3.8.100 \(\int \frac {A+B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) [800]

Optimal. Leaf size=233 \[ -\frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]

[Out]

-(2*A*a^2+A*b^2-A*c^2-3*B*a*b+3*C*a*c)*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*(-B*c+b*C-(A*c-C*a)*cosh(x)-(A*b-B*a)*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^2+1/2*(-a*(B*c-C*b)-(
3*a*A*c-a^2*C-2*c*(B*b-C*c))*cosh(x)-(3*a*A*b-a^2*B-2*b*(B*b-C*c))*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sin
h(x))

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Rubi [A]
time = 0.35, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3235, 3232, 3203, 632, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right ) \left (2 a^2 A-3 a b B+3 a c C+A b^2-A c^2\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b (b B-c C)\right )+\cosh (x) \left (a^2 (-C)+3 a A c-2 c (b B-c C)\right )+a (B c-b C)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}-\frac {\sinh (x) (A b-a B)+\cosh (x) (A c-a C)-b C+B c}{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])/(a + b*Cosh[x] + c*Sinh[x])^3,x]

[Out]

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

Rule 3235

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

Rubi steps

\begin {align*} \int \frac {A+B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx &=-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-a 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 A-b B+c C)+(A b-a B) \cosh (x)+(A c-a C) \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx}{2 \left (a^2-b^2+c^2\right )}\\ &=-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)} \, dx}{2 \left (a^2-b^2+c^2\right )^2}\\ &=-\frac {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\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 {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \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 A+A b^2-3 a b B-A c^2+3 a c C\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 A+A b^2-3 a b B-A c^2+3 a c C\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 {B c-b C+(A c-a C) \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)+\left (3 a A c-a^2 C-2 c (b B-c C)\right ) \cosh (x)+\left (3 a A b-a^2 B-2 b (b B-c C)\right ) \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 = 0.69, size = 465, normalized size = 2.00 \begin {gather*} \frac {\left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\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 {6 a^3 A c+3 a A b^2 c-9 a^2 b B c-3 a A c^3-2 a^4 C+4 a^2 b^2 C-2 b^4 C+5 a^2 c^2 C+4 b^2 c^2 C-2 c^4 C+2 b c \left (2 a^2 A+A b^2-3 a b B-A c^2+3 a c C\right ) \cosh (x)+c \left (3 a A \left (-b^2+c^2\right )+a^2 (b B-c C)+2 \left (b^2-c^2\right ) (b B-c C)\right ) \cosh (2 x)-8 a^2 A b^2 \sinh (x)+2 A b^4 \sinh (x)+4 a^3 b B \sinh (x)+2 a b^3 B \sinh (x)+12 a^2 A c^2 \sinh (x)-2 A b^2 c^2 \sinh (x)-8 a b B c^2 \sinh (x)-4 a^3 c C \sinh (x)-2 a b^2 c C \sinh (x)+8 a c^3 C \sinh (x)-3 a A b^3 \sinh (2 x)+a^2 b^2 B \sinh (2 x)+2 b^4 B \sinh (2 x)+3 a A b c^2 \sinh (2 x)-2 b^2 B c^2 \sinh (2 x)-a^2 b c C \sinh (2 x)-2 b^3 c C \sinh (2 x)+2 b c^3 C \sinh (2 x)}{4 b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[x] + C*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^3,x]

[Out]

