3.752 \(\int \frac {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx\)

Optimal. Leaf size=140 \[ -\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{6 c^4 (a \cosh (x)+a+c \sinh (x))^2}+\frac {a \left (5 a^2-3 c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^7}-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{6 c^6 (a \cosh (x)+a+c \sinh (x))}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3} \]

[Out]

1/2*a*(5*a^2-3*c^2)*ln(a+c*tanh(1/2*x))/c^7+1/3*(-c*cosh(x)-a*sinh(x))/c^2/(a+a*cosh(x)+c*sinh(x))^3-5/6*(a*c*
cosh(x)+a^2*sinh(x))/c^4/(a+a*cosh(x)+c*sinh(x))^2+1/6*(-c*(15*a^2-4*c^2)*cosh(x)-a*(15*a^2-4*c^2)*sinh(x))/c^
6/(a+a*cosh(x)+c*sinh(x))

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Rubi [A]  time = 0.21, antiderivative size = 140, 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 = {3129, 3156, 3153, 3124, 31} \[ \frac {a \left (5 a^2-3 c^2\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^7}-\frac {a \left (15 a^2-4 c^2\right ) \sinh (x)+c \left (15 a^2-4 c^2\right ) \cosh (x)}{6 c^6 (a \cosh (x)+a+c \sinh (x))}-\frac {5 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{6 c^4 (a \cosh (x)+a+c \sinh (x))^2}-\frac {a \sinh (x)+c \cosh (x)}{3 c^2 (a \cosh (x)+a+c \sinh (x))^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cosh[x] + c*Sinh[x])^(-4),x]

[Out]

(a*(5*a^2 - 3*c^2)*Log[a + c*Tanh[x/2]])/(2*c^7) - (c*Cosh[x] + a*Sinh[x])/(3*c^2*(a + a*Cosh[x] + c*Sinh[x])^
3) - (5*(a*c*Cosh[x] + a^2*Sinh[x]))/(6*c^4*(a + a*Cosh[x] + c*Sinh[x])^2) - (c*(15*a^2 - 4*c^2)*Cosh[x] + a*(
15*a^2 - 4*c^2)*Sinh[x])/(6*c^6*(a + a*Cosh[x] + c*Sinh[x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3124

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 3129

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 3153

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 3156

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 {1}{(a+a \cosh (x)+c \sinh (x))^4} \, dx &=-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {\int \frac {-3 a+2 a \cosh (x)+2 c \sinh (x)}{(a+a \cosh (x)+c \sinh (x))^3} \, dx}{3 c^2}\\ &=-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {5 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{6 c^4 (a+a \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 \left (5 a^2-2 c^2\right )-5 a^2 \cosh (x)-5 a c \sinh (x)}{(a+a \cosh (x)+c \sinh (x))^2} \, dx}{6 c^4}\\ &=-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {5 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{6 c^4 (a+a \cosh (x)+c \sinh (x))^2}-\frac {c \left (15 a^2-4 c^2\right ) \cosh (x)+a \left (15 a^2-4 c^2\right ) \sinh (x)}{6 c^6 (a+a \cosh (x)+c \sinh (x))}-\frac {\left (a \left (3-\frac {5 a^2}{c^2}\right )\right ) \int \frac {1}{a+a \cosh (x)+c \sinh (x)} \, dx}{2 c^4}\\ &=-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {5 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{6 c^4 (a+a \cosh (x)+c \sinh (x))^2}-\frac {c \left (15 a^2-4 c^2\right ) \cosh (x)+a \left (15 a^2-4 c^2\right ) \sinh (x)}{6 c^6 (a+a \cosh (x)+c \sinh (x))}-\frac {\left (a \left (3-\frac {5 a^2}{c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{c^4}\\ &=-\frac {a \left (3-\frac {5 a^2}{c^2}\right ) \log \left (a+c \tanh \left (\frac {x}{2}\right )\right )}{2 c^5}-\frac {c \cosh (x)+a \sinh (x)}{3 c^2 (a+a \cosh (x)+c \sinh (x))^3}-\frac {5 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{6 c^4 (a+a \cosh (x)+c \sinh (x))^2}-\frac {c \left (15 a^2-4 c^2\right ) \cosh (x)+a \left (15 a^2-4 c^2\right ) \sinh (x)}{6 c^6 (a+a \cosh (x)+c \sinh (x))}\\ \end {align*}

