3.778 \(\int \frac {1}{(-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{3/2}} \, dx\)

Optimal. Leaf size=159 \[ -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt {2} \left (b^2-c^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3116, 3115, 2649, 204} \[ -\frac {b \sinh (x)+c \cosh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b^2-c^2} \sinh \left (x+i \tan ^{-1}(b,-i c)\right )}{\sqrt {2} \sqrt {-\sqrt {b^2-c^2}+\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}}\right )}{2 \sqrt {2} \left (b^2-c^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(-3/2),x]

[Out]

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]*Cos
h[x + I*ArcTan[b, (-I)*c]]])]/(2*Sqrt[2]*(b^2 - c^2)^(3/4)) - (c*Cosh[x] + b*Sinh[x])/(2*Sqrt[b^2 - c^2]*(-Sqr
t[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 )^{3/2}} \, dx &=-\frac {c \cosh (x)+b \sinh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {\int \frac {1}{\sqrt {-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)}} \, dx}{4 \sqrt {b^2-c^2}}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {\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}{4 \sqrt {b^2-c^2}}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{2 \sqrt {b^2-c^2} \left (-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x)\right )^{3/2}}-\frac {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 )}{2 \sqrt {b^2-c^2}}\\ &=\frac {\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 )}{2 \sqrt {2} \left (b^2-c^2\right )^{3/4}}-\frac {c \cosh (x)+b \sinh (x)}{2 \sqrt {b^2-c^2} \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])^(-3/2),x]

[Out]

$Aborted

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fricas [B]  time = 0.59, size = 2137, normalized size = 13.44 \[ \text {result too large to display} \]

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))^(3/2),x, algorithm="fricas")

[Out]

-1/4*((sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^6 + 6*sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)*s
inh(x)^5 + sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*sinh(x)^6 - 3*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^4
+ 3*(5*sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^2 - sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3))*sinh(x)^4 + 4
*(5*sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^3 - 3*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x))*sinh(x)
^3 + 3*sqrt(2)*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^2 + 3*(5*sqrt(2)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^4
- 6*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^2 + sqrt(2)*(b^3 - b^2*c - b*c^2 + c^3))*sinh(x)^2 + 6*(sqrt(2
)*(b^3 + 3*b^2*c + 3*b*c^2 + c^3)*cosh(x)^5 - 2*sqrt(2)*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 + sqrt(2)*(b^3 -
b^2*c - b*c^2 + c^3)*cosh(x))*sinh(x) - sqrt(2)*(b^3 - 3*b^2*c + 3*b*c^2 - c^3))*(b^2 - c^2)^(1/4)*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 - 2*sqrt(1/2)*(sqrt(2)*(b + c)*cos
h(x)^3 + 3*sqrt(2)*(b + c)*cosh(x)*sinh(x)^2 + sqrt(2)*(b + c)*sinh(x)^3 + sqrt(2)*(b - c)*cosh(x) + (3*sqrt(2
)*(b + c)*cosh(x)^2 + sqrt(2)*(b - c))*sinh(x) + 2*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*si
nh(x)^2)*sqrt(b^2 - c^2))*(b^2 - c^2)^(1/4)*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 + 2*b*c - c^2 + 2*((b + c)*c
osh(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)*s
inh(x))*sqrt(b^2 - c^2))/((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))) + 4*sqrt(1/2)*(4*(b^3 + b^
2*c - b*c^2 - c^3)*cosh(x)^4 + 16*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)*sinh(x)^3 + 4*(b^3 + b^2*c - b*c^2 - c^3
)*sinh(x)^4 + 4*(b^3 - b^2*c - b*c^2 + c^3)*cosh(x)^2 + 4*(b^3 - b^2*c - b*c^2 + c^3 + 6*(b^3 + b^2*c - b*c^2
- c^3)*cosh(x)^2)*sinh(x)^2 + 8*(2*(b^3 + b^2*c - b*c^2 - c^3)*cosh(x)^3 + (b^3 - b^2*c - b*c^2 + c^3)*cosh(x)
)*sinh(x) + ((b^2 + 2*b*c + c^2)*cosh(x)^5 + 5*(b^2 + 2*b*c + c^2)*cosh(x)*sinh(x)^4 + (b^2 + 2*b*c + c^2)*sin
h(x)^5 + 6*(b^2 - c^2)*cosh(x)^3 + 2*(5*(b^2 + 2*b*c + c^2)*cosh(x)^2 + 3*b^2 - 3*c^2)*sinh(x)^3 + 2*(5*(b^2 +
2*b*c + c^2)*cosh(x)^3 + 9*(b^2 - c^2)*cosh(x))*sinh(x)^2 + (b^2 - 2*b*c + c^2)*cosh(x) + (5*(b^2 + 2*b*c + c
^2)*cosh(x)^4 + 18*(b^2 - c^2)*cosh(x)^2 + b^2 - 2*b*c + c^2)*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^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*cosh(x)^6 + 6*(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)^5 + (b^5 + 3*b^4*c + 2*b^3*c^2 - 2*b^2*c^3 - 3*b*c^4 - c^5)*sinh(x
)^6 - b^5 + 3*b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + 3*b*c^4 - c^5 - 3*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 +
c^5)*cosh(x)^4 - 3*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5 - 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)*sinh(x)^4 + 4*(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)^3 - 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 + 3*(b^5 - b^4*c - 2*b^3*c^2 +
2*b^2*c^3 + b*c^4 - c^5)*cosh(x)^2 + 3*(b^5 - b^4*c - 2*b^3*c^2 + 2*b^2*c^3 + b*c^4 - c^5 + 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)^4 - 6*(b^5 + b^4*c - 2*b^3*c^2 - 2*b^2*c^3 + b*c^4 + c^5)*cos
h(x)^2)*sinh(x)^2 + 6*((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 - 2*(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))*si
nh(x))

