3.828 \(\int \frac {\sinh ^2(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx\)

Optimal. Leaf size=309 \[ -\frac {\sqrt {2} \left (\frac {b^2-2 a c}{\sqrt {4 a c-b^2}}+i b\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (-\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {\sqrt {2} \left (-\frac {b^2-2 a c}{\sqrt {4 a c-b^2}}+i b\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}+\frac {x}{c} \]

[Out]

x/c-arctan(1/2*(2*I*c-(I*b+(4*a*c-b^2)^(1/2))*tanh(1/2*x))*2^(1/2)/(b^2-2*(a-c)*c-I*b*(4*a*c-b^2)^(1/2))^(1/2)
)*2^(1/2)*(I*b+(2*a*c-b^2)/(4*a*c-b^2)^(1/2))/c/(b^2-2*(a-c)*c-I*b*(4*a*c-b^2)^(1/2))^(1/2)-arctan(1/2*(2*I*c-
(I*b-(4*a*c-b^2)^(1/2))*tanh(1/2*x))*2^(1/2)/(b^2-2*(a-c)*c+I*b*(4*a*c-b^2)^(1/2))^(1/2))*2^(1/2)*(I*b+(-2*a*c
+b^2)/(4*a*c-b^2)^(1/2))/c/(b^2-2*(a-c)*c+I*b*(4*a*c-b^2)^(1/2))^(1/2)

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Rubi [A]  time = 1.08, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3256, 3292, 2660, 618, 204} \[ -\frac {\sqrt {2} \left (\frac {b^2-2 a c}{\sqrt {4 a c-b^2}}+i b\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (-\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {\sqrt {2} \left (-\frac {b^2-2 a c}{\sqrt {4 a c-b^2}}+i b\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}+\frac {x}{c} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^2/(a + b*Sinh[x] + c*Sinh[x]^2),x]

[Out]

x/c - (Sqrt[2]*(I*b + (b^2 - 2*a*c)/Sqrt[-b^2 + 4*a*c])*ArcTan[((2*I)*c - (I*b - Sqrt[-b^2 + 4*a*c])*Tanh[x/2]
)/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]])])/(c*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a
*c]]) - (Sqrt[2]*(I*b - (b^2 - 2*a*c)/Sqrt[-b^2 + 4*a*c])*ArcTan[((2*I)*c - (I*b + Sqrt[-b^2 + 4*a*c])*Tanh[x/
2])/(Sqrt[2]*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]])])/(c*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4
*a*c]])

Rule 204

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

Rule 618

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 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3256

Int[sin[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)]^(n_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]
^(n2_.))^(p_), x_Symbol] :> Int[ExpandTrig[sin[d + e*x]^m*(a + b*sin[d + e*x]^n + c*sin[d + e*x]^(2*n))^p, x],
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegersQ[m, n, p]

Rule 3292

Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)] + (c_.)*sin[(d_.) + (e_.)*(x
_)]^2), x_Symbol] :> Module[{q = Rt[b^2 - 4*a*c, 2]}, Dist[B + (b*B - 2*A*c)/q, Int[1/(b + q + 2*c*Sin[d + e*x
]), x], x] + Dist[B - (b*B - 2*A*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]), x], x]] /; FreeQ[{a, b, c, d, e, A, B
}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {\sinh ^2(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx &=-\int \left (-\frac {1}{c}+\frac {-a-b \sinh (x)}{c \left (-a-b \sinh (x)-c \sinh ^2(x)\right )}\right ) \, dx\\ &=\frac {x}{c}-\frac {\int \frac {-a-b \sinh (x)}{-a-b \sinh (x)-c \sinh ^2(x)} \, dx}{c}\\ &=\frac {x}{c}-\frac {\left (i b-\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{i b+\sqrt {-b^2+4 a c}+2 i c \sinh (x)} \, dx}{c}-\frac {\left (i b+\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{i b-\sqrt {-b^2+4 a c}+2 i c \sinh (x)} \, dx}{c}\\ &=\frac {x}{c}-\frac {\left (2 \left (i b-\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{i b+\sqrt {-b^2+4 a c}+4 i c x-\left (i b+\sqrt {-b^2+4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{c}-\frac {\left (2 \left (i b+\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{i b-\sqrt {-b^2+4 a c}+4 i c x-\left (i b-\sqrt {-b^2+4 a c}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{c}\\ &=\frac {x}{c}+\frac {\left (4 \left (i b-\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-8 \left (b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}\right )-x^2} \, dx,x,4 i c+2 \left (-i b-\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )\right )}{c}+\frac {\left (4 \left (i b+\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-8 \left (b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}\right )-x^2} \, dx,x,4 i c+2 \left (-i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )\right )}{c}\\ &=\frac {x}{c}-\frac {\sqrt {2} \left (i b+\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right ) \tan ^{-1}\left (\frac {2 i c-\left (i b-\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}\right )}{c \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}-\frac {\sqrt {2} \left (i b-\frac {b^2-2 a c}{\sqrt {-b^2+4 a c}}\right ) \tan ^{-1}\left (\frac {2 i c-\left (i b+\sqrt {-b^2+4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\right )}{c \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\\ \end {align*}

