3.705 \(\int \frac {\cosh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx\)

Optimal. Leaf size=104 \[ \frac {a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}+\frac {2 a b}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}+\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}-\frac {b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3086, 3483, 3529, 3531, 3530} \[ \frac {a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}+\frac {2 a b}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}+\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}-\frac {b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 + 3*b^2)*x)/(a^2 - b^2)^3 - (b*(3*a^2 + b^2)*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + b/(2*(a^2 - b
^2)*(a + b*Tanh[x])^2) + (2*a*b)/((a^2 - b^2)^2*(a + b*Tanh[x]))

Rule 3086

Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps

\begin {align*} \int \frac {\cosh ^3(x)}{(a \cosh (x)+b \sinh (x))^3} \, dx &=\int \frac {1}{(a+b \tanh (x))^3} \, dx\\ &=\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}+\frac {\int \frac {a-b \tanh (x)}{(a+b \tanh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}+\frac {\int \frac {a^2+b^2-2 a b \tanh (x)}{a+b \tanh (x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}-\frac {\left (i b \left (3 a^2+b^2\right )\right ) \int \frac {-i b-i a \tanh (x)}{a+b \tanh (x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac {a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}-\frac {b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {b}{2 \left (a^2-b^2\right ) (a+b \tanh (x))^2}+\frac {2 a b}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}\\ \end {align*}

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Mathematica [A]  time = 1.05, size = 119, normalized size = 1.14 \[ \frac {a x \left (a^2+3 b^2\right )}{(a-b)^3 (a+b)^3}+\frac {\left (-3 a^2 b-b^3\right ) \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {b^3}{2 (a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))^2}-\frac {3 b^2 \sinh (x)}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 + 3*b^2)*x)/((a - b)^3*(a + b)^3) + ((-3*a^2*b - b^3)*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - b^3/
(2*(a - b)^2*(a + b)^2*(a*Cosh[x] + b*Sinh[x])^2) - (3*b^2*Sinh[x])/((a - b)^2*(a + b)^2*(a*Cosh[x] + b*Sinh[x
]))

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fricas [B]  time = 0.48, size = 1269, normalized size = 12.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^4 + 4*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^
2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)*sinh(x)^3 + (a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*sinh(
x)^4 + 6*a^3*b^2 - 12*a^2*b^3 + 6*a*b^4 + 2*(3*a^3*b^2 - a^2*b^3 - 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 2*a^3*b^2
- 2*a^2*b^3 - 3*a*b^4 - b^5)*x)*cosh(x)^2 + 2*(3*a^3*b^2 - a^2*b^3 - 3*a*b^4 + b^5 + 3*(a^5 + 5*a^4*b + 10*a^3
*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^2 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5)*x)*si
nh(x)^2 + (a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*x - (3*a^4*b - 6*a^3*b^2 + 4*a^2*b^3 - 2*a*b^4 +
b^5 + (3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^4 + 4*(3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*
b^4 + b^5)*cosh(x)*sinh(x)^3 + (3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*sinh(x)^4 + 2*(3*a^4*b - 2*a^
2*b^3 - b^5)*cosh(x)^2 + 2*(3*a^4*b - 2*a^2*b^3 - b^5 + 3*(3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*co
sh(x)^2)*sinh(x)^2 + 4*((3*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + 2*a*b^4 + b^5)*cosh(x)^3 + (3*a^4*b - 2*a^2*b^3 - b
^5)*cosh(x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) + 4*((a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5)*x*cosh(x)^3 + (3*a^3*b^2 - a^2*b^3 - 3*a*b^4 + b^5 + (a^5 + 3*a^4*b + 2*a^3*b^2 - 2*
a^2*b^3 - 3*a*b^4 - b^5)*x)*cosh(x))*sinh(x))/(a^8 - 2*a^7*b - 2*a^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 +
2*a*b^7 - b^8 + (a^8 + 2*a^7*b - 2*a^6*b^2 - 6*a^5*b^3 + 6*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - b^8)*cosh(x)^4 + 4
*(a^8 + 2*a^7*b - 2*a^6*b^2 - 6*a^5*b^3 + 6*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - b^8)*cosh(x)*sinh(x)^3 + (a^8 + 2*
a^7*b - 2*a^6*b^2 - 6*a^5*b^3 + 6*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - b^8)*sinh(x)^4 + 2*(a^8 - 4*a^6*b^2 + 6*a^4*
b^4 - 4*a^2*b^6 + b^8)*cosh(x)^2 + 2*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8 + 3*(a^8 + 2*a^7*b - 2*a^6
*b^2 - 6*a^5*b^3 + 6*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - b^8)*cosh(x)^2)*sinh(x)^2 + 4*((a^8 + 2*a^7*b - 2*a^6*b^2
- 6*a^5*b^3 + 6*a^3*b^5 + 2*a^2*b^6 - 2*a*b^7 - b^8)*cosh(x)^3 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b
^8)*cosh(x))*sinh(x))

