3.8.44 \(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) [744]

Optimal. Leaf size=146 \[ -\frac {\left (2 a^2+b^2-c^2\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]

[Out]

-(2*a^2+b^2-c^2)*arctanh((c-(a-b)*tanh(1/2*x))/(a^2-b^2+c^2)^(1/2))/(a^2-b^2+c^2)^(5/2)+1/2*(-c*cosh(x)-b*sinh
(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^2-3/2*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sinh
(x))

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Rubi [A]
time = 0.13, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3208, 3232, 3203, 632, 212} \begin {gather*} -\frac {\left (2 a^2+b^2-c^2\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {3 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*a^2 + b^2 - c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(5/2)) - (c*C
osh[x] + b*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) - (3*(a*c*Cosh[x] + a*b*Sinh[x]))/(2*(
a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))

Rule 212

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 3203

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[2*(f/e), Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 3208

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 + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]

Rule 3232

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)
*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 -
b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[
a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]

Rubi steps

\begin {align*} \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx &=-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {\int \frac {-2 a+b \cosh (x)+c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (a^2-b^2+c^2\right )}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2+b^2-c^2\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)} \, dx}{2 \left (a^2-b^2+c^2\right )^2}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2+b^2-c^2\right ) \text {Subst}\left (\int \frac {1}{a+b+2 c x-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^2}\\ &=-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {\left (2 \left (2 a^2+b^2-c^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2-b^2+c^2\right )-x^2} \, dx,x,2 c+2 (-a+b) \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2+c^2\right )^2}\\ &=-\frac {\left (2 a^2+b^2-c^2\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 183, normalized size = 1.25 \begin {gather*} \frac {1}{2} \left (\frac {2 \left (2 a^2+b^2-c^2\right ) \text {ArcTan}\left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {-a c+\left (b^2-c^2\right ) \sinh (x)}{b \left (-a^2+b^2-c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {c \left (2 a^2+b^2-c^2\right )-3 a \left (b^2-c^2\right ) \sinh (x)}{b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-3),x]

[Out]

((2*(2*a^2 + b^2 - c^2)*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/(-a^2 + b^2 - c^2)^(5/2) + (-
(a*c) + (b^2 - c^2)*Sinh[x])/(b*(-a^2 + b^2 - c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) + (c*(2*a^2 + b^2 - c^2) - 3
*a*(b^2 - c^2)*Sinh[x])/(b*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])))/2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(576\) vs. \(2(138)=276\).
time = 1.62, size = 577, normalized size = 3.95

method	result	size
default	\(-\frac {2 \left (-\frac {\left (4 a^{3} b -7 a^{2} b^{2}+5 a^{2} c^{2}+2 a \,b^{3}-2 a \,c^{2} b +b^{4}-3 b^{2} c^{2}+2 c^{4}\right ) \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right )}-\frac {c \left (4 a^{4}-12 a^{3} b +13 a^{2} b^{2}-7 a^{2} c^{2}-6 a \,b^{3}+6 a \,c^{2} b +b^{4}+b^{2} c^{2}-2 c^{4}\right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{4} b -5 a^{3} b^{2}+11 a^{3} c^{2}-3 a^{2} b^{3}-3 a^{2} b \,c^{2}+5 a \,b^{4}-7 a \,b^{2} c^{2}+2 a \,c^{4}-b^{5}-b^{3} c^{2}+2 b \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {c \left (4 a^{4}-3 a^{2} b^{2}+a^{2} c^{2}-b^{4}+b^{2} c^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{2}}-\frac {\left (2 a^{2}+b^{2}-c^{2}\right ) \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\)	\(577\)
risch	\(\frac {2 a^{2} b \,{\mathrm e}^{3 x}+2 a^{2} c \,{\mathrm e}^{3 x}+b^{3} {\mathrm e}^{3 x}+b^{2} c \,{\mathrm e}^{3 x}-b \,c^{2} {\mathrm e}^{3 x}-{\mathrm e}^{3 x} c^{3}+6 a^{3} {\mathrm e}^{2 x}+3 \,{\mathrm e}^{2 x} a \,b^{2}-3 a \,c^{2} {\mathrm e}^{2 x}+10 a^{2} b \,{\mathrm e}^{x}-10 \,{\mathrm e}^{x} a^{2} c -b^{3} {\mathrm e}^{x}+{\mathrm e}^{x} b^{2} c +{\mathrm e}^{x} b \,c^{2}-{\mathrm e}^{x} c^{3}+3 a \,b^{2}-6 a b c +3 a \,c^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +2 a \,{\mathrm e}^{x}+b -c \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) b^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) c^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) b^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) c^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}\)	\(973\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*cosh(x)+c*sinh(x))^3,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*(4*a^3*b-7*a^2*b^2+5*a^2*c^2+2*a*b^3-2*a*b*c^2+b^4-3*b^2*c^2+2*c^4)/(a-b)/(a^4-2*a^2*b^2+2*a^2*c^2+b^
4-2*b^2*c^2+c^4)*tanh(1/2*x)^3-1/2*c*(4*a^4-12*a^3*b+13*a^2*b^2-7*a^2*c^2-6*a*b^3+6*a*b*c^2+b^4+b^2*c^2-2*c^4)
/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tanh(1/2*x)^2+1/2*(4*a^4*b-5*a^3*b^2+11*a^3*c^2-3
*a^2*b^3-3*a^2*b*c^2+5*a*b^4-7*a*b^2*c^2+2*a*c^4-b^5-b^3*c^2+2*b*c^4)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c
^4)/(a^2-2*a*b+b^2)*tanh(1/2*x)+1/2*c*(4*a^4-3*a^2*b^2+a^2*c^2-b^4+b^2*c^2)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2
*c^2+c^4)/(a^2-2*a*b+b^2))/(a*tanh(1/2*x)^2-b*tanh(1/2*x)^2-2*c*tanh(1/2*x)-a-b)^2-(2*a^2+b^2-c^2)/(a^4-2*a^2*
b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-b^2+a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3633 vs. \(2 (138) = 276\).
time = 0.45, size = 7379, normalized size = 50.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="fricas")

