3.829 \(\int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx\)

Optimal. Leaf size=363 \[ \frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}+i \left (\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}-i \left (-\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {b x}{c^2}+\frac {\cosh (x)}{c} \]

[Out]

-b*x/c^2+cosh(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-3*I*a*b*c/(4*a*c-b^2)^(1/2))+b^3/(4*a*c-b^2)^(1/2))/c^2/(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*tanh(1/2*x)+(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+3*I*a*b*c/(4*a*c-b^2)^(1/2))+b^3/(4*a*c-b^2)^(1/2))/c^
2/(b^2-2*(a-c)*c+I*b*(4*a*c-b^2)^(1/2))^(1/2)

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Rubi [A]  time = 4.68, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3256, 2638, 3292, 2660, 618, 204} \[ \frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}+i \left (\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {4 a c-b^2}}-i \left (-\frac {3 i a b c}{\sqrt {4 a c-b^2}}-a c+b^2\right )\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{c^2 \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}-\frac {b x}{c^2}+\frac {\cosh (x)}{c} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(a + b*Sinh[x] + c*Sinh[x]^2),x]

[Out]

-((b*x)/c^2) + (Sqrt[2]*(b^3/Sqrt[-b^2 + 4*a*c] + I*(b^2 - a*c + ((3*I)*a*b*c)/Sqrt[-b^2 + 4*a*c]))*ArcTan[((2
*I)*c - I*b*Tanh[x/2] + 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^2*Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*c]]) - (Sqrt[2]*(b^3/Sqrt[-b^2 + 4*a*c] - I*(b^2 - a*c
- ((3*I)*a*b*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^2*Sqrt[b^2 - 2*(a - c)*c - I*b*Sqrt[-b^2 + 4*a*c]]) + Cosh[x]/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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx &=i \int \left (\frac {i b}{c^2}-\frac {i \sinh (x)}{c}+\frac {-i a b-i b^2 \left (1-\frac {a c}{b^2}\right ) \sinh (x)}{c^2 \left (a+b \sinh (x)+c \sinh ^2(x)\right )}\right ) \, dx\\ &=-\frac {b x}{c^2}+\frac {i \int \frac {-i a b-i b^2 \left (1-\frac {a c}{b^2}\right ) \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx}{c^2}+\frac {\int \sinh (x) \, dx}{c}\\ &=-\frac {b x}{c^2}+\frac {\cosh (x)}{c}-\frac {\left (i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \int \frac {1}{-i b-\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx}{c^2}+\frac {\left (i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\right ) \int \frac {1}{-i b+\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx}{c^2}\\ &=-\frac {b x}{c^2}+\frac {\cosh (x)}{c}-\frac {\left (2 i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b 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^2}+\frac {\left (2 i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b 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^2}\\ &=-\frac {b x}{c^2}+\frac {\cosh (x)}{c}+\frac {\left (4 i \left (b^2-a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b 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^2}-\frac {\left (4 i \left (-b^2+a c+\frac {i b^3}{\sqrt {-b^2+4 a c}}-\frac {3 i a b 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^2}\\ &=-\frac {b x}{c^2}+\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}+i \left (b^2-a c+\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\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^2 \sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}-\frac {\sqrt {2} \left (\frac {b^3}{\sqrt {-b^2+4 a c}}-i \left (b^2-a c-\frac {3 i a b c}{\sqrt {-b^2+4 a c}}\right )\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^2 \sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}+\frac {\cosh (x)}{c}\\ \end {align*}

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Mathematica [A]  time = 0.94, size = 326, normalized size = 0.90 \[ \frac {\frac {\sqrt {2} \left (b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}+3 a b c-b^3\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}-b\right )+2 c}{\sqrt {2 b \sqrt {b^2-4 a c}+4 c (a-c)-2 b^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}+\frac {\sqrt {2} \left (b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}-3 a b c+b^3\right ) \tan ^{-1}\left (\frac {2 c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}-b x+c \cosh (x)}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^3/(a + b*Sinh[x] + c*Sinh[x]^2),x]

[Out]

(-(b*x) + (Sqrt[2]*(-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*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^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*A
rcTan[(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]])])/(Sq
rt[b^2 - 4*a*c]*Sqrt[-b^2 + 2*(a - c)*c - b*Sqrt[b^2 - 4*a*c]]) + c*Cosh[x])/c^2

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fricas [B]  time = 1.39, size = 6680, normalized size = 18.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="fricas")

[Out]