((2*a^2*A + A*b^2 - 3*a*b*B - A*c^2 + 3*a*c*C)*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/(-a^2
+ b^2 - c^2)^(5/2) + (6*a^3*A*c + 3*a*A*b^2*c - 9*a^2*b*B*c - 3*a*A*c^3 - 2*a^4*C + 4*a^2*b^2*C - 2*b^4*C + 5*
a^2*c^2*C + 4*b^2*c^2*C - 2*c^4*C + 2*b*c*(2*a^2*A + A*b^2 - 3*a*b*B - A*c^2 + 3*a*c*C)*Cosh[x] + c*(3*a*A*(-b
^2 + c^2) + a^2*(b*B - c*C) + 2*(b^2 - c^2)*(b*B - c*C))*Cosh[2*x] - 8*a^2*A*b^2*Sinh[x] + 2*A*b^4*Sinh[x] + 4
*a^3*b*B*Sinh[x] + 2*a*b^3*B*Sinh[x] + 12*a^2*A*c^2*Sinh[x] - 2*A*b^2*c^2*Sinh[x] - 8*a*b*B*c^2*Sinh[x] - 4*a^
3*c*C*Sinh[x] - 2*a*b^2*c*C*Sinh[x] + 8*a*c^3*C*Sinh[x] - 3*a*A*b^3*Sinh[2*x] + a^2*b^2*B*Sinh[2*x] + 2*b^4*B*
Sinh[2*x] + 3*a*A*b*c^2*Sinh[2*x] - 2*b^2*B*c^2*Sinh[2*x] - a^2*b*c*C*Sinh[2*x] - 2*b^3*c*C*Sinh[2*x] + 2*b*c^
3*C*Sinh[2*x])/(4*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. \(1083\) vs. \(2(228)=456\).
time = 1.80, size = 1084, normalized size = 4.65