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Mathematica [B]  time = 0.63, size = 300, normalized size = 2.14 \[ \frac {192 \left (3 a c^2-5 a^3\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+192 a \left (5 a^2-3 c^2\right ) \log \left (a \cosh \left (\frac {x}{2}\right )+c \sinh \left (\frac {x}{2}\right )\right )-\frac {c \text {sech}^6\left (\frac {x}{2}\right ) \left (150 a^6 \sinh (x)+120 a^6 \sinh (2 x)+30 a^6 \sinh (3 x)+75 a^5 c \cosh (3 x)-150 a^5 c-255 a^4 c^2 \sinh (x)-72 a^4 c^2 \sinh (2 x)+37 a^4 c^2 \sinh (3 x)-35 a^3 c^3 \cosh (3 x)+130 a^3 c^3+129 a^2 c^4 \sinh (x)+36 a^2 c^4 \sinh (2 x)-27 a^2 c^4 \sinh (3 x)+\left (-75 a^5 c+75 a^3 c^3+12 a c^5\right ) \cosh (x)+6 a c \left (25 a^4-15 a^2 c^2+4 c^4\right ) \cosh (2 x)+4 a c^5 \cosh (3 x)-24 a c^5-12 c^6 \sinh (x)+4 c^6 \sinh (3 x)\right )}{a \left (a+c \tanh \left (\frac {x}{2}\right )\right )^3}}{384 c^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cosh[x] + c*Sinh[x])^(-4),x]

[Out]

(192*(-5*a^3 + 3*a*c^2)*Log[Cosh[x/2]] + 192*a*(5*a^2 - 3*c^2)*Log[a*Cosh[x/2] + c*Sinh[x/2]] - (c*Sech[x/2]^6
*(-150*a^5*c + 130*a^3*c^3 - 24*a*c^5 + (-75*a^5*c + 75*a^3*c^3 + 12*a*c^5)*Cosh[x] + 6*a*c*(25*a^4 - 15*a^2*c
^2 + 4*c^4)*Cosh[2*x] + 75*a^5*c*Cosh[3*x] - 35*a^3*c^3*Cosh[3*x] + 4*a*c^5*Cosh[3*x] + 150*a^6*Sinh[x] - 255*
a^4*c^2*Sinh[x] + 129*a^2*c^4*Sinh[x] - 12*c^6*Sinh[x] + 120*a^6*Sinh[2*x] - 72*a^4*c^2*Sinh[2*x] + 36*a^2*c^4
*Sinh[2*x] + 30*a^6*Sinh[3*x] + 37*a^4*c^2*Sinh[3*x] - 27*a^2*c^4*Sinh[3*x] + 4*c^6*Sinh[3*x]))/(a*(a + c*Tanh
[x/2])^3))/(384*c^7)

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fricas [B]  time = 0.52, size = 4015, normalized size = 28.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="fricas")

[Out]