[Out]

[1/2*(6*a^3*b^2 - 6*a*b^4 + 6*a^3*c^2 - 12*a*b*c^3 + 6*a*c^4 + 2*(2*a^4*b - a^2*b^3 - b^5 - b*c^4 - c^5 + (a^2
+ 2*b^2)*c^3 + (a^2*b + 2*b^3)*c^2 + (2*a^4 - a^2*b^2 - b^4)*c)*cosh(x)^3 + 2*(2*a^4*b - a^2*b^3 - b^5 - b*c^
4 - c^5 + (a^2 + 2*b^2)*c^3 + (a^2*b + 2*b^3)*c^2 + (2*a^4 - a^2*b^2 - b^4)*c)*sinh(x)^3 + 6*(2*a^5 - a^3*b^2
- a*b^4 - a*c^4 + (a^3 + 2*a*b^2)*c^2)*cosh(x)^2 + 6*(2*a^5 - a^3*b^2 - a*b^4 - a*c^4 + (a^3 + 2*a*b^2)*c^2 +
(2*a^4*b - a^2*b^3 - b^5 - b*c^4 - c^5 + (a^2 + 2*b^2)*c^3 + (a^2*b + 2*b^3)*c^2 + (2*a^4 - a^2*b^2 - b^4)*c)*
cosh(x))*sinh(x)^2 - ((2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b*c^3 - c^4 + 2*(2*a^2*b + b^3)*c)*cosh(x)^4 + (2*a^2*b
^2 + b^4 + 2*a^2*c^2 - 2*b*c^3 - c^4 + 2*(2*a^2*b + b^3)*c)*sinh(x)^4 + 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b*c^3
- c^4 + 4*(2*a^3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x)^3 + 4*(2*a^3*b + a*b^3 - a*b*c^2 - a
*c^3 + (2*a^3 + a*b^2)*c + (2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b*c^3 - c^4 + 2*(2*a^2*b + b^3)*c)*cosh(x))*sinh(x
)^3 + 2*(4*a^4 + 4*a^2*b^2 + b^4 + c^4 - 2*(2*a^2 + b^2)*c^2)*cosh(x)^2 + 2*(4*a^4 + 4*a^2*b^2 + b^4 + c^4 - 2
*(2*a^2 + b^2)*c^2 + 3*(2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b*c^3 - c^4 + 2*(2*a^2*b + b^3)*c)*cosh(x)^2 + 6*(2*a^
3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a*b^2)*c)*cosh(x))*sinh(x)^2 - 2*(2*a^2*b + b^3)*c + 4*(2*a^3*b + a*b
^3 - a*b*c^2 + a*c^3 - (2*a^3 + a*b^2)*c)*cosh(x) + 4*(2*a^3*b + a*b^3 - a*b*c^2 + a*c^3 + (2*a^2*b^2 + b^4 +
2*a^2*c^2 - 2*b*c^3 - c^4 + 2*(2*a^2*b + b^3)*c)*cosh(x)^3 + 3*(2*a^3*b + a*b^3 - a*b*c^2 - a*c^3 + (2*a^3 + a
*b^2)*c)*cosh(x)^2 - (2*a^3 + a*b^2)*c + (4*a^4 + 4*a^2*b^2 + b^4 + c^4 - 2*(2*a^2 + b^2)*c^2)*cosh(x))*sinh(x
))*sqrt(a^2 - b^2 + c^2)*log(((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 + 2*b*c + c^2)*sinh(x)^2 + 2*a^2 - b^2 + c^
2 + 2*(a*b + a*c)*cosh(x) + 2*(a*b + a*c + (b^2 + 2*b*c + c^2)*cosh(x))*sinh(x) + 2*sqrt(a^2 - b^2 + c^2)*((b
+ c)*cosh(x) + (b + c)*sinh(x) + a))/((b + c)*cosh(x)^2 + (b + c)*sinh(x)^2 + 2*a*cosh(x) + 2*((b + c)*cosh(x)
+ a)*sinh(x) + b - c)) - 12*(a^3*b - a*b^3)*c + 2*(10*a^4*b - 11*a^2*b^3 + b^5 + b*c^4 - c^5 - (11*a^2 - 2*b^
2)*c^3 + (11*a^2*b - 2*b^3)*c^2 - (10*a^4 - 11*a^2*b^2 + b^4)*c)*cosh(x) + 2*(10*a^4*b - 11*a^2*b^3 + b^5 + b*
c^4 - c^5 - (11*a^2 - 2*b^2)*c^3 + (11*a^2*b - 2*b^3)*c^2 + 3*(2*a^4*b - a^2*b^3 - b^5 - b*c^4 - c^5 + (a^2 +
2*b^2)*c^3 + (a^2*b + 2*b^3)*c^2 + (2*a^4 - a^2*b^2 - b^4)*c)*cosh(x)^2 - (10*a^4 - 11*a^2*b^2 + b^4)*c + 6*(2
*a^5 - a^3*b^2 - a*b^4 - a*c^4 + (a^3 + 2*a*b^2)*c^2)*cosh(x))*sinh(x))/(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8
