3.831 \(\int \frac {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx\)
Optimal. Leaf size=300 \[ \frac {\sqrt {2} \left (-\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}+\frac {\sqrt {2} \left (\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}} \]
[Out]
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*e+(-b*e+2*c*d)/(4*a*c-b^2)^(1/2))/(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*t
anh(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*e+(b
*e-2*c*d)/(4*a*c-b^2)^(1/2))/(b^2-2*(a-c)*c+I*b*(4*a*c-b^2)^(1/2))^(1/2)
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Rubi [A] time = 0.76, antiderivative size = 300, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used =
{3292, 2660, 618, 204} \[ \frac {\sqrt {2} \left (-\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \sqrt {4 a c-b^2}-i b \tanh \left (\frac {x}{2}\right )+2 i c}{\sqrt {2} \sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}+\frac {\sqrt {2} \left (\frac {2 c d-b e}{\sqrt {4 a c-b^2}}+i e\right ) \tan ^{-1}\left (\frac {2 i c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {4 a c-b^2}+i b\right )}{\sqrt {2} \sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}}\right )}{\sqrt {-i b \sqrt {4 a c-b^2}-2 c (a-c)+b^2}} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*Sinh[x])/(a + b*Sinh[x] + c*Sinh[x]^2),x]
[Out]
(Sqrt[2]*(I*e - (2*c*d - b*e)/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]])])/Sqrt[b^2 - 2*(a - c)*c + I*b*Sqrt[-b^2 + 4*a*
c]] + (Sqrt[2]*(I*e + (2*c*d - b*e)/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]])])/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 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 {d+e \sinh (x)}{a+b \sinh (x)+c \sinh ^2(x)} \, dx &=\left (-i e-\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{-i b-\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx+\left (-i e+\frac {2 c d-b e}{\sqrt {-b^2+4 a c}}\right ) \int \frac {1}{-i b+\sqrt {-b^2+4 a c}-2 i c \sinh (x)} \, dx\\ &=-\left (\left (2 \left (i e-\frac {2 c d-b e}{\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 )\right )-\left (2 \left (i e+\frac {2 c d-b e}{\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 )\\ &=\left (4 \left (i e-\frac {2 c d-b e}{\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 )+\left (4 \left (i e+\frac {2 c d-b e}{\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 )\\ &=\frac {\sqrt {2} \left (i e-\frac {2 c d-b e}{\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 )}{\sqrt {b^2-2 (a-c) c+i b \sqrt {-b^2+4 a c}}}+\frac {\sqrt {2} \left (i e+\frac {2 c d-b e}{\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 )}{\sqrt {b^2-2 (a-c) c-i b \sqrt {-b^2+4 a c}}}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 258, normalized size = 0.86 \[ \frac {\sqrt {2} \left (\frac {\left (e \left (\sqrt {b^2-4 a c}-b\right )+2 c d\right ) \tan ^{-1}\left (\frac {\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}-b\right )+2 c}{\sqrt {2 b \sqrt {b^2-4 a c}+4 c (a-c)-2 b^2}}\right )}{\sqrt {b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}+\frac {\left (e \left (\sqrt {b^2-4 a c}+b\right )-2 c d\right ) \tan ^{-1}\left (\frac {2 c-\tanh \left (\frac {x}{2}\right ) \left (\sqrt {b^2-4 a c}+b\right )}{\sqrt {2} \sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt {-b \sqrt {b^2-4 a c}+2 c (a-c)-b^2}}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*Sinh[x])/(a + b*Sinh[x] + c*Sinh[x]^2),x]
[Out]
(Sqrt[2]*(((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*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]] + ((-2*c*d + (b + Sqrt
[b^2 - 4*a*c])*e)*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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fricas [B] time = 4.50, size = 6841, normalized size = 22.80 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*sinh(x))/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^
2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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(-2*b^2*c*d^4 + 2*a*b^2*e^4 + 2*(b^3 + 2*a*b*
c - 2*b*c^2)*d^3*e - 6*(a*b^2 - b^2*c)*d^2*e^2 + 2*(2*a^2*b - b^3 - 2*a*b*c)*d*e^3 + sqrt(2)*((b^4 - 4*a*b^2*c
)*d^3 - 3*(a*b^3 + 4*a*b*c^2 - (4*a^2*b + b^3)*c)*d^2*e + (2*a^2*b^2 - b^4 - 8*a^3*c - 8*a*c^3 + 2*(8*a^2 + b^
2)*c^2)*d*e^2 + (a*b^3 + 4*a*b*c^2 - (4*a^2*b + b^3)*c)*e^3 - ((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)*d - (a^3*b^3 + a*b^
5 - 4*a*b*c^4 + (4*a^2*b + b^3)*c^3 + (4*a^3*b - 5*a*b^3)*c^2 - (4*a^4*b + 5*a^2*b^3 - b^5)*c)*e)*sqrt((b^2*d^
4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*
e^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*d^4 + b^2*e^4 - 4*(a*b -
b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)) - 4*(b*c^2*d^4 - a*b*c*e^4
- (b^2*c + 2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*cosh(x) - 4
*(b*c