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giac [B]  time = 0.15, size = 251, normalized size = 2.41 \[ -\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {9 \, a^{3} b e^{\left (4 \, x\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, a b^{3} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 18 \, a^{3} b e^{\left (2 \, x\right )} - 6 \, a^{2} b^{2} e^{\left (2 \, x\right )} - 10 \, a b^{3} e^{\left (2 \, x\right )} - 2 \, b^{4} e^{\left (2 \, x\right )} + 9 \, a^{3} b - 15 \, a^{2} b^{2} + 3 \, a b^{3} + 3 \, b^{4}}{2 \, {\left (a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a^2*b + b^3)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) + x/(a^3 - 3*a^2*
b + 3*a*b^2 - b^3) + 1/2*(9*a^3*b*e^(4*x) + 9*a^2*b^2*e^(4*x) + 3*a*b^3*e^(4*x) + 3*b^4*e^(4*x) + 18*a^3*b*e^(
2*x) - 6*a^2*b^2*e^(2*x) - 10*a*b^3*e^(2*x) - 2*b^4*e^(2*x) + 9*a^3*b - 15*a^2*b^2 + 3*a*b^3 + 3*b^4)/((a^5 -
a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*(a*e^(2*x) + b*e^(2*x) + a - b)^2)

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maple [B]  time = 0.29, size = 494, normalized size = 4.75 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{3}}-\frac {6 b^{2} a^{3} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}+\frac {8 b^{4} a \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}-\frac {2 b^{6} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2} a}-\frac {10 b^{3} a^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}+\frac {12 b^{5} \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}-\frac {2 b^{7} \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2} a^{2}}-\frac {6 b^{2} a^{3} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}+\frac {8 b^{4} a \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}-\frac {2 b^{6} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2} a}-\frac {3 b \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right ) a^{2}}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {b^{3} \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a+b)^3*ln(tanh(1/2*x)-1)-6*b^2/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2*a^3*tanh(1/2*x)^3+8*b
^4/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2*a*tanh(1/2*x)^3-2*b^6/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x
)*b+a*tanh(1/2*x)^2)^2/a*tanh(1/2*x)^3-10*b^3/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2*a^2*tanh(1
/2*x)^2+12*b^5/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2*tanh(1/2*x)^2-2*b^7/(a-b)^3/(a+b)^3/(a+2*
tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2/a^2*tanh(1/2*x)^2-6*b^2/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2
*a^3*tanh(1/2*x)+8*b^4/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2*a*tanh(1/2*x)-2*b^6/(a-b)^3/(a+b)
^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)^2/a*tanh(1/2*x)-3*b/(a-b)^3/(a+b)^3*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^
2)*a^2-b^3/(a-b)^3/(a+b)^3*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+1/(a-b)^3*ln(tanh(1/2*x)+1)

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maxima [B]  time = 0.47, size = 292, normalized size = 2.81 \[ -\frac {{\left (3 \, a^{2} b + b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {2 \, {\left (3 \, a^{2} b^{2} + 3 \, a b^{3} + {\left (3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )}\right )}}{a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} + 2 \, {\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, x\right )} + {\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, x\right )}} + \frac {x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a^2*b + b^3)*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 2*(3*a^2*b^2 + 3*a*b^3 +
(3*a^2*b^2 - 2*a*b^3 - b^4)*e^(-2*x))/(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 -
b^7 + 2*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*e^(-2*x) + (a^7 - 3*a^6*b
+ a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*e^(-4*x)) + x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)

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mupad [B]  time = 1.63, size = 159, normalized size = 1.53 \[ \frac {x}{{\left (a-b\right )}^3}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (3\,a^2\,b+b^3\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {2\,\left (3\,a\,b^2-b^3\right )}{{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )}+\frac {2\,b^3}{{\left (a+b\right )}^3\,\left (a-b\right )\,\left ({\mathrm {e}}^{4\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2+2\,{\mathrm {e}}^{2\,x}\,\left (a+b\right )\,\left (a-b\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(a - b)^3 - (log(a - b + a*exp(2*x) + b*exp(2*x))*(3*a^2*b + b^3))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2) + (2*
(3*a*b^2 - b^3))/((a + b)^3*(a - b)^2*(a - b + exp(2*x)*(a + b))) + (2*b^3)/((a + b)^3*(a - b)*(exp(4*x)*(a +
b)^2 + (a - b)^2 + 2*exp(2*x)*(a + b)*(a - b)))