method	result	size
default	\(-\frac {2 \left (-\frac {\left (4 A \,a^{3} b -7 A \,a^{2} b^{2}+5 A \,a^{2} c^{2}+2 A a \,b^{3}-2 A a b \,c^{2}+A \,b^{4}-3 A \,b^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}+3 B \,a^{3} b -2 B \,a^{2} b^{2}-4 B \,a^{2} c^{2}+3 B a \,b^{3}-2 B \,b^{4}+4 B \,b^{2} c^{2}-2 B \,c^{4}-3 C \,a^{3} c +6 C \,a^{2} b c -3 C a \,b^{2} c \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 {\left (4 A \,a^{4} c -12 A \,a^{3} b c +13 A \,a^{2} b^{2} c -7 A \,a^{2} c^{3}-6 A a \,b^{3} c +6 A a b \,c^{3}+A \,b^{4} c +A \,b^{2} c^{3}-2 A \,c^{5}+2 B \,a^{4} c -9 B \,a^{3} b c +14 B \,a^{2} b^{2} c +4 B \,a^{2} c^{3}-9 B a \,b^{3} c +2 B \,b^{4} c -4 B \,b^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+2 C \,a^{4} b +4 C \,a^{3} b^{2}+5 C \,a^{3} c^{2}-4 C \,a^{2} b^{3}-14 C \,a^{2} b \,c^{2}-2 C a \,b^{4}+13 C a \,b^{2} c^{2}-2 C a \,c^{4}+2 C \,b^{5}-4 C \,b^{3} c^{2}+2 C b \,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 \,a^{4} b -5 A \,a^{3} b^{2}+11 A \,a^{3} c^{2}-3 A \,a^{2} b^{3}-3 A \,a^{2} b \,c^{2}+5 A a \,b^{4}-7 A a \,b^{2} c^{2}+2 A a \,c^{4}-A \,b^{5}-A \,b^{3} c^{2}+2 A b \,c^{4}-2 B \,a^{5}+3 B \,a^{4} b -B \,a^{3} b^{2}-4 B \,a^{3} c^{2}-B \,a^{2} b^{3}-8 B \,a^{2} b \,c^{2}+3 B a \,b^{4}+8 B a \,b^{2} c^{2}-2 B a \,c^{4}-2 B \,b^{5}+4 B \,b^{3} c^{2}-2 B b \,c^{4}-5 C \,a^{4} c +5 C \,a^{3} b c +5 C \,a^{2} b^{2} c +4 C \,a^{2} c^{3}-5 C a \,b^{3} c -4 C a b \,c^{3}\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 {4 A \,a^{4} c -3 A \,a^{2} b^{2} c +A \,a^{2} c^{3}-A \,b^{4} c +A \,b^{2} c^{3}-5 B \,a^{3} b c +5 B a \,b^{3} c -2 B a b \,c^{3}-2 C \,a^{5}+4 C \,a^{3} b^{2}+C \,a^{3} c^{2}-2 C a \,b^{4}-C a \,b^{2} c^{2}}{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} A +A \,b^{2}-A \,c^{2}-3 B a b +3 C a c \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}}}\)	\(1084\)
risch	\(\text {Expression too large to display}\)	\(1966\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-1/2*(4*A*a^3*b-7*A*a^2*b^2+5*A*a^2*c^2+2*A*a*b^3-2*A*a*b*c^2+A*b^4-3*A*b^2*c^2+2*A*c^4-2*B*a^4+3*B*a^3*b-
2*B*a^2*b^2-4*B*a^2*c^2+3*B*a*b^3-2*B*b^4+4*B*b^2*c^2-2*B*c^4-3*C*a^3*c+6*C*a^2*b*c-3*C*a*b^2*c)/(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*(4*A*a^4*c-12*A*a^3*b*c+13*A*a^2*b^2*c-7*A*a^2*c^3-6*A*
a*b^3*c+6*A*a*b*c^3+A*b^4*c+A*b^2*c^3-2*A*c^5+2*B*a^4*c-9*B*a^3*b*c+14*B*a^2*b^2*c+4*B*a^2*c^3-9*B*a*b^3*c+2*B
*b^4*c-4*B*b^2*c^3+2*B*c^5-2*C*a^5+2*C*a^4*b+4*C*a^3*b^2+5*C*a^3*c^2-4*C*a^2*b^3-14*C*a^2*b*c^2-2*C*a*b^4+13*C
*a*b^2*c^2-2*C*a*c^4+2*C*b^5-4*C*b^3*c^2+2*C*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)^2+1/2*(4*A*a^4*b-5*A*a^3*b^2+11*A*a^3*c^2-3*A*a^2*b^3-3*A*a^2*b*c^2+5*A*a*b^4-7*A*a*b^2*c^2+2*A*
a*c^4-A*b^5-A*b^3*c^2+2*A*b*c^4-2*B*a^5+3*B*a^4*b-B*a^3*b^2-4*B*a^3*c^2-B*a^2*b^3-8*B*a^2*b*c^2+3*B*a*b^4+8*B*
a*b^2*c^2-2*B*a*c^4-2*B*b^5+4*B*b^3*c^2-2*B*b*c^4-5*C*a^4*c+5*C*a^3*b*c+5*C*a^2*b^2*c+4*C*a^2*c^3-5*C*a*b^3*c-