- 2*b*c^7 + c^8 + (3*a^2 - 2*b^2)*c^6 - 6*(a^2*b - b^3)*c^5 + 3*(a^4 - a^2*b^2)*c^4 + (a^6*b^2 - 3*a^4*b^4 +
3*a^2*b^6 - b^8 + 2*b*c^7 + c^8 + (3*a^2 - 2*b^2)*c^6 + 6*(a^2*b - b^3)*c^5 + 3*(a^4 - a^2*b^2)*c^4 + 6*(a^4*b
- 2*a^2*b^3 + b^5)*c^3 + (a^6 - 3*a^2*b^4 + 2*b^6)*c^2 + 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c)*cosh(x)^4
+ (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + 2*b*c^7 + c^8 + (3*a^2 - 2*b^2)*c^6 + 6*(a^2*b - b^3)*c^5 + 3*(a^4
- a^2*b^2)*c^4 + 6*(a^4*b - 2*a^2*b^3 + b^5)*c^3 + (a^6 - 3*a^2*b^4 + 2*b^6)*c^2 + 2*(a^6*b - 3*a^4*b^3 + 3*a
^2*b^5 - b^7)*c)*sinh(x)^4 - 6*(a^4*b - 2*a^2*b^3 + b^5)*c^3 + 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7 + a*b*
c^6 + a*c^7 + 3*(a^3 - a*b^2)*c^5 + 3*(a^3*b - a*b^3)*c^4 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^3 + 3*(a^5*b - 2*a^3
*b^3 + a*b^5)*c^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c)*cosh(x)^3 + 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*
b^7 + a*b*c^6 + a*c^7 + 3*(a^3 - a*b^2)*c^5 + 3*(a^3*b - a*b^3)*c^4 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^3 + 3*(a^5
*b - 2*a^3*b^3 + a*b^5)*c^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c + (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8
+ 2*b*c^7 + c^8 + (3*a^2 - 2*b^2)*c^6 + 6*(a^2*b - b^3)*c^5 + 3*(a^4 - a^2*b^2)*c^4 + 6*(a^4*b - 2*a^2*b^3 +
b^5)*c^3 + (a^6 - 3*a^2*b^4 + 2*b^6)*c^2 + 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c)*cosh(x))*sinh(x)^3 + (a^
6 - 3*a^2*b^4 + 2*b^6)*c^2 + 2*(2*a^8 - 5*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 - b^8 - c^8 - (a^2 - 4*b^2)*c^6 + 3*(a
^4 + a^2*b^2 - 2*b^4)*c^4 + (5*a^6 - 6*a^4*b^2 - 3*a^2*b^4 + 4*b^6)*c^2)*cosh(x)^2 + 2*(2*a^8 - 5*a^6*b^2 + 3*
a^4*b^4 + a^2*b^6 - b^8 - c^8 - (a^2 - 4*b^2)*c^6 + 3*(a^4 + a^2*b^2 - 2*b^4)*c^4 + (5*a^6 - 6*a^4*b^2 - 3*a^2
*b^4 + 4*b^6)*c^2 + 3*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8 + 2*b*c^7 + c^8 + (3*a^2 - 2*b^2)*c^6 + 6*(a^2*b
- b^3)*c^5 + 3*(a^4 - a^2*b^2)*c^4 + 6*(a^4*b - 2*a^2*b^3 + b^5)*c^3 + (a^6 - 3*a^2*b^4 + 2*b^6)*c^2 + 2*(a^6*
b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c)*cosh(x)^2 + 6*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7 + a*b*c^6 + a*c^7 + 3
*(a^3 - a*b^2)*c^5 + 3*(a^3*b - a*b^3)*c^4 + 3*(a^5 - 2*a^3*b^2 + a*b^4)*c^3 + 3*(a^5*b - 2*a^3*b^3 + a*b^5)*c
^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c)*cosh(x))*sinh(x)^2 - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*c +
4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7 + a*b*c^6 - a*c^7 - 3*(a^3 - a*b^2)*c^5 + 3*(a^3*b - a*b^3)*c^4 - 3*
(a^5 - 2*a^3*b^2 + a*b^4)*c^3 + 3*(a^5*b - 2*a^3*b^3 + a*b^5)*c^2 - (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*c)*c
osh(x) + 4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b...

