3.826 \(\int \frac{1}{a+b \sinh (x)+c \sinh ^2(x)} \, dx\)

Optimal. Leaf size=271 \[ \frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{2 i c-\tanh \left (\frac{x}{2}\right ) \left (\sqrt{4 a c-b^2}+i b\right )}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt{4 a c-b^2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right ) \sqrt{4 a c-b^2}-i b \tanh \left (\frac{x}{2}\right )+2 i c}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt{4 a c-b^2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}} \]

[Out]

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

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Rubi [A]  time = 0.915188, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3248, 2660, 618, 204} \[ \frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{2 i c-\tanh \left (\frac{x}{2}\right ) \left (\sqrt{4 a c-b^2}+i b\right )}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt{4 a c-b^2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}-\frac{2 \sqrt{2} c \tan ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right ) \sqrt{4 a c-b^2}-i b \tanh \left (\frac{x}{2}\right )+2 i c}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt{4 a c-b^2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a-c)+b^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[x] + c*Sinh[x]^2)^(-1),x]

[Out]

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

Rule 3248

Int[((a_.) + (b_.)*sin[(d_.) + (e_.)*(x_)]^(n_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]^(n2_.))^(-1), x_Symbol] :> Mo
dule[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[1/(b - q + 2*c*Sin[d + e*x]^n), x], x] - Dist[(2*c)/q, Int[1/
(b + q + 2*c*Sin[d + e*x]^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && 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 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 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])

Rubi steps

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

Mathematica [A]  time = 0.712136, size = 217, normalized size = 0.8 \[ \frac{2 \sqrt{2} c \left (\frac{\tan ^{-1}\left (\frac{\tanh \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}-b\right )+2 c}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a-c)-2 b^2}}\right )}{\sqrt{b \sqrt{b^2-4 a c}+2 c (a-c)-b^2}}-\frac{\tan ^{-1}\left (\frac{2 c-\tanh \left (\frac{x}{2}\right ) \left (\sqrt{b^2-4 a c}+b\right )}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt{-b \sqrt{b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[x] + c*Sinh[x]^2)^(-1),x]

[Out]

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

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Maple [C]  time = 0.037, size = 74, normalized size = 0.3 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{4}-2\,b{{\it \_Z}}^{3}+ \left ( -2\,a+4\,c \right ){{\it \_Z}}^{2}+2\,b{\it \_Z}+a \right ) }{\frac{-{{\it \_R}}^{2}+1}{2\,{{\it \_R}}^{3}a-3\,b{{\it \_R}}^{2}-2\,{\it \_R}\,a+4\,c{\it \_R}+b}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -{\it \_R} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sinh(x)+c*sinh(x)^2),x)

[Out]

sum((-_R^2+1)/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{c \sinh \left (x\right )^{2} + b \sinh \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="maxima")

[Out]

integrate(1/(c*sinh(x)^2 + b*sinh(x) + a), x)

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Fricas [B]  time = 3.68895, size = 7035, normalized size = 25.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x)+c*sinh(x)**2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{c \sinh \left (x\right )^{2} + b \sinh \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="giac")

[Out]

integrate(1/(c*sinh(x)^2 + b*sinh(x) + a), x)