4*C*a*b*c^3)/(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*(4*A*a^4*c-3*A*a^2*b^
2*c+A*a^2*c^3-A*b^4*c+A*b^2*c^3-5*B*a^3*b*c+5*B*a*b^3*c-2*B*a*b*c^3-2*C*a^5+4*C*a^3*b^2+C*a^3*c^2-2*C*a*b^4-C*
a*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*t
anh(1/2*x)-a-b)^2-(2*A*a^2+A*b^2-A*c^2-3*B*a*b+3*C*a*c)/(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((A+B*cosh(x)+C*sinh(x))/(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. 6850 vs. \(2 (223) = 446\).
time = 0.56, size = 13813, normalized size = 59.28 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(2*B*a^4*b^2 - 6*A*a^3*b^3 + 2*B*a^2*b^4 + 6*A*a*b^5 - 4*B*b^6 - 4*C*c^6 - 2*(3*A*a - 2*(B + C)*b)*c^5 -
2*(C*a^2 - 3*A*a*b + 2*(B - 2*C)*b^2)*c^4 - 2*(3*A*a^3 - (B + C)*a^2*b - 6*A*a*b^2 + 4*(B + C)*b^3)*c^3 - 2*(
2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2*b^4 + 3*B*a*b^5 - A*b^6 - A*c^6 + (3*C*a - 2*A*b)*c^5 + (A*a^2 - 3*(B - 2*C)
*a*b + A*b^2)*c^4 + (3*C*a^3 + 2*A*a^2*b - 6*B*a*b^2 + 4*A*b^3)*c^3 + (2*A*a^4 - 3*(B - 2*C)*a^3*b - 6*C*a*b^3
+ A*b^4)*c^2 + (4*A*a^4*b - 3*(2*B - C)*a^3*b^2 - 2*A*a^2*b^3 + 3*(2*B - C)*a*b^4 - 2*A*b^5)*c)*cosh(x)^3 - 2
*(2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2*b^4 + 3*B*a*b^5 - A*b^6 - A*c^6 + (3*C*a - 2*A*b)*c^5 + (A*a^2 - 3*(B - 2*
C)*a*b + A*b^2)*c^4 + (3*C*a^3 + 2*A*a^2*b - 6*B*a*b^2 + 4*A*b^3)*c^3 + (2*A*a^4 - 3*(B - 2*C)*a^3*b - 6*C*a*b
^3 + A*b^4)*c^2 + (4*A*a^4*b - 3*(2*B - C)*a^3*b^2 - 2*A*a^2*b^3 + 3*(2*B - C)*a*b^4 - 2*A*b^5)*c)*sinh(x)^3 +
2*(C*a^4 + 3*A*a^3*b - (B - C)*a^2*b^2 - 6*A*a*b^3 + 2*(2*B - C)*b^4)*c^2 + 2*(2*(B + C)*a^6 - 6*A*a^5*b + 3*
(B - 2*C)*a^4*b^2 + 3*A*a^3*b^3 - 3*(B - 2*C)*a^2*b^4 + 3*A*a*b^5 - 2*(B + C)*b^6 + 3*A*a*c^5 + 2*(B + C)*c^6
+ 3*((2*B - C)*a^2 + A*a*b - 2*(B + C)*b^2)*c^4 - 3*(A*a^3 - 3*(B - C)*a^2*b + 2*A*a*b^2)*c^3 + 3*((2*B - C)*a
^4 - A*a^3*b - (B + C)*a^2*b^2 - 2*A*a*b^3 + 2*(B + C)*b^4)*c^2 - 3*(2*A*a^5 - 3*(B - C)*a^4*b - A*a^3*b^2 + 3
*(B - C)*a^2*b^3 - A*a*b^4)*c)*cosh(x)^2 + 2*(2*(B + C)*a^6 - 6*A*a^5*b + 3*(B - 2*C)*a^4*b^2 + 3*A*a^3*b^3 -
3*(B - 2*C)*a^2*b^4 + 3*A*a*b^5 - 2*(B + C)*b^6 + 3*A*a*c^5 + 2*(B + C)*c^6 + 3*((2*B - C)*a^2 + A*a*b - 2*(B
+ C)*b^2)*c^4 - 3*(A*a^3 - 3*(B - C)*a^2*b + 2*A*a*b^2)*c^3 + 3*((2*B - C)*a^4 - A*a^3*b - (B + C)*a^2*b^2 - 2
*A*a*b^3 + 2*(B + C)*b^4)*c^2 - 3*(2*A*a^5 - 3*(B - C)*a^4*b - A*a^3*b^2 + 3*(B - C)*a^2*b^3 - A*a*b^4)*c - 3*
(2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2*b^4 + 3*B*a*b^5 - A*b^6 - A*c^6 + (3*C*a - 2*A*b)*c^5 + (A*a^2 - 3*(B - 2*C
)*a*b + A*b^2)*c^4 + (3*C*a^3 + 2*A*a^2*b - 6*B*a*b^2 + 4*A*b^3)*c^3 + (2*A*a^4 - 3*(B - 2*C)*a^3*b - 6*C*a*b^
3 + A*b^4)*c^2 + (4*A*a^4*b - 3*(2*B - C)*a^3*b^2 - 2*A*a^2*b^3 + 3*(2*B - C)*a*b^4 - 2*A*b^5)*c)*cosh(x))*sin
h(x)^2 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 - A*c^5 + (3*C*a + A*b)*c^4 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 - A*c