1/6*(30*a^5*c - 90*a^4*c^2 + 82*a^3*c^3 - 6*a^2*c^4 - 24*a*c^5 + 8*c^6 + 6*(5*a^5*c + 10*a^4*c^2 + 2*a^3*c^3 -
6*a^2*c^4 - 3*a*c^5)*cosh(x)^5 + 6*(5*a^5*c + 10*a^4*c^2 + 2*a^3*c^3 - 6*a^2*c^4 - 3*a*c^5)*sinh(x)^5 + 30*(5
*a^5*c + 5*a^4*c^2 - 3*a^3*c^3 - 3*a^2*c^4)*cosh(x)^4 + 30*(5*a^5*c + 5*a^4*c^2 - 3*a^3*c^3 - 3*a^2*c^4 + (5*a
^5*c + 10*a^4*c^2 + 2*a^3*c^3 - 6*a^2*c^4 - 3*a*c^5)*cosh(x))*sinh(x)^4 + 4*(75*a^5*c - 65*a^3*c^3 + 12*a*c^5)
*cosh(x)^3 + 4*(75*a^5*c - 65*a^3*c^3 + 12*a*c^5 + 15*(5*a^5*c + 10*a^4*c^2 + 2*a^3*c^3 - 6*a^2*c^4 - 3*a*c^5)
*cosh(x)^2 + 30*(5*a^5*c + 5*a^4*c^2 - 3*a^3*c^3 - 3*a^2*c^4)*cosh(x))*sinh(x)^3 + 12*(25*a^5*c - 25*a^4*c^2 -
10*a^3*c^3 + 10*a^2*c^4 + 2*a*c^5 - 2*c^6)*cosh(x)^2 + 12*(25*a^5*c - 25*a^4*c^2 - 10*a^3*c^3 + 10*a^2*c^4 +
2*a*c^5 - 2*c^6 + 5*(5*a^5*c + 10*a^4*c^2 + 2*a^3*c^3 - 6*a^2*c^4 - 3*a*c^5)*cosh(x)^3 + 15*(5*a^5*c + 5*a^4*c
^2 - 3*a^3*c^3 - 3*a^2*c^4)*cosh(x)^2 + (75*a^5*c - 65*a^3*c^3 + 12*a*c^5)*cosh(x))*sinh(x)^2 + 30*(5*a^5*c -
10*a^4*c^2 + 4*a^3*c^3 + 2*a^2*c^4 - a*c^5)*cosh(x) + 3*((5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^
4 - 3*a*c^5)*cosh(x)^6 + (5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*sinh(x)^6 + 5*a^6 -
15*a^5*c + 12*a^4*c^2 + 4*a^3*c^3 - 9*a^2*c^4 + 3*a*c^5 + 6*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2
*c^4)*cosh(x)^5 + 6*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4 + (5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4
*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x))*sinh(x)^5 + 3*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^
4 + 3*a*c^5)*cosh(x)^4 + 3*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5 + 5*(5*a^6 + 15*
a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^2 + 10*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3
- 3*a^2*c^4)*cosh(x))*sinh(x)^4 + 4*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4)*cosh(x)^3 + 4*(25*a^6 - 30*a^4*c^2 + 9*
a^2*c^4 + 5*(5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^3 + 15*(5*a^6 + 10*a^5*c
+ 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^2 + 3*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 +
3*a*c^5)*cosh(x))*sinh(x)^3 + 3*(25*a^6 - 25*a^5*c - 20*a^4*c^2 + 20*a^3*c^3 + 3*a^2*c^4 - 3*a*c^5)*cosh(x)^2
+ 3*(25*a^6 - 25*a^5*c - 20*a^4*c^2 + 20*a^3*c^3 + 3*a^2*c^4 - 3*a*c^5 + 5*(5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4
*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^4 + 20*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^
3 + 6*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5)*cosh(x)^2 + 4*(25*a^6 - 30*a^4*c^2 +
9*a^2*c^4)*cosh(x))*sinh(x)^2 + 6*(5*a^6 - 10*a^5*c + 2*a^4*c^2 + 6*a^3*c^3 - 3*a^2*c^4)*cosh(x) + 6*(5*a^6 -
10*a^5*c + 2*a^4*c^2 + 6*a^3*c^3 - 3*a^2*c^4 + (5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c
^5)*cosh(x)^5 + 5*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^4 + 2*(25*a^6 + 25*a^5*c - 20
*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5)*cosh(x)^3 + 2*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4)*cosh(x)^2 + (25*a
^6 - 25*a^5*c - 20*a^4*c^2 + 20*a^3*c^3 + 3*a^2*c^4 - 3*a*c^5)*cosh(x))*sinh(x))*log((a + c)*cosh(x) + (a + c)
*sinh(x) + a - c) - 3*((5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^6 + (5*a^6 +
15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*sinh(x)^6 + 5*a^6 - 15*a^5*c + 12*a^4*c^2 + 4*a^3*c^3
- 9*a^2*c^4 + 3*a*c^5 + 6*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^5 + 6*(5*a^6 + 10*a^
