3.720 \(\int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx\)
Optimal. Leaf size=261 \[ -\frac {2 a b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {a^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a^2 \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {4 a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {2 a b^3 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {3 a^2 b^3 \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {2 a^4 b \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {2 a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))} \]
[Out]
-2*a^4*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(7/2)-3*a^2*b^3*arctan((b*cosh(x)+a*sinh(x))/
(a^2-b^2)^(1/2))/(a^2-b^2)^(7/2)-4*a^2*b^2*cosh(x)/(a^2-b^2)^3-a^2*cosh(x)/(a^2-b^2)^2+1/3*a^2*cosh(x)^3/(a^2-
b^2)^2+1/3*b^2*cosh(x)^3/(a^2-b^2)^2+2*a^3*b*sinh(x)/(a^2-b^2)^3+2*a*b^3*sinh(x)/(a^2-b^2)^3-2/3*a*b*sinh(x)^3
/(a^2-b^2)^2-a^3*b^2/(a^2-b^2)^3/(a*cosh(x)+b*sinh(x))
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Rubi [A] time = 1.03, antiderivative size = 261, normalized size of antiderivative = 1.00,
number of steps used = 33, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600,
Rules used = {3111, 3109, 2565, 30, 2564, 2637, 2638, 3074, 206, 2633, 3099, 3154}
\[ -\frac {2 a b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {2 a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {2 a b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {a^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a^2 \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {4 a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}-\frac {2 a^4 b \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {3 a^2 b^3 \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[(Cosh[x]^2*Sinh[x]^3)/(a*Cosh[x] + b*Sinh[x])^2,x]
[Out]
(-2*a^4*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - (3*a^2*b^3*ArcTan[(b*Cosh[x] +
a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - (4*a^2*b^2*Cosh[x])/(a^2 - b^2)^3 - (a^2*Cosh[x])/(a^2 - b^2)
^2 + (a^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) + (b^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) + (2*a^3*b*Sinh[x])/(a^2 - b^2)^3 +
(2*a*b^3*Sinh[x])/(a^2 - b^2)^3 - (2*a*b*Sinh[x]^3)/(3*(a^2 - b^2)^2) - (a^3*b^2)/((a^2 - b^2)^3*(a*Cosh[x] +
b*Sinh[x]))
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2564
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])
Rule 2565
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&& !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Rule 2633
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3074
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Dist[d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]
Rule 3099
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(a*Sin[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2)/
(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
Rule 3109
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m
- 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Rule 3111
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*
x] + b*Sin[c + d*x])^(p + 1), x], x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n*(a*Cos[c +
d*x] + b*Sin[c + d*x])^(p + 1), x], x] - Dist[(a*b)/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1
)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] &&
IGtQ[n, 0] && ILtQ[p, 0]
Rule 3154
Int[((A_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(
x_)])^2, x_Symbol] :> -Simp[(b*C + (a*C - c*A)*Cos[d + e*x] + b*A*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Co
s[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - 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, C}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - c*C, 0]
Rubi steps
\begin {align*} \int \frac {\cosh ^2(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac {a \int \frac {\cosh (x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac {b \int \frac {\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac {(a b) \int \frac {\cosh (x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac {a^2 \int \sinh ^3(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {(a b) \int \cosh (x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac {\left (a^2 b\right ) \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {b^2 \int \cosh ^2(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac {\left (a b^2\right ) \int \frac {\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+2 \left (\frac {a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {\left (a^4 b\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a^2 b^2\right ) \int \sinh (x) \, dx}{\left (a^2-b^2\right )^3}\right )-\frac {\left (a^2 b^3\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-2 \left (\frac {\left (a^2 b^2\right ) \int \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac {\left (a b^3\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (a^2 b^3\right ) \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-\frac {a^2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}-2 \frac {(i a b) \operatorname {Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {a^2 \cosh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {2 a b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+2 \left (-\frac {a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}-\frac {\left (i a^4 b\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )-\frac {\left (i a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}-2 \left (\frac {a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac {\left (i a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )\\ &=-\frac {a^2 b^3 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {a^2 \cosh (x)}{\left (a^2-b^2\right )^2}+\frac {a^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac {b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {2 a b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac {a^3 b^2}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+2 \left (-\frac {a^4 b \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac {a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac {a^3 b \sinh (x)}{\left (a^2-b^2\right )^3}\right )-2 \left (\frac {a^2 b^3 \tan ^{-1}\left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {a^2 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac {a b^3 \sinh (x)}{\left (a^2-b^2\right )^3}\right )\\ \end {align*}
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Mathematica [A] time = 3.94, size = 474, normalized size = 1.82 \[ \frac {1}{16} \left (\frac {32 a b \left (a^2+b^2\right ) \sinh (x)}{(a-b)^3 (a+b)^3}-\frac {4 \left (a^2+b^2\right ) \cosh (x)}{(a-b)^2 (a+b)^2}+\frac {4 \left (a^2+b^2\right ) \cosh (3 x)}{3 (a-b)^2 (a+b)^2}-\frac {6 b \left (3 a^2+b^2\right ) \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}-\frac {a \left (a^2+3 b^2\right )}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}-\frac {8 \left (a^4+6 a^2 b^2+b^4\right ) \cosh (x)}{(a-b)^3 (a+b)^3}-\frac {10 b \left (5 a^4+10 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac {a \left (a^4+10 a^2 b^2+5 b^4\right )}{(a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))}+\frac {2 \left (2 b^2 \sqrt {a+b} \sinh (x) \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )+2 a b \sqrt {a+b} \cosh (x) \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )+a \sqrt {a-b} (a+b)\right )}{(a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))}+\frac {8 a b \sinh (x)}{(a-b)^2 (a+b)^2}-\frac {8 a b \sinh (3 x)}{3 (a-b)^2 (a+b)^2}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(Cosh[x]^2*Sinh[x]^3)/(a*Cosh[x] + b*Sinh[x])^2,x]
[Out]
((-6*b*(3*a^2 + b^2)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(5/2)*(a + b)^(5/2)) - (10*
b*(5*a^4 + 10*a^2*b^2 + b^4)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(7/2)*(a + b)^(7/2)
) - (4*(a^2 + b^2)*Cosh[x])/((a - b)^2*(a + b)^2) - (8*(a^4 + 6*a^2*b^2 + b^4)*Cosh[x])/((a - b)^3*(a + b)^3)
+ (4*(a^2 + b^2)*Cosh[3*x])/(3*(a - b)^2*(a + b)^2) + (8*a*b*Sinh[x])/((a - b)^2*(a + b)^2) + (32*a*b*(a^2 + b
^2)*Sinh[x])/((a - b)^3*(a + b)^3) - (a*(a^2 + 3*b^2))/((a - b)^2*(a + b)^2*(a*Cosh[x] + b*Sinh[x])) - (a*(a^4
+ 10*a^2*b^2 + 5*b^4))/((a - b)^3*(a + b)^3*(a*Cosh[x] + b*Sinh[x])) + (2*(a*Sqrt[a - b]*(a + b) + 2*a*b*Sqrt
[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Cosh[x] + 2*b^2*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2
])/(Sqrt[a - b]*Sqrt[a + b])]*Sinh[x]))/((a - b)^(3/2)*(a + b)^2*(a*Cosh[x] + b*Sinh[x])) - (8*a*b*Sinh[3*x])/
(3*(a - b)^2*(a + b)^2))/16
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fricas [B] time = 0.63, size = 5061, normalized size = 19.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(x)^2*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x, algorithm="fricas")
[Out]
[1/24*((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)*cosh(x)^8 + 8*(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)*cosh(x)*sinh(x)^7 + (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)*sinh(x)^8 + 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*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7
)*cosh(x)^6 - 2*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7 - 14*(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)*cosh(x)^2)*sinh(x)^6 + 4*(14*(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)*cosh(x)^3 - 3*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 1
7*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7)*cosh(x))*sinh(x)^5 - 6*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7
*a*b^6)*cosh(x)^4 - 2*(9*a^7 + 81*a^5*b^2 - 69*a^3*b^4 - 21*a*b^6 - 35*(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)*cosh(x)^4 + 15*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*
a^2*b^5 + 6*a*b^6 - b^7)*cosh(x)^2)*sinh(x)^4 + 8*(7*(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)*cosh(x)^5 - 5*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 -
b^7)*cosh(x)^3 - 3*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x))*sinh(x)^3 - 2*(4*a^7 + 9*a^6*b - 2*a^
5*b^2 - 17*a^4*b^3 - 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7)*cosh(x)^2 - 2*(4*a^7 + 9*a^6*b - 2*a^5*b^2 - 17*a^
4*b^3 - 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7 - 14*(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)*cosh(x)^6 + 15*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 -
b^7)*cosh(x)^4 + 18*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x)^2)*sinh(x)^2 + 24*((2*a^5*b + 2*a^4*b^
2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^5 + 5*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)*sinh(x)^4 + (2*
a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*sinh(x)^5 + (2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4)*cosh(x)^3
+ (2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4 + 10*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^2)*
sinh(x)^3 + (10*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^3 + 3*(2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 -
3*a^2*b^4)*cosh(x))*sinh(x)^2 + (5*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^4 + 3*(2*a^5*b - 2*a