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Mathematica [A]  time = 0.61, size = 283, normalized size = 0.92 \[ \frac {-\frac {\sqrt {2} \left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}-b\right )+2 c}{\sqrt {2 b \sqrt {b^2-4 a c}+4 c (a-c)-2 b^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}-\frac {\sqrt {2} \left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \tan ^{-1}\left (\frac {2 c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}+x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^2/(a + b*Sinh[x] + c*Sinh[x]^2),x]

[Out]

(x - (Sqrt[2]*(-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (-b + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/Sqrt[-2*b
^2 + 4*(a - c)*c + 2*b*Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b^2 + 2*(a - c)*c + b*Sqrt[b^2 - 4*a*c]])
- (Sqrt[2]*(b^2 - 2*a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(2*c - (b + Sqrt[b^2 - 4*a*c])*Tanh[x/2])/(Sqrt[2]*Sqrt
[-b^2 + 2*(a - c)*c - b*Sqrt[b^2 - 4*a*c]])])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b^2 + 2*(a - c)*c - b*Sqrt[b^2 - 4*a*c]
]))/c

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fricas [B]  time = 0.84, size = 4943, normalized size = 16.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="fricas")

[Out]

1/2*(sqrt(2)*c*sqrt(-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c + (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^
3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)
*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3
*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^
3 - (a^2*b^2 + b^4)*c^2))*log(-2*a^4*b^2 - 2*a^2*b^4 + 4*a^3*b^2*c + sqrt(2)*(8*a^2*b^2*c^3 - 2*(2*a^3*b^2 + 3
*a*b^4)*c^2 + (a^2*b^4 + b^6)*c + (8*a^2*c^7 - 6*(4*a^3 + a*b^2)*c^6 + (24*a^4 + 22*a^2*b^2 + b^4)*c^5 - 2*(4*
a^5 + 9*a^3*b^2 + 4*a*b^4)*c^4 + (2*a^4*b^2 + 3*a^2*b^4 + b^6)*c^3)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b
^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2
+ b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))*sqrt(-(a^2*b^2 + b^4 + 2*
a^2*c^2 - 2*(a^3 + 2*a*b^2)*c + (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*
sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(
2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b
^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)) + 4*(2*a^3*b*c
^2 - (a^4*b + a^2*b^3)*c)*cosh(x) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*sinh(x) - 2*(4*a^3*c^5 - (8*a^4 + a^
2*b^2)*c^4 + 2*(2*a^5 + 3*a^3*b^2)*c^3 - (a^4*b^2 + a^2*b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2
*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 +
b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4))) - sqrt(2)*c*sqrt(-(a^2*b^2
+ b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c + (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 +
b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*
c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2
+ 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2))*log(
-2*a^4*b^2 - 2*a^2*b^4 + 4*a^3*b^2*c - sqrt(2)*(8*a^2*b^2*c^3 - 2*(2*a^3*b^2 + 3*a*b^4)*c^2 + (a^2*b^4 + b^6)*
c + (8*a^2*c^7 - 6*(4*a^3 + a*b^2)*c^6 + (24*a^4 + 22*a^2*b^2 + b^4)*c^5 - 2*(4*a^5 + 9*a^3*b^2 + 4*a*b^4)*c^4
+ (2*a^4*b^2 + 3*a^2*b^4 + b^6)*c^3)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c
)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b
^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))*sqrt(-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c
+ (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b
^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4
+ 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8
*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*cosh
(x) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*sinh(x) - 2*(4*a^3*c^5 - (8*a^4 + a^2*b^2)*c^4 + 2*(2*a^5 + 3*a^3*
b^2)*c^3 - (a^4*b^2 + a^2*b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/
(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2
+ 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4))) + sqrt(2)*c*sqrt(-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2
*a*b^2)*c - (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a
^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 -
2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*
a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2))*log(-2*a^4*b^2 - 2*a^2*b^4 + 4*a^3
*b^2*c + sqrt(2)*(8*a^2*b^2*c^3 - 2*(2*a^3*b^2 + 3*a*b^4)*c^2 + (a^2*b^4 + b^6)*c - (8*a^2*c^7 - 6*(4*a^3 + a*
b^2)*c^6 + (24*a^4 + 22*a^2*b^2 + b^4)*c^5 - 2*(4*a^5 + 9*a^3*b^2 + 4*a*b^4)*c^4 + (2*a^4*b^2 + 3*a^2*b^4 + b^
6)*c^3)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^
8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 +
2*a^2*b^4 + b^6)*c^4)))*sqrt(-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c - (4*a*c^5 - (8*a^2 + b^2)*c^4
+ 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^
2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a
^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3
*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*cosh(x) + 4*(2*a^3*b*c^2 - (a^4*b
+ a^2*b^3)*c)*sinh(x) + 2*(4*a^3*c^5 - (8*a^4 + a^2*b^2)*c^4 + 2*(2*a^5 + 3*a^3*b^2)*c^3 - (a^4*b^2 + a^2*b^4)
*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8
+ 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2
*a^2*b^4 + b^6)*c^4))) - sqrt(2)*c*sqrt(-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c - (4*a*c^5 - (8*a^2
+ b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 -
4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*
c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*
(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*c^2))*log(-2*a^4*b^2 - 2*a^2*b^4 + 4*a^3*b^2*c - sqrt(2)*(8*a^2*b^2*c^
3 - 2*(2*a^3*b^2 + 3*a*b^4)*c^2 + (a^2*b^4 + b^6)*c - (8*a^2*c^7 - 6*(4*a^3 + a*b^2)*c^6 + (24*a^4 + 22*a^2*b^
2 + b^4)*c^5 - 2*(4*a^5 + 9*a^3*b^2 + 4*a*b^4)*c^4 + (2*a^4*b^2 + 3*a^2*b^4 + b^6)*c^3)*sqrt(-(a^4*b^2 + 2*a^2
*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2
*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))*sqrt(
-(a^2*b^2 + b^4 + 2*a^2*c^2 - 2*(a^3 + 2*a*b^2)*c - (4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (
a^2*b^2 + b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a
^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5
- (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4)))/(4*a*c^5 - (8*a^2 + b^2)*c^4 + 2*(2*a^3 + 3*a*b^2)*c^3 - (a^2*b^2 + b^4)*
c^2)) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*cosh(x) + 4*(2*a^3*b*c^2 - (a^4*b + a^2*b^3)*c)*sinh(x) + 2*(4*a
^3*c^5 - (8*a^4 + a^2*b^2)*c^4 + 2*(2*a^5 + 3*a^3*b^2)*c^3 - (a^4*b^2 + a^2*b^4)*c^2)*sqrt(-(a^4*b^2 + 2*a^2*b
^4 + b^6 + 4*a^2*b^2*c^2 - 4*(a^3*b^2 + a*b^4)*c)/(4*a*c^9 - (16*a^2 + b^2)*c^8 + 12*(2*a^3 + a*b^2)*c^7 - 2*(
8*a^4 + 11*a^2*b^2 + b^4)*c^6 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^4))) + 2*x)/
c