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sympy [A]  time = 4.39, size = 3840, normalized size = 36.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(log(sinh(x)) - cosh(x)**2/(2*sinh(x)**2)), Eq(a, 0) & Eq(b, 0)), (3*x*sinh(x)**3/(-24*b**3*sin
h(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) - 9*x*sinh(x)**2*cosh(
x)/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 9*x*
sinh(x)*cosh(x)**2/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*co
sh(x)**3) - 3*x*cosh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2 + 24
*b**3*cosh(x)**3) - sinh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(x)*cosh(x)**2
+ 24*b**3*cosh(x)**3) + 6*sinh(x)*cosh(x)**2/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3*sinh(
x)*cosh(x)**2 + 24*b**3*cosh(x)**3) - 9*cosh(x)**3/(-24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) - 72*b**3
*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3), Eq(a, -b)), (3*x*sinh(x)**3/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**
2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 9*x*sinh(x)**2*cosh(x)/(24*b**3*sinh(x)**3 + 72
*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 9*x*sinh(x)*cosh(x)**2/(24*b**3*
sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + 3*x*cosh(x)**3/(2
4*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) + sinh(x)**3
/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)**3) - 6*sinh(
x)*cosh(x)**2/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cosh(x)*
*3) - 9*cosh(x)**3/(24*b**3*sinh(x)**3 + 72*b**3*sinh(x)**2*cosh(x) + 72*b**3*sinh(x)*cosh(x)**2 + 24*b**3*cos
h(x)**3), Eq(a, b)), ((log(sinh(x)) - cosh(x)**2/(2*sinh(x)**2))/b**3, Eq(a, 0)), (2*a**5*x*cosh(x)**2/(2*a**8
*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x
)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(
x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + 4*a**4*b*x*sinh(x)*cosh(x)/(2
*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*s
inh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*
sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 6*a**4*b*log(cosh(x) + b
*sinh(x)/a)*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*co
sh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(
x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) +
3*a**4*b*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh
(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)
*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + 2
*a**3*b**2*x*sinh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*c
osh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh
(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2)
+ 6*a**3*b**2*x*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**
2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*s
inh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**
2) - 12*a**3*b**2*log(cosh(x) + b*sinh(x)/a)*sinh(x)*cosh(x)/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2
*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**
4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*
sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + 12*a**2*b**3*x*sinh(x)*cosh(x)/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*co
sh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**
2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 -
4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 6*a**2*b**3*log(cosh(x) + b*sinh(x)/a)*sinh(x)**2/(2*a**8*cosh
(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cos
h(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2
- 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 2*a**2*b**3*log(cosh(x) + b*sinh(x
)/a)*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**
2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh
(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 2*a**2
*b**3*sinh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)*
*2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cos
h(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 4*a**
2*b**3*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)
**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*co
sh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + 6*a*
b**4*x*sinh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)
**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*co
sh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) - 4*a*
b**4*log(cosh(x) + b*sinh(x)/a)*sinh(x)*cosh(x)/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2*a**6*b**2*si
nh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**4*b**4*cosh(x
)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*sinh(x)*cosh(
x) - 2*b**8*sinh(x)**2) - 2*b**5*log(cosh(x) + b*sinh(x)/a)*sinh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*c
osh(x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)*
*2 + 6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 -
4*a*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + 2*b**5*sinh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(
x) + 2*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 +
6*a**4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a
*b**7*sinh(x)*cosh(x) - 2*b**8*sinh(x)**2) + b**5*cosh(x)**2/(2*a**8*cosh(x)**2 + 4*a**7*b*sinh(x)*cosh(x) + 2
*a**6*b**2*sinh(x)**2 - 6*a**6*b**2*cosh(x)**2 - 12*a**5*b**3*sinh(x)*cosh(x) - 6*a**4*b**4*sinh(x)**2 + 6*a**
4*b**4*cosh(x)**2 + 12*a**3*b**5*sinh(x)*cosh(x) + 6*a**2*b**6*sinh(x)**2 - 2*a**2*b**6*cosh(x)**2 - 4*a*b**7*
sinh(x)*cosh(x) - 2*b**8*sinh(x)**2), True))

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