^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4)*cosh(x)^2)*sinh(x))*sqrt(-a^2 + b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x
)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)
*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 4*(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)*cosh(x)^7 - 3*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6
- b^7)*cosh(x)^5 - 6*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x)^3 - (4*a^7 + 9*a^6*b - 2*a^5*b^2 - 17
*a^4*b^3 - 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7)*cosh(x))*sinh(x))/((a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*
a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^5 + 5*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3
+ 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)*sinh(x)^4 + (a^9 + a^8*b - 4*a^7*b^2 -
4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*sinh(x)^5 + (a^9 - a^8*b - 4*a^7*b^2
+ 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^3 + (a^9 - a^8*b - 4*a^7*b^
2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9 + 10*(a^9 + a^8*b - 4*a^7*b^2 - 4*
a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^2)*sinh(x)^3 + (10*(a^9 + a^8*b
- 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^3 + 3*(a^9 - a
^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x))*sinh(x)^2
+ (5*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh
(x)^4 + 3*(a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*
cosh(x)^2)*sinh(x)), 1/24*((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)*cosh(x)
^8 + 8*(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)*cosh(x)*sinh(x)^7 + (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)*sinh(x)^8 + 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*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2
*b^5 + 6*a*b^6 - b^7)*cosh(x)^6 - 2*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^
6 - b^7 - 14*(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)*cosh(x)^2)*sinh(x)^6
+ 4*(14*(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)*cosh(x)^3 - 3*(4*a^7 - 9*a
^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7)*cosh(x))*sinh(x)^5 - 6*(3*a^7 + 27*a^5*
b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x)^4 - 2*(9*a^7 + 81*a^5*b^2 - 69*a^3*b^4 - 21*a*b^6 - 35*(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)*cosh(x)^4 + 15*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*
b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7)*cosh(x)^2)*sinh(x)^4 + 8*(7*(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)*cosh(x)^5 - 5*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7
*a^2*b^5 + 6*a*b^6 - b^7)*cosh(x)^3 - 3*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x))*sinh(x)^3 - 2*(4*
a^7 + 9*a^6*b - 2*a^5*b^2 - 17*a^4*b^3 - 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7)*cosh(x)^2 - 2*(4*a^7 + 9*a^6*b
- 2*a^5*b^2 - 17*a^4*b^3 - 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7 - 14*(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)*cosh(x)^6 + 15*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*
a^2*b^5 + 6*a*b^6 - b^7)*cosh(x)^4 + 18*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x)^2)*sinh(x)^2 + 48*
((2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^5 + 5*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cos
h(x)*sinh(x)^4 + (2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*sinh(x)^5 + (2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 -
3*a^2*b^4)*cosh(x)^3 + (2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4 + 10*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*
a^2*b^4)*cosh(x)^2)*sinh(x)^3 + (10*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^3 + 3*(2*a^5*b - 2*a
^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4)*cosh(x))*sinh(x)^2 + (5*(2*a^5*b + 2*a^4*b^2 + 3*a^3*b^3 + 3*a^2*b^4)*cosh(x)^
4 + 3*(2*a^5*b - 2*a^4*b^2 + 3*a^3*b^3 - 3*a^2*b^4)*cosh(x)^2)*sinh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)
/((a + b)*cosh(x) + (a + b)*sinh(x))) + 4*(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)*cosh(x)^7 - 3*(4*a^7 - 9*a^6*b - 2*a^5*b^2 + 17*a^4*b^3 - 8*a^3*b^4 - 7*a^2*b^5 + 6*a*b^6 - b^7)*co
sh(x)^5 - 6*(3*a^7 + 27*a^5*b^2 - 23*a^3*b^4 - 7*a*b^6)*cosh(x)^3 - (4*a^7 + 9*a^6*b - 2*a^5*b^2 - 17*a^4*b^3
- 8*a^3*b^4 + 7*a^2*b^5 + 6*a*b^6 + b^7)*cosh(x))*sinh(x))/((a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 +
6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^5 + 5*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b
^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)*sinh(x)^4 + (a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3
+ 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*sinh(x)^5 + (a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b
^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^3 + (a^9 - a^8*b - 4*a^7*b^2 + 4*a^6
*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9 + 10*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 +
6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^2)*sinh(x)^3 + (10*(a^9 + a^8*b - 4*a^7*
b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^3 + 3*(a^9 - a^8*b - 4*
a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x))*sinh(x)^2 + (5*(a^
9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^4 + 3
*(a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^2
)*sinh(x))]