^5 + 3*(C*a - A*b)*c^4 + (2*A*a^2 - 3*(B - 3*C)*a*b - 2*A*b^2)*c^3 + (6*A*a^2*b - 9*(B - C)*a*b^2 + 2*A*b^3)*c
^2 + 3*(2*A*a^2*b^2 - (3*B - C)*a*b^3 + A*b^4)*c)*cosh(x)^4 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 - A*c^5 + 3*(C*
a - A*b)*c^4 + (2*A*a^2 - 3*(B - 3*C)*a*b - 2*A*b^2)*c^3 + (6*A*a^2*b - 9*(B - C)*a*b^2 + 2*A*b^3)*c^2 + 3*(2*
A*a^2*b^2 - (3*B - C)*a*b^3 + A*b^4)*c)*sinh(x)^4 + (2*A*a^2 - 3*(B + C)*a*b + 2*A*b^2)*c^3 + 4*(2*A*a^3*b^2 -
3*B*a^2*b^3 + A*a*b^4 - A*a*c^4 + (3*C*a^2 - 2*A*a*b)*c^3 + (2*A*a^3 - 3*(B - 2*C)*a^2*b)*c^2 + (4*A*a^3*b -
3*(2*B - C)*a^2*b^2 + 2*A*a*b^3)*c)*cosh(x)^3 + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 - A*a*c^4 + (3*C*a^2 -
2*A*a*b)*c^3 + (2*A*a^3 - 3*(B - 2*C)*a^2*b)*c^2 + (4*A*a^3*b - 3*(2*B - C)*a^2*b^2 + 2*A*a*b^3)*c + (2*A*a^2*
b^3 - 3*B*a*b^4 + A*b^5 - A*c^5 + 3*(C*a - A*b)*c^4 + (2*A*a^2 - 3*(B - 3*C)*a*b - 2*A*b^2)*c^3 + (6*A*a^2*b -
9*(B - C)*a*b^2 + 2*A*b^3)*c^2 + 3*(2*A*a^2*b^2 - (3*B - C)*a*b^3 + A*b^4)*c)*cosh(x))*sinh(x)^3 - (2*A*a^2*b
- 3*(B - C)*a*b^2 + 2*A*b^3)*c^2 + 2*(4*A*a^4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 + A*c^5 - (3*
C*a - A*b)*c^4 - (4*A*a^2 - 3*(B - C)*a*b + 2*A*b^2)*c^3 + (6*C*a^3 - 4*A*a^2*b + 3*(B + C)*a*b^2 - 2*A*b^3)*c
^2 + (4*A*a^4 - 6*(B - C)*a^3*b + 4*A*a^2*b^2 - 3*(B - C)*a*b^3 + A*b^4)*c)*cosh(x)^2 + 2*(4*A*a^4*b - 6*B*a^3
*b^2 + 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 + A*c^5 - (3*C*a - A*b)*c^4 - (4*A*a^2 - 3*(B - C)*a*b + 2*A*b^2)*c^3 +
(6*C*a^3 - 4*A*a^2*b + 3*(B + C)*a*b^2 - 2*A*b^3)*c^2 + 3*(2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 - A*c^5 + 3*(C*a -
A*b)*c^4 + (2*A*a^2 - 3*(B - 3*C)*a*b - 2*A*b^2)*c^3 + (6*A*a^2*b - 9*(B - C)*a*b^2 + 2*A*b^3)*c^2 + 3*(2*A*a
^2*b^2 - (3*B - C)*a*b^3 + A*b^4)*c)*cosh(x)^2 + (4*A*a^4 - 6*(B - C)*a^3*b + 4*A*a^2*b^2 - 3*(B - C)*a*b^3 +
A*b^4)*c + 6*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 - A*a*c^4 + (3*C*a^2 - 2*A*a*b)*c^3 + (2*A*a^3 - 3*(B - 2*C)
*a^2*b)*c^2 + (4*A*a^3*b - 3*(2*B - C)*a^2*b^2 + 2*A*a*b^3)*c)*cosh(x))*sinh(x)^2 - (2*A*a^2*b^2 - 3*(B + C)*a
*b^3 + A*b^4)*c + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 + 3*C*a^2*b^2*c - 3*C*a^2*c^3 + A*a*c^4 - (2*A*a^3 -
3*B*a^2*b + 2*A*a*b^2)*c^2)*cosh(x) + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 + 3*C*a^2*b^2*c - 3*C*a^2*c^3 + A
*a*c^4 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 - A*c^5 + 3*(C*a - A*b)*c^4 + (2*A*a^2 - 3*(B - 3*C)*a*b - 2*A*b^2)*
c^3 + (6*A*a^2*b - 9*(B - C)*a*b^2 + 2*A*b^3)*c^2 + 3*(2*A*a^2*b^2 - (3*B - C)*a*b^3 + A*b^4)*c)*cosh(x)^3 - (
2*A*a^3 - 3*B*a^2*b + 2*A*a*b^2)*c^2 + 3*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 - A*a*c^4 + (3*C*a^2 - 2*A*a*b)*
c^3 + (2*A*a^3 - 3*(B - 2*C)*a^2*b)*c^2 + (4*A*a^3*b - 3*(2*B - C)*a^2*b^2 + 2*A*a*b^3)*c)*cosh(x)^2 + (4*A*a^
4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 + A*c^5 - (3*C*a - A*b)*c^4 - (4*A*a^2 - 3*(B - C)*a*b + 2
*A*b^2)*c^3 + (6*C*a^3 - 4*A*a^2*b + 3*(B + C)*...

