large to display} \]

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)^(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)*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)*sin
h(x)^3+(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)+(b^2-c^2)^(1/2)))*(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^
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)))*(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^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)*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)))*(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)+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)))*(-(b^2-c^2)^(1/2)*sinh(x)^3+(b^2-c^2)^(1/2)*sinh(x)^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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

________________________________________________________________________________________