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giac [A] time = 0.15, size = 310, normalized size = 1.19 \[ -\frac {2 \, a^{3} b^{2} e^{x}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} - \frac {{\left (9 \, a e^{\left (2 \, x\right )} + 3 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} - \frac {2 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {a^{4} e^{\left (3 \, x\right )} + 4 \, a^{3} b e^{\left (3 \, x\right )} + 6 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 4 \, a b^{3} e^{\left (3 \, x\right )} + b^{4} e^{\left (3 \, x\right )} - 9 \, a^{4} e^{x} - 24 \, a^{3} b e^{x} - 18 \, a^{2} b^{2} e^{x} + 3 \, b^{4} e^{x}}{24 \, {\left (a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(x)^2*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x, algorithm="giac")
[Out]
-2*a^3*b^2*e^x/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*e^(2*x) + b*e^(2*x) + a - b)) - 1/24*(9*a*e^(2*x) + 3*b
*e^(2*x) - a + b)*e^(-3*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 2*(2*a^4*b + 3*a^2*b^3)*arctan((a*e^x + b*e^x)/sq
rt(a^2 - b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(a^2 - b^2)) + 1/24*(a^4*e^(3*x) + 4*a^3*b*e^(3*x) + 6
*a^2*b^2*e^(3*x) + 4*a*b^3*e^(3*x) + b^4*e^(3*x) - 9*a^4*e^x - 24*a^3*b*e^x - 18*a^2*b^2*e^x + 3*b^4*e^x)/(a^6
+ 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)
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maple [A] time = 0.27, size = 326, normalized size = 1.25 \[ -\frac {1}{3 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {a}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {b}{2 \left (a +b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2 a^{2} b^{3} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {2 a^{3} b^{2}}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {4 a^{4} b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {a^{2}-b^{2}}}-\frac {6 a^{2} b^{3} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {a^{2}-b^{2}}}+\frac {1}{3 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a}{2 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b}{2 \left (a -b \right )^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(x)^2*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x)
[Out]
-1/3/(a+b)^2/(tanh(1/2*x)-1)^3-1/2/(a+b)^2/(tanh(1/2*x)-1)^2+1/2/(a+b)^3/(tanh(1/2*x)-1)*a-1/2/(a+b)^3/(tanh(1
/2*x)-1)*b-2*a^2/(a-b)^3/(a+b)^3*b^3*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)-2*a^3*b^2/(a-b)^3/(a+b)^3
/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)-4*a^4*b/(a-b)^3/(a+b)^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/
(a^2-b^2)^(1/2))-6*a^2*b^3/(a-b)^3/(a+b)^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))+1
/3/(a-b)^2/(tanh(1/2*x)+1)^3-1/2/(a-b)^2/(tanh(1/2*x)+1)^2-1/2/(a-b)^3/(tanh(1/2*x)+1)*a-1/2/(a-b)^3/(tanh(1/2
*x)+1)*b
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(x)^2*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,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(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?
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mupad [B] time = 1.95, size = 592, normalized size = 2.27 \[ \frac {{\mathrm {e}}^{3\,x}}{24\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-3\,x}}{24\,{\left (a-b\right )}^2}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\left (2\,a^4\,b\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}+3\,a^2\,b^3\,\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}\right )}{a^7\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+b^7\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-3\,a^2\,b^5\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+3\,a^3\,b^4\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}+3\,a^4\,b^3\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-3\,a^5\,b^2\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-a\,b^6\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}-a^6\,b\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}}\right )\,\sqrt {4\,a^8\,b^2+12\,a^6\,b^4+9\,a^4\,b^6}}{\sqrt {a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}-\frac {{\mathrm {e}}^x\,\left (3\,a-b\right )}{8\,{\left (a+b\right )}^3}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a+b\right )}{8\,{\left (a-b\right )}^3}-\frac {2\,a^3\,b^2\,{\mathrm {e}}^x}{{\left (a+b\right )}^3\,{\left (a-b\right )}^3\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((cosh(x)^2*sinh(x)^3)/(a*cosh(x) + b*sinh(x))^2,x)
[Out]
exp(3*x)/(24*(a + b)^2) + exp(-3*x)/(24*(a - b)^2) - (2*atan((exp(x)*(2*a^4*b*(a^14 - b^14 + 7*a^2*b^12 - 21*a
^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)^(1/2) + 3*a^2*b^3*(a^14 - b^14 + 7*a^2*b^12 - 21
*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)^(1/2)))/(a^7*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b
^2)^(1/2) + b^7*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - 3*a^2*b^5*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1
/2) + 3*a^3*b^4*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) + 3*a^4*b^3*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1
/2) - 3*a^5*b^2*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2) - a*b^6*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2)
- a^6*b*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2)))*(9*a^4*b^6 + 12*a^6*b^4 + 4*a^8*b^2)^(1/2))/(a^14 - b^14
+ 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)^(1/2) - (exp(x)*(3*a - b))/(8
*(a + b)^3) - (exp(-x)*(3*a + b))/(8*(a - b)^3) - (2*a^3*b^2*exp(x))/((a + b)^3*(a - b)^3*(a - b + exp(2*x)*(a
+ b)))
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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(cosh(x)**2*sinh(x)**3/(a*cosh(x)+b*sinh(x))**2,x)
[Out]
Timed out
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