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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))^(3/2),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: 0.6Unable to divide, perhaps due to rounding e
rror%%%{%%%{8,[4,0]%%%}+%%%{16,[3,1]%%%}+%%%{-8,[3,0]%%%}+%%%{-8,[2,1]%%%}+%%%{-16,[1,3]%%%}+%%%{8,[1,2]%%%}+%
%%{-8,[0,4]%%%}+%%%{8,[0,3]%%%},[6,1]%%%}+%%%{%%{[%%%{12,[1,0]%%%}+%%%{12,[0,1]%%%},0,%%%{-36,[3,0]%%%}+%%%{-3
6,[2,1]%%%}+%%%{-12,[2,0]%%%}+%%%{36,[1,2]%%%}+%%%{36,[0,3]%%%}+%%%{12,[0,2]%%%},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]
%%%}+%%%{%%%{24,[4,0]%%%}+%%%{-24,[3,0]%%%}+%%%{-48,[2,2]%%%}+%%%{24,[2,1]%%%}+%%%{24,[1,2]%%%}+%%%{24,[0,4]%%
%}+%%%{-24,[0,3]%%%},[2,1]%%%}+%%%{%%{[%%%{4,[1,0]%%%}+%%%{-4,[0,1]%%%},0,%%%{-12,[3,0]%%%}+%%%{12,[2,1]%%%}+%
%%{-4,[2,0]%%%}+%%%{12,[1,2]%%%}+%%%{8,[1,1]%%%}+%%%{-12,[0,3]%%%}+%%%{-4,[0,2]%%%},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]%%%} / %%%{%%%{8,[9,0]%%%}+%%%{24,[8,1]%%%}+%%%{-24,[8,0]%%%}+%%%{-48,[7,1]%%%}+%%%{24,[7,0]%%%}+%%%{-64,[6
,3]%%%}+%%%{48,[6,2]%%%}+%%%{24,[6,1]%%%}+%%%{-8,[6,0]%%%}+%%%{-48,[5,4]%%%}+%%%{144,[5,3]%%%}+%%%{-72,[5,2]%%
%}+%%%{48,[4,5]%%%}+%%%{-72,[4,3]%%%}+%%%{24,[4,2]%%%}+%%%{64,[3,6]%%%}+%%%{-144,[3,5]%%%}+%%%{72,[3,4]%%%}+%%
%{-48,[2,6]%%%}+%%%{72,[2,5]%%%}+%%%{-24,[2,4]%%%}+%%%{-24,[1,8]%%%}+%%%{48,[1,7]%%%}+%%%{-24,[1,6]%%%}+%%%{-8
,[0,9]%%%}+%%%{24,[0,8]%%%}+%%%{-24,[0,7]%%%}+%%%{8,[0,6]%%%},[6,0]%%%}+%%%{%%{[%%%{12,[6,0]%%%}+%%%{24,[5,1]%
%%}+%%%{-24,[5,0]%%%}+%%%{-12,[4,2]%%%}+%%%{-24,[4,1]%%%}+%%%{12,[4,0]%%%}+%%%{-48,[3,3]%%%}+%%%{48,[3,2]%%%}+
%%%{-12,[2,4]%%%}+%%%{48,[2,3]%%%}+%%%{-24,[2,2]%%%}+%%%{24,[1,5]%%%}+%%%{-24,[1,4]%%%}+%%%{12,[0,6]%%%}+%%%{-
24,[0,5]%%%}+%%%{12,[0,4]%%%},0,%%%{-36,[8,0]%%%}+%%%{-72,[7,1]%%%}+%%%{60,[7,0]%%%}+%%%{72,[6,2]%%%}+%%%{60,[
6,1]%%%}+%%%{-12,[6,0]%%%}+%%%{216,[5,3]%%%}+%%%{-180,[5,2]%%%}+%%%{-12,[5,0]%%%}+%%%{-180,[4,3]%%%}+%%%{36,[4
