3.9.29 \(\int \frac {\sinh ^3(x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx\) [829]

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

[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 = 3.46, 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 = {3337, 2718, 3373, 2739, 632, 210} \begin {gather*} \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 ) \text {ArcTan}\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 ) \text {ArcTan}\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} \end {gather*}

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 210

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

Rule 632

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 2718

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

Rule 2739

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 3337

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 3373

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.56, size = 326, normalized size = 0.90 \begin {gather*} \frac {-b x+\frac {\sqrt {2} \left (-b^3+3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) \text {ArcTan}\left (\frac {2 c+\left (-b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {-2 b^2+4 (a-c) c+2 b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b^2+2 (a-c) c+b \sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) \text {ArcTan}\left (\frac {2 c-\left (b+\sqrt {b^2-4 a c}\right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {2} \sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b^2+2 (a-c) c-b \sqrt {b^2-4 a c}}}+c \cosh (x)}{c^2} \end {gather*}

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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.34, size = 144, normalized size = 0.40

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 \textit {\_Z} b +a \right )}{\sum }\frac {\left (-a b \,\textit {\_R}^{2}+2 \left (-a c +b^{2}\right ) \textit {\_R} +a b \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 \textit {\_R}^{2} b -2 \textit {\_R} a +4 \textit {\_R} c +b}}{c^{2}}+\frac {1}{c \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{c^{2}}-\frac {1}{c \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{c^{2}}\)	\(144\)
risch	\(\text {Expression too large to display}\)	\(2135\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*sum((-a*b*_R^2+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*_Z*b+a))+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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6680 vs. \(2 (297) = 594\).
time = 1.23, size = 6680, normalized size = 18.40 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [A]
time = 0.87, size = 24, normalized size = 0.07 \begin {gather*} -\frac {b x}{c^{2}} + \frac {e^{\left (-x\right )}}{2 \, c} + \frac {e^{x}}{2 \, c} \end {gather*}

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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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

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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