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giac [A]  time = 2.11, size = 5, normalized size = 0.02 \[ \frac {x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="giac")

[Out]

x/c

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maple [C]  time = 0.25, size = 108, normalized size = 0.35 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{c}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{c}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 b \textit {\_Z} +a \right )}{\sum }\frac {\left (\textit {\_R}^{2} a -2 \textit {\_R} b -a \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 b \,\textit {\_R}^{2}-2 \textit {\_R} a +4 \textit {\_R} c +b}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(a+b*sinh(x)+c*sinh(x)^2),x)

[Out]

-1/c*ln(tanh(1/2*x)-1)+1/c*ln(tanh(1/2*x)+1)+1/c*sum((_R^2*a-2*_R*b-a)/(2*_R^3*a-3*_R^2*b-2*_R*a+4*_R*c+b)*ln(
tanh(1/2*x)-_R),_R=RootOf(a*_Z^4-2*b*_Z^3+(-2*a+4*c)*_Z^2+2*b*_Z+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x}{c} - \frac {1}{4} \, \int \frac {8 \, {\left (b e^{\left (3 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - b e^{x}\right )}}{c^{2} e^{\left (4 \, x\right )} + 2 \, b c e^{\left (3 \, x\right )} - 2 \, b c e^{x} + c^{2} + 2 \, {\left (2 \, a c - c^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="maxima")

[Out]

x/c - 1/4*integrate(8*(b*e^(3*x) + 2*a*e^(2*x) - b*e^x)/(c^2*e^(4*x) + 2*b*c*e^(3*x) - 2*b*c*e^x + c^2 + 2*(2*
a*c - c^2)*e^(2*x)), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(a + c*sinh(x)^2 + b*sinh(x)),x)

[Out]

\text{Hanged}

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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(sinh(x)**2/(a+b*sinh(x)+c*sinh(x)**2),x)

[Out]

Timed out

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