,2]%%%}+%%%{12,[4,1]%%%}+%%%{-216,[3,5]%%%}+%%%{180,[3,4]%%%}+%%%{24,[3,2]%%%}+%%%{-72,[2,6]%%%}+%%%{180,[2,5]
%%%}+%%%{-36,[2,4]%%%}+%%%{-24,[2,3]%%%}+%%%{72,[1,7]%%%}+%%%{-60,[1,6]%%%}+%%%{-12,[1,4]%%%}+%%%{36,[0,8]%%%}
+%%%{-60,[0,7]%%%}+%%%{12,[0,6]%%%}+%%%{12,[0,5]%%%},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]%%%}+%%%{%%%{24,[9,0]%%%}+%%
%{24,[8,1]%%%}+%%%{-72,[8,0]%%%}+%%%{-96,[7,2]%%%}+%%%{72,[7,0]%%%}+%%%{-96,[6,3]%%%}+%%%{288,[6,2]%%%}+%%%{-7
2,[6,1]%%%}+%%%{-24,[6,0]%%%}+%%%{144,[5,4]%%%}+%%%{-216,[5,2]%%%}+%%%{48,[5,1]%%%}+%%%{144,[4,5]%%%}+%%%{-432
,[4,4]%%%}+%%%{216,[4,3]%%%}+%%%{24,[4,2]%%%}+%%%{-96,[3,6]%%%}+%%%{216,[3,4]%%%}+%%%{-96,[3,3]%%%}+%%%{-96,[2
,7]%%%}+%%%{288,[2,6]%%%}+%%%{-216,[2,5]%%%}+%%%{24,[2,4]%%%}+%%%{24,[1,8]%%%}+%%%{-72,[1,6]%%%}+%%%{48,[1,5]%
%%}+%%%{24,[0,9]%%%}+%%%{-72,[0,8]%%%}+%%%{72,[0,7]%%%}+%%%{-24,[0,6]%%%},[2,0]%%%}+%%%{%%{[%%%{4,[6,0]%%%}+%%
%{-8,[5,0]%%%}+%%%{-12,[4,2]%%%}+%%%{8,[4,1]%%%}+%%%{4,[4,0]%%%}+%%%{16,[3,2]%%%}+%%%{-8,[3,1]%%%}+%%%{12,[2,4
]%%%}+%%%{-16,[2,3]%%%}+%%%{-8,[1,4]%%%}+%%%{8,[1,3]%%%}+%%%{-4,[0,6]%%%}+%%%{8,[0,5]%%%}+%%%{-4,[0,4]%%%},0,%
%%{-12,[8,0]%%%}+%%%{20,[7,0]%%%}+%%%{48,[6,2]%%%}+%%%{-20,[6,1]%%%}+%%%{-4,[6,0]%%%}+%%%{-60,[5,2]%%%}+%%%{8,
[5,1]%%%}+%%%{-4,[5,0]%%%}+%%%{-72,[4,4]%%%}+%%%{60,[4,3]%%%}+%%%{4,[4,2]%%%}+%%%{12,[4,1]%%%}+%%%{60,[3,4]%%%
}+%%%{-16,[3,3]%%%}+%%%{-8,[3,2]%%%}+%%%{48,[2,6]%%%}+%%%{-60,[2,5]%%%}+%%%{4,[2,4]%%%}+%%%{-8,[2,3]%%%}+%%%{-
20,[1,6]%%%}+%%%{8,[1,5]%%%}+%%%{12,[1,4]%%%}+%%%{-12,[0,8]%%%}+%%%{20,[0,7]%%%}+%%%{-4,[0,6]%%%}+%%%{-4,[0,5]
%%%},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 = 1.48, size = 415, normalized size = 2.61 \[ \frac {\sqrt {2}\, \arctanh \left (\frac {\cosh \relax (x ) \sqrt {2}}{2}\right )}{2 \sqrt {b^{2}-c^{2}}\, \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}}-\frac {\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}\, \sqrt {2}\, \sqrt {b^{2}-c^{2}}\, \left (\ln \left (-\frac {2 \left (\cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}\, \sinh \relax (x )-\sinh \relax (x ) \sqrt {b^{2}-c^{2}}+\cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}-\sqrt {b^{2}-c^{2}}-\sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}\right )}{\cosh \relax (x )-\sqrt {2}}\right )-\ln \left (\frac {2 \cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}\, \sinh \relax (x )+2 \sinh \relax (x ) \sqrt {b^{2}-c^{2}}+2 \cosh \relax (x ) \sqrt {b^{2}-c^{2}}\, \sqrt {2}+2 \sqrt {b^{2}-c^{2}}+2 \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right ) \left (\sinh ^{2}\relax (x )\right )}}{\cosh \relax (x )+\sqrt {2}}\right )\right )}{4 \left (b -c \right ) \left (b +c \right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh \relax (x )+1\right )}\, \sinh \relax (x ) \sqrt {-\frac {\sinh \relax (x ) b^{2}-\sinh \relax (x ) c^{2}+b^{2}-c^{2}}{\sqrt {b^{2}-c^{2}}}}} \]

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))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \cosh \relax (x) + c \sinh \relax (x) - \sqrt {b^{2} - c^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]

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))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cosh(x) + c*sinh(x) - sqrt(b^2 - c^2))^(-3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,\mathrm {cosh}\relax (x)-\sqrt {b^2-c^2}+c\,\mathrm {sinh}\relax (x)\right )}^{3/2}} \,d x \]

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))^(3/2),x)

[Out]

int(1/(b*cosh(x) - (b^2 - c^2)^(1/2) + c*sinh(x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \cosh {\relax (x )} + c \sinh {\relax (x )} - \sqrt {b^{2} - c^{2}}\right )^{\frac {3}{2}}}\, dx \]

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))**(3/2),x)

[Out]

Integral((b*cosh(x) + c*sinh(x) - sqrt(b**2 - c**2))**(-3/2), x)

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