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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/(a+b*cosh(x)+c*sinh(x))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (138) = 276\).
time = 0.42, size = 304, normalized size = 2.08 \begin {gather*} \frac {{\left (2 \, a^{2} + b^{2} - c^{2}\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} + \frac {2 \, a^{2} b e^{\left (3 \, x\right )} + b^{3} e^{\left (3 \, x\right )} + 2 \, a^{2} c e^{\left (3 \, x\right )} + b^{2} c e^{\left (3 \, x\right )} - b c^{2} e^{\left (3 \, x\right )} - c^{3} e^{\left (3 \, x\right )} + 6 \, a^{3} e^{\left (2 \, x\right )} + 3 \, a b^{2} e^{\left (2 \, x\right )} - 3 \, a c^{2} e^{\left (2 \, x\right )} + 10 \, a^{2} b e^{x} - b^{3} e^{x} - 10 \, a^{2} c e^{x} + b^{2} c e^{x} + b c^{2} e^{x} - c^{3} e^{x} + 3 \, a b^{2} - 6 \, a b c + 3 \, a c^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="giac")

[Out]

(2*a^2 + b^2 - c^2)*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c^2))/((a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2
*b^2*c^2 + c^4)*sqrt(-a^2 + b^2 - c^2)) + (2*a^2*b*e^(3*x) + b^3*e^(3*x) + 2*a^2*c*e^(3*x) + b^2*c*e^(3*x) - b
*c^2*e^(3*x) - c^3*e^(3*x) + 6*a^3*e^(2*x) + 3*a*b^2*e^(2*x) - 3*a*c^2*e^(2*x) + 10*a^2*b*e^x - b^3*e^x - 10*a
^2*c*e^x + b^2*c*e^x + b*c^2*e^x - c^3*e^x + 3*a*b^2 - 6*a*b*c + 3*a*c^2)/((a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2
- 2*b^2*c^2 + c^4)*(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + b - c)^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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