-1/2*(2*b*x*cosh(x) - c*cosh(x)^2 - sqrt(2)*(c^2*cosh(x) + c^2*sinh(x))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*
a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c + (4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a
^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6
*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 1
2*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*
a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4))*log(2*a^5
*b^4 + 2*a^3*b^6 + 6*a^5*b^2*c^2 - 4*(a^6*b^2 + 2*a^4*b^4)*c + sqrt(2)*(12*a^4*b*c^5 - (20*a^5*b + 31*a^3*b^3)
*c^4 + (8*a^6*b + 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 + 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 + 2*a^2*b^
7 + b^9)*c + (12*a^2*b*c^9 - 7*(4*a^3*b + a*b^3)*c^8 + (20*a^4*b + 27*a^2*b^3 + b^5)*c^7 - (4*a^5*b + 13*a^3*b
^3 + 9*a*b^5)*c^6 + (a^4*b^3 + 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a
^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(
4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*
b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)
*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c + (4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^
4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^
4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2
)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c
^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)) + 4*(3*a^5*b*c^3 - 2*(a^6
*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*cosh(x) + 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5*b^3 +
a^3*b^5)*c)*sinh(x) + 2*(4*a^4*c^7 - (8*a^5 + a^3*b^2)*c^6 + 2*(2*a^6 + 3*a^4*b^2)*c^5 - (a^5*b^2 + a^3*b^4)*
c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*
b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b
^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)
*c^8))) + sqrt(2)*(c^2*cosh(x) + c^2*sinh(x))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(
2*a^3*b^2 + 3*a*b^4)*c + (4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(
a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2
*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2
*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a
*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4))*log(2*a^5*b^4 + 2*a^3*b^6 + 6*a^5*b
^2*c^2 - 4*(a^6*b^2 + 2*a^4*b^4)*c - sqrt(2)*(12*a^4*b*c^5 - (20*a^5*b + 31*a^3*b^3)*c^4 + (8*a^6*b + 33*a^4*b
^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 + 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 + 2*a^2*b^7 + b^9)*c + (12*a^2*b*c^9
- 7*(4*a^3*b + a*b^3)*c^8 + (20*a^4*b + 27*a^2*b^3 + b^5)*c^7 - (4*a^5*b + 13*a^3*b^3 + 9*a*b^5)*c^6 + (a^4*b
^3 + 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 +
2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*
c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*
b^2 + 2*a^2*b^4 + b^6)*c^8)))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*
b^4)*c + (4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*
b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a
^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2
*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 +
b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)) + 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5
*b^3 + a^3*b^5)*c)*cosh(x) + 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*sinh(x) + 2*(
4*a^4*c^7 - (8*a^5 + a^3*b^2)*c^6 + 2*(2*a^6 + 3*a^4*b^2)*c^5 - (a^5*b^2 + a^3*b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^
2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*
(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a
^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8))) - sqrt(2)*(c^2*cos
h(x) + c^2*sinh(x))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c - (
4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10
+ 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3
*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4
)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 +
2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4))*log(2*a^5*b^4 + 2*a^3*b^6 + 6*a^5*b^2*c^2 - 4*(a^6*b^2 + 2*a^
4*b^4)*c + sqrt(2)*(12*a^4*b*c^5 - (20*a^5*b + 31*a^3*b^3)*c^4 + (8*a^6*b + 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(
2*a^5*b^3 + 5*a^3*b^5 + 3*a*b^7)*c^2 + (a^4*b^5 + 2*a^2*b^7 + b^9)*c - (12*a^2*b*c^9 - 7*(4*a^3*b + a*b^3)*c^8
+ (20*a^4*b + 27*a^2*b^3 + b^5)*c^7 - (4*a^5*b + 13*a^3*b^3 + 9*a*b^5)*c^6 + (a^4*b^3 + 2*a^2*b^5 + b^7)*c^5)
*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4
+ 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*
c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8
)))*sqrt(-(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c - (4*a*c^7 - (8*a^2
+ b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4
- 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b
^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5
+ 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*
b^2)*c^5 - (a^2*b^2 + b^4)*c^4)) + 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*cosh(x)
+ 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*sinh(x) - 2*(4*a^4*c^7 - (8*a^5 + a^3*b
^2)*c^6 + 2*(2*a^6 + 3*a^4*b^2)*c^5 - (a^5*b^2 + a^3*b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c
^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a
*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^
5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8))) + sqrt(2)*(c^2*cosh(x) + c^2*sinh(x))*sqrt(-
(a^2*b^4 + b^6 - 2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c - (4*a*c^7 - (8*a^2 + b^2)*c^
6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5
*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*
a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^
2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 -
(a^2*b^2 + b^4)*c^4))*log(2*a^5*b^4 + 2*a^3*b^6 + 6*a^5*b^2*c^2 - 4*(a^6*b^2 + 2*a^4*b^4)*c - sqrt(2)*(12*a^4
*b*c^5 - (20*a^5*b + 31*a^3*b^3)*c^4 + (8*a^6*b + 33*a^4*b^3 + 27*a^2*b^5)*c^3 - 3*(2*a^5*b^3 + 5*a^3*b^5 + 3*
a*b^7)*c^2 + (a^4*b^5 + 2*a^2*b^7 + b^9)*c - (12*a^2*b*c^9 - 7*(4*a^3*b + a*b^3)*c^8 + (20*a^4*b + 27*a^2*b^3
+ b^5)*c^7 - (4*a^5*b + 13*a^3*b^3 + 9*a*b^5)*c^6 + (a^4*b^3 + 2*a^2*b^5 + b^7)*c^5)*sqrt(-(a^4*b^6 + 2*a^2*b^
8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5
*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b
^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))*sqrt(-(a^2*b^4 + b^6 -
2*a^3*c^3 + (2*a^4 + 9*a^2*b^2)*c^2 - 2*(2*a^3*b^2 + 3*a*b^4)*c - (4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3
*a*b^2)*c^5 - (a^2*b^2 + b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b^4
)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a^2
+ b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c^9
- (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8)))/(4*a*c^7 - (8*a^2 + b^2)*c^6 + 2*(2*a^3 + 3*a*b^2)*c^5 - (a^2*b^2 + b^4)
*c^4)) + 4*(3*a^5*b*c^3 - 2*(a^6*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*cosh(x) + 4*(3*a^5*b*c^3 - 2*(a^6
*b + 2*a^4*b^3)*c^2 + (a^5*b^3 + a^3*b^5)*c)*sinh(x) - 2*(4*a^4*c^7 - (8*a^5 + a^3*b^2)*c^6 + 2*(2*a^6 + 3*a^4
*b^2)*c^5 - (a^5*b^2 + a^3*b^4)*c^4)*sqrt(-(a^4*b^6 + 2*a^2*b^8 + b^10 + 9*a^4*b^2*c^4 - 12*(a^5*b^2 + 2*a^3*b
^4)*c^3 + 2*(2*a^6*b^2 + 11*a^4*b^4 + 11*a^2*b^6)*c^2 - 4*(a^5*b^4 + 3*a^3*b^6 + 2*a*b^8)*c)/(4*a*c^13 - (16*a
^2 + b^2)*c^12 + 12*(2*a^3 + a*b^2)*c^11 - 2*(8*a^4 + 11*a^2*b^2 + b^4)*c^10 + 4*(a^5 + 3*a^3*b^2 + 2*a*b^4)*c
^9 - (a^4*b^2 + 2*a^2*b^4 + b^6)*c^8))) - c*sinh(x)^2 + 2*(b*x - c*cosh(x))*sinh(x) - c)/(c^2*cosh(x) + c^2*si
nh(x))

