3.8.79 \(\int \frac {1}{(-\sqrt {b^2-c^2}+b \cosh (x)+c \sinh (x))^{5/2}} \, dx\) [779]

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

[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 = 0.12, 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 = {3195, 3194, 2728, 210} \begin {gather*} -\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}} \end {gather*}

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 210

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

Rule 2728

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 3194

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 3195

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) \text {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 \begin {gather*} \text {\$Aborted} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(983\) vs. \(2(180)=360\).
time = 3.51, size = 984, normalized size = 4.66

method	result	size
default	\(-\frac {2 \sqrt {2}\, \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \arctanh \left (\frac {\cosh \left (x \right ) \sqrt {2}}{2}\right ) \left (\sinh ^{2}\left (x \right )\right )+2 \sqrt {2}\, \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \arctanh \left (\frac {\cosh \left (x \right ) \sqrt {2}}{2}\right ) \sinh \left (x \right )+\sqrt {2}\, \ln \left (\frac {2 \cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+2 \cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}+2 \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}+2 \sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+2 \sqrt {b^{2}-c^{2}}}{\cosh \left (x \right )+\sqrt {2}}\right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}\, \sinh \left (x \right )-\sqrt {2}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}\, \ln \left (-\frac {2 \left (\cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+\cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}-\sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}\right )}{\cosh \left (x \right )-\sqrt {2}}\right ) \sinh \left (x \right )+4 \cosh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sinh \left (x \right )+\sqrt {2}\, \ln \left (\frac {2 \cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+2 \cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}+2 \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}+2 \sinh \left (x \right ) \sqrt {b^{2}-c^{2}}+2 \sqrt {b^{2}-c^{2}}}{\cosh \left (x \right )+\sqrt {2}}\right ) \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}-\sqrt {2}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}\, \ln \left (-\frac {2 \left (\cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}\, \sinh \left (x \right )+\cosh \left (x \right ) \sqrt {2}\, \sqrt {b^{2}-c^{2}}-\sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{3}\left (x \right )\right )-\sqrt {b^{2}-c^{2}}\, \left (\sinh ^{2}\left (x \right )\right )}-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}\right )}{\cosh \left (x \right )-\sqrt {2}}\right )}{8 \left (\sinh \left (x \right )+1\right ) \sinh \left (x \right ) \sqrt {-\sinh \left (x \right ) \sqrt {b^{2}-c^{2}}-\sqrt {b^{2}-c^{2}}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )}{\sqrt {b^{2}-c^{2}}}+\frac {-b^{2}+c^{2}}{\sqrt {b^{2}-c^{2}}}}\, \left (b^{2}-c^{2}\right )}\)	\(984\)

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,method=_RETURNVERBOSE)

[Out]

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

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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(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5675 vs. \(2 (176) = 352\).
time = 0.50, size = 5675, normalized size = 26.90 \begin {gather*} \text {Too large to display} \end {gather*}

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)*cos...

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs((-

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

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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