5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4 + (5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*co
sh(x))*sinh(x)^5 + 3*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5)*cosh(x)^4 + 3*(25*a^6
+ 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5 + 5*(5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9
*a^2*c^4 - 3*a*c^5)*cosh(x)^2 + 10*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x))*sinh(x)^4 +
4*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4)*cosh(x)^3 + 4*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4 + 5*(5*a^6 + 15*a^5*c + 12
*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^3 + 15*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c
^4)*cosh(x)^2 + 3*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5)*cosh(x))*sinh(x)^3 + 3*(
25*a^6 - 25*a^5*c - 20*a^4*c^2 + 20*a^3*c^3 + 3*a^2*c^4 - 3*a*c^5)*cosh(x)^2 + 3*(25*a^6 - 25*a^5*c - 20*a^4*c
^2 + 20*a^3*c^3 + 3*a^2*c^4 - 3*a*c^5 + 5*(5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*co
sh(x)^4 + 20*(5*a^6 + 10*a^5*c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^3 + 6*(25*a^6 + 25*a^5*c - 20*a^4*
c^2 - 20*a^3*c^3 + 3*a^2*c^4 + 3*a*c^5)*cosh(x)^2 + 4*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4)*cosh(x))*sinh(x)^2 + 6
*(5*a^6 - 10*a^5*c + 2*a^4*c^2 + 6*a^3*c^3 - 3*a^2*c^4)*cosh(x) + 6*(5*a^6 - 10*a^5*c + 2*a^4*c^2 + 6*a^3*c^3
- 3*a^2*c^4 + (5*a^6 + 15*a^5*c + 12*a^4*c^2 - 4*a^3*c^3 - 9*a^2*c^4 - 3*a*c^5)*cosh(x)^5 + 5*(5*a^6 + 10*a^5*
c + 2*a^4*c^2 - 6*a^3*c^3 - 3*a^2*c^4)*cosh(x)^4 + 2*(25*a^6 + 25*a^5*c - 20*a^4*c^2 - 20*a^3*c^3 + 3*a^2*c^4
+ 3*a*c^5)*cosh(x)^3 + 2*(25*a^6 - 30*a^4*c^2 + 9*a^2*c^4)*cosh(x)^2 + (25*a^6 - 25*a^5*c - 20*a^4*c^2 + 20*a^
3*c^3 + 3*a^2*c^4 - 3*a*c^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 6*(25*a^5*c - 50*a^4*c^2 + 20*a^3*
c^3 + 10*a^2*c^4 - 5*a*c^5 + 5*(5*a^5*c + 10*a^4*c^2 + 2*a^3*c^3 - 6*a^2*c^4 - 3*a*c^5)*cosh(x)^4 + 20*(5*a^5*
c + 5*a^4*c^2 - 3*a^3*c^3 - 3*a^2*c^4)*cosh(x)^3 + 2*(75*a^5*c - 65*a^3*c^3 + 12*a*c^5)*cosh(x)^2 + 4*(25*a^5*
c - 25*a^4*c^2 - 10*a^3*c^3 + 10*a^2*c^4 + 2*a*c^5 - 2*c^6)*cosh(x))*sinh(x))/(a^3*c^7 - 3*a^2*c^8 + 3*a*c^9 -
c^10 + (a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)*cosh(x)^6 + (a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)*sinh(x)^6 +
6*(a^3*c^7 + 2*a^2*c^8 + a*c^9)*cosh(x)^5 + 6*(a^3*c^7 + 2*a^2*c^8 + a*c^9 + (a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 +
c^10)*cosh(x))*sinh(x)^5 + 3*(5*a^3*c^7 + 5*a^2*c^8 - a*c^9 - c^10)*cosh(x)^4 + 3*(5*a^3*c^7 + 5*a^2*c^8 - a*c
^9 - c^10 + 5*(a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)*cosh(x)^2 + 10*(a^3*c^7 + 2*a^2*c^8 + a*c^9)*cosh(x))*sin
h(x)^4 + 4*(5*a^3*c^7 - 3*a*c^9)*cosh(x)^3 + 4*(5*a^3*c^7 - 3*a*c^9 + 5*(a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)
*cosh(x)^3 + 15*(a^3*c^7 + 2*a^2*c^8 + a*c^9)*cosh(x)^2 + 3*(5*a^3*c^7 + 5*a^2*c^8 - a*c^9 - c^10)*cosh(x))*si
nh(x)^3 + 3*(5*a^3*c^7 - 5*a^2*c^8 - a*c^9 + c^10)*cosh(x)^2 + 3*(5*a^3*c^7 - 5*a^2*c^8 - a*c^9 + c^10 + 5*(a^
3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)*cosh(x)^4 + 20*(a^3*c^7 + 2*a^2*c^8 + a*c^9)*cosh(x)^3 + 6*(5*a^3*c^7 + 5*
a^2*c^8 - a*c^9 - c^10)*cosh(x)^2 + 4*(5*a^3*c^7 - 3*a*c^9)*cosh(x))*sinh(x)^2 + 6*(a^3*c^7 - 2*a^2*c^8 + a*c^
9)*cosh(x) + 6*(a^3*c^7 - 2*a^2*c^8 + a*c^9 + (a^3*c^7 + 3*a^2*c^8 + 3*a*c^9 + c^10)*cosh(x)^5 + 5*(a^3*c^7 +
2*a^2*c^8 + a*c^9)*cosh(x)^4 + 2*(5*a^3*c^7 + 5*a^2*c^8 - a*c^9 - c^10)*cosh(x)^3 + 2*(5*a^3*c^7 - 3*a*c^9)*co
sh(x)^2 + (5*a^3*c^7 - 5*a^2*c^8 - a*c^9 + c^10)*cosh(x))*sinh(x))