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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((A+B*cosh(x)+C*sinh(x))/(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. 819 vs. \(2 (223) = 446\).
time = 0.43, size = 819, normalized size = 3.52 \begin {gather*} \frac {{\left (2 \, A a^{2} - 3 \, B a b + A b^{2} + 3 \, C a c - A 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 a^{2} b^{2} e^{\left (3 \, x\right )} - 3 \, B a b^{3} e^{\left (3 \, x\right )} + A b^{4} e^{\left (3 \, x\right )} + 4 \, A a^{2} b c e^{\left (3 \, x\right )} - 6 \, B a b^{2} c e^{\left (3 \, x\right )} + 3 \, C a b^{2} c e^{\left (3 \, x\right )} + 2 \, A b^{3} c e^{\left (3 \, x\right )} + 2 \, A a^{2} c^{2} e^{\left (3 \, x\right )} - 3 \, B a b c^{2} e^{\left (3 \, x\right )} + 6 \, C a b c^{2} e^{\left (3 \, x\right )} + 3 \, C a c^{3} e^{\left (3 \, x\right )} - 2 \, A b c^{3} e^{\left (3 \, x\right )} - A c^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} - 2 \, C a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} - 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} + 4 \, C a^{2} b^{2} e^{\left (2 \, x\right )} + 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} - 2 \, C b^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} c e^{\left (2 \, x\right )} - 9 \, B a^{2} b c e^{\left (2 \, x\right )} + 9 \, C a^{2} b c e^{\left (2 \, x\right )} + 3 \, A a b^{2} c e^{\left (2 \, x\right )} - 4 \, B a^{2} c^{2} e^{\left (2 \, x\right )} + 5 \, C a^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a b c^{2} e^{\left (2 \, x\right )} + 4 \, B b^{2} c^{2} e^{\left (2 \, x\right )} + 4 \, C b^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a c^{3} e^{\left (2 \, x\right )} - 2 \, B c^{4} e^{\left (2 \, x\right )} - 2 \, C c^{4} e^{\left (2 \, x\right )} - 4 \, B a^{3} b e^{x} + 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} + 4 \, C a^{3} c e^{x} + 5 \, C a b^{2} c e^{x} - 10 \, A a^{2} c^{2} e^{x} + 5 \, B a b c^{2} e^{x} + 2 \, A b^{2} c^{2} e^{x} - 5 \, C a c^{3} e^{x} - A c^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} - 2 \, B b^{4} + B a^{2} b c + C a^{2} b c - 3 \, A a b^{2} c + 2 \, B b^{3} c + 2 \, C b^{3} c - C a^{2} c^{2} - 3 \, A a b c^{2} + 2 \, B b^{2} c^{2} - 2 \, C b^{2} c^{2} + 3 \, A a c^{3} - 2 \, B b c^{3} - 2 \, C b c^{3} + 2 \, C c^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + a^{4} c - 2 \, a^{2} b^{2} c + b^{4} c + 2 \, a^{2} b c^{2} - 2 \, b^{3} c^{2} + 2 \, a^{2} c^{3} - 2 \, b^{2} c^{3} + b c^{4} + c^{5}\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((A+B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="giac")