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giac [A]  time = 6.46, size = 24, normalized size = 0.07 \[ -\frac {b x}{c^{2}} + \frac {e^{\left (-x\right )}}{2 \, c} + \frac {e^{x}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="giac")

[Out]

-b*x/c^2 + 1/2*e^(-x)/c + 1/2*e^x/c

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maple [C]  time = 0.23, size = 144, normalized size = 0.40 \[ -\frac {1}{c \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{c^{2}}+\frac {1}{c \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{c^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 b \textit {\_Z} +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} a b +2 \left (-a c +b^{2}\right ) \textit {\_R} +a b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 b \,\textit {\_R}^{2}-2 \textit {\_R} a +4 \textit {\_R} c +b}}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^3/(a+b*sinh(x)+c*sinh(x)^2),x)

[Out]

-1/c/(tanh(1/2*x)-1)+b/c^2*ln(tanh(1/2*x)-1)+1/c/(tanh(1/2*x)+1)-b/c^2*ln(tanh(1/2*x)+1)+1/c^2*sum((-_R^2*a*b+
2*(-a*c+b^2)*_R+a*b)/(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 \[ -\frac {{\left (2 \, b x e^{x} - c e^{\left (2 \, x\right )} - c\right )} e^{\left (-x\right )}}{2 \, c^{2}} - \frac {1}{8} \, \int -\frac {16 \, {\left (2 \, a b e^{\left (2 \, x\right )} + {\left (b^{2} - a c\right )} e^{\left (3 \, x\right )} - {\left (b^{2} - a c\right )} e^{x}\right )}}{c^{3} e^{\left (4 \, x\right )} + 2 \, b c^{2} e^{\left (3 \, x\right )} - 2 \, b c^{2} e^{x} + c^{3} + 2 \, {\left (2 \, a c^{2} - c^{3}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^3/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="maxima")

[Out]

-1/2*(2*b*x*e^x - c*e^(2*x) - c)*e^(-x)/c^2 - 1/8*integrate(-16*(2*a*b*e^(2*x) + (b^2 - a*c)*e^(3*x) - (b^2 -
a*c)*e^x)/(c^3*e^(4*x) + 2*b*c^2*e^(3*x) - 2*b*c^2*e^x + c^3 + 2*(2*a*c^2 - c^3)*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)^3/(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)**3/(a+b*sinh(x)+c*sinh(x)**2),x)

[Out]

Timed out

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