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giac [B]  time = 0.16, size = 377, normalized size = 2.69 \[ \frac {{\left (5 \, a^{4} + 5 \, a^{3} c - 3 \, a^{2} c^{2} - 3 \, a c^{3}\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{2 \, {\left (a c^{7} + c^{8}\right )}} - \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, c^{7}} + \frac {15 \, a^{5} e^{\left (5 \, x\right )} + 30 \, a^{4} c e^{\left (5 \, x\right )} + 6 \, a^{3} c^{2} e^{\left (5 \, x\right )} - 18 \, a^{2} c^{3} e^{\left (5 \, x\right )} - 9 \, a c^{4} e^{\left (5 \, x\right )} + 75 \, a^{5} e^{\left (4 \, x\right )} + 75 \, a^{4} c e^{\left (4 \, x\right )} - 45 \, a^{3} c^{2} e^{\left (4 \, x\right )} - 45 \, a^{2} c^{3} e^{\left (4 \, x\right )} + 150 \, a^{5} e^{\left (3 \, x\right )} - 130 \, a^{3} c^{2} e^{\left (3 \, x\right )} + 24 \, a c^{4} e^{\left (3 \, x\right )} + 150 \, a^{5} e^{\left (2 \, x\right )} - 150 \, a^{4} c e^{\left (2 \, x\right )} - 60 \, a^{3} c^{2} e^{\left (2 \, x\right )} + 60 \, a^{2} c^{3} e^{\left (2 \, x\right )} + 12 \, a c^{4} e^{\left (2 \, x\right )} - 12 \, c^{5} e^{\left (2 \, x\right )} + 75 \, a^{5} e^{x} - 150 \, a^{4} c e^{x} + 60 \, a^{3} c^{2} e^{x} + 30 \, a^{2} c^{3} e^{x} - 15 \, a c^{4} e^{x} + 15 \, a^{5} - 45 \, a^{4} c + 41 \, a^{3} c^{2} - 3 \, a^{2} c^{3} - 12 \, a c^{4} + 4 \, c^{5}}{3 \, {\left (a e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + a - c\right )}^{3} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="giac")