[Out]

(2*A*a^2 - 3*B*a*b + A*b^2 + 3*C*a*c - A*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*a^2*b^2*e^(3*x) - 3*B*a*b^3*e^(3*x) +
A*b^4*e^(3*x) + 4*A*a^2*b*c*e^(3*x) - 6*B*a*b^2*c*e^(3*x) + 3*C*a*b^2*c*e^(3*x) + 2*A*b^3*c*e^(3*x) + 2*A*a^2
*c^2*e^(3*x) - 3*B*a*b*c^2*e^(3*x) + 6*C*a*b*c^2*e^(3*x) + 3*C*a*c^3*e^(3*x) - 2*A*b*c^3*e^(3*x) - A*c^4*e^(3*
x) - 2*B*a^4*e^(2*x) - 2*C*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) - 5*B*a^2*b^2*e^(2*x) + 4*C*a^2*b^2*e^(2*x) + 3*A*a
*b^3*e^(2*x) - 2*B*b^4*e^(2*x) - 2*C*b^4*e^(2*x) + 6*A*a^3*c*e^(2*x) - 9*B*a^2*b*c*e^(2*x) + 9*C*a^2*b*c*e^(2*
x) + 3*A*a*b^2*c*e^(2*x) - 4*B*a^2*c^2*e^(2*x) + 5*C*a^2*c^2*e^(2*x) - 3*A*a*b*c^2*e^(2*x) + 4*B*b^2*c^2*e^(2*
x) + 4*C*b^2*c^2*e^(2*x) - 3*A*a*c^3*e^(2*x) - 2*B*c^4*e^(2*x) - 2*C*c^4*e^(2*x) - 4*B*a^3*b*e^x + 10*A*a^2*b^
2*e^x - 5*B*a*b^3*e^x - A*b^4*e^x + 4*C*a^3*c*e^x + 5*C*a*b^2*c*e^x - 10*A*a^2*c^2*e^x + 5*B*a*b*c^2*e^x + 2*A
*b^2*c^2*e^x - 5*C*a*c^3*e^x - A*c^4*e^x - B*a^2*b^2 + 3*A*a*b^3 - 2*B*b^4 + B*a^2*b*c + C*a^2*b*c - 3*A*a*b^2
*c + 2*B*b^3*c + 2*C*b^3*c - C*a^2*c^2 - 3*A*a*b*c^2 + 2*B*b^2*c^2 - 2*C*b^2*c^2 + 3*A*a*c^3 - 2*B*b*c^3 - 2*C
*b*c^3 + 2*C*c^4)/((a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 2*a^2*b^2*c + b^4*c + 2*a^2*b*c^2 - 2*b^3*c^2 + 2*a^2*c^
3 - 2*b^2*c^3 + b*c^4 + c^5)*(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 = -0.00 \begin {gather*} \int \frac {A+B\,\mathrm {cosh}\left (x\right )+C\,\mathrm {sinh}\left (x\right )}{{\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((A + B*cosh(x) + C*sinh(x))/(a + b*cosh(x) + c*sinh(x))^3,x)

[Out]

int((A + B*cosh(x) + C*sinh(x))/(a + b*cosh(x) + c*sinh(x))^3, x)

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