[Out]

1/2*(5*a^4 + 5*a^3*c - 3*a^2*c^2 - 3*a*c^3)*log(abs(a*e^x + c*e^x + a - c))/(a*c^7 + c^8) - 1/2*(5*a^3 - 3*a*c
^2)*log(e^x + 1)/c^7 + 1/3*(15*a^5*e^(5*x) + 30*a^4*c*e^(5*x) + 6*a^3*c^2*e^(5*x) - 18*a^2*c^3*e^(5*x) - 9*a*c
^4*e^(5*x) + 75*a^5*e^(4*x) + 75*a^4*c*e^(4*x) - 45*a^3*c^2*e^(4*x) - 45*a^2*c^3*e^(4*x) + 150*a^5*e^(3*x) - 1
30*a^3*c^2*e^(3*x) + 24*a*c^4*e^(3*x) + 150*a^5*e^(2*x) - 150*a^4*c*e^(2*x) - 60*a^3*c^2*e^(2*x) + 60*a^2*c^3*
e^(2*x) + 12*a*c^4*e^(2*x) - 12*c^5*e^(2*x) + 75*a^5*e^x - 150*a^4*c*e^x + 60*a^3*c^2*e^x + 30*a^2*c^3*e^x - 1
5*a*c^4*e^x + 15*a^5 - 45*a^4*c + 41*a^3*c^2 - 3*a^2*c^3 - 12*a*c^4 + 4*c^5)/((a*e^(2*x) + c*e^(2*x) + 2*a*e^x
+ a - c)^3*c^6)

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maple [A]  time = 0.30, size = 250, normalized size = 1.79 \[ -\frac {\tanh ^{3}\left (\frac {x}{2}\right )}{24 c^{4}}+\frac {a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{4 c^{5}}-\frac {5 a^{2} \tanh \left (\frac {x}{2}\right )}{4 c^{6}}+\frac {3 \tanh \left (\frac {x}{2}\right )}{8 c^{4}}+\frac {a^{6}}{24 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {a^{4}}{8 c^{5} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{3}}+\frac {a^{2}}{8 c^{3} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {1}{24 c \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {3 a^{5}}{8 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {3 a^{3}}{4 c^{5} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {3 a}{8 c^{3} \left (a +c \tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {5 a^{3} \ln \left (a +c \tanh \left (\frac {x}{2}\right )\right )}{2 c^{7}}-\frac {3 a \ln \left (a +c \tanh \left (\frac {x}{2}\right )\right )}{2 c^{5}}+\frac {15 a^{4}}{8 c^{7} \left (a +c \tanh \left (\frac {x}{2}\right )\right )}-\frac {9 a^{2}}{4 c^{5} \left (a +c \tanh \left (\frac {x}{2}\right )\right )}+\frac {3}{8 c^{3} \left (a +c \tanh \left (\frac {x}{2}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*cosh(x)+c*sinh(x))^4,x)

[Out]

-1/24/c^4*tanh(1/2*x)^3+1/4/c^5*a*tanh(1/2*x)^2-5/4/c^6*a^2*tanh(1/2*x)+3/8/c^4*tanh(1/2*x)+1/24/c^7/(a+c*tanh
(1/2*x))^3*a^6-1/8/c^5/(a+c*tanh(1/2*x))^3*a^4+1/8/c^3/(a+c*tanh(1/2*x))^3*a^2-1/24/c/(a+c*tanh(1/2*x))^3-3/8*
a^5/c^7/(a+c*tanh(1/2*x))^2+3/4*a^3/c^5/(a+c*tanh(1/2*x))^2-3/8*a/c^3/(a+c*tanh(1/2*x))^2+5/2*a^3/c^7*ln(a+c*t
anh(1/2*x))-3/2*a/c^5*ln(a+c*tanh(1/2*x))+15/8/c^7/(a+c*tanh(1/2*x))*a^4-9/4/c^5/(a+c*tanh(1/2*x))*a^2+3/8/c^3
/(a+c*tanh(1/2*x))

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maxima [B]  time = 0.38, size = 487, normalized size = 3.48 \[ -\frac {15 \, a^{5} + 45 \, a^{4} c + 41 \, a^{3} c^{2} + 3 \, a^{2} c^{3} - 12 \, a c^{4} - 4 \, c^{5} + 15 \, {\left (5 \, a^{5} + 10 \, a^{4} c + 4 \, a^{3} c^{2} - 2 \, a^{2} c^{3} - a c^{4}\right )} e^{\left (-x\right )} + 6 \, {\left (25 \, a^{5} + 25 \, a^{4} c - 10 \, a^{3} c^{2} - 10 \, a^{2} c^{3} + 2 \, a c^{4} + 2 \, c^{5}\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (75 \, a^{5} - 65 \, a^{3} c^{2} + 12 \, a c^{4}\right )} e^{\left (-3 \, x\right )} + 15 \, {\left (5 \, a^{5} - 5 \, a^{4} c - 3 \, a^{3} c^{2} + 3 \, a^{2} c^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (5 \, a^{5} - 10 \, a^{4} c + 2 \, a^{3} c^{2} + 6 \, a^{2} c^{3} - 3 \, a c^{4}\right )} e^{\left (-5 \, x\right )}}{3 \, {\left (a^{3} c^{6} + 3 \, a^{2} c^{7} + 3 \, a c^{8} + c^{9} + 6 \, {\left (a^{3} c^{6} + 2 \, a^{2} c^{7} + a c^{8}\right )} e^{\left (-x\right )} + 3 \, {\left (5 \, a^{3} c^{6} + 5 \, a^{2} c^{7} - a c^{8} - c^{9}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (5 \, a^{3} c^{6} - 3 \, a c^{8}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (5 \, a^{3} c^{6} - 5 \, a^{2} c^{7} - a c^{8} + c^{9}\right )} e^{\left (-4 \, x\right )} + 6 \, {\left (a^{3} c^{6} - 2 \, a^{2} c^{7} + a c^{8}\right )} e^{\left (-5 \, x\right )} + {\left (a^{3} c^{6} - 3 \, a^{2} c^{7} + 3 \, a c^{8} - c^{9}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{2 \, c^{7}} - \frac {{\left (5 \, a^{3} - 3 \, a c^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^4,x, algorithm="maxima")

[Out]

-1/3*(15*a^5 + 45*a^4*c + 41*a^3*c^2 + 3*a^2*c^3 - 12*a*c^4 - 4*c^5 + 15*(5*a^5 + 10*a^4*c + 4*a^3*c^2 - 2*a^2
*c^3 - a*c^4)*e^(-x) + 6*(25*a^5 + 25*a^4*c - 10*a^3*c^2 - 10*a^2*c^3 + 2*a*c^4 + 2*c^5)*e^(-2*x) + 2*(75*a^5
- 65*a^3*c^2 + 12*a*c^4)*e^(-3*x) + 15*(5*a^5 - 5*a^4*c - 3*a^3*c^2 + 3*a^2*c^3)*e^(-4*x) + 3*(5*a^5 - 10*a^4*
c + 2*a^3*c^2 + 6*a^2*c^3 - 3*a*c^4)*e^(-5*x))/(a^3*c^6 + 3*a^2*c^7 + 3*a*c^8 + c^9 + 6*(a^3*c^6 + 2*a^2*c^7 +
a*c^8)*e^(-x) + 3*(5*a^3*c^6 + 5*a^2*c^7 - a*c^8 - c^9)*e^(-2*x) + 4*(5*a^3*c^6 - 3*a*c^8)*e^(-3*x) + 3*(5*a^
3*c^6 - 5*a^2*c^7 - a*c^8 + c^9)*e^(-4*x) + 6*(a^3*c^6 - 2*a^2*c^7 + a*c^8)*e^(-5*x) + (a^3*c^6 - 3*a^2*c^7 +
3*a*c^8 - c^9)*e^(-6*x)) + 1/2*(5*a^3 - 3*a*c^2)*log(-(a - c)*e^(-x) - a - c)/c^7 - 1/2*(5*a^3 - 3*a*c^2)*log(
e^(-x) + 1)/c^7

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+a\,\mathrm {cosh}\relax (x)+c\,\mathrm {sinh}\relax (x)\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + a*cosh(x) + c*sinh(x))^4,x)

[Out]

int(1/(a + a*cosh(x) + c*sinh(x))^4, x)

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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/(a+a*cosh(x)+c*sinh(x))**4,x)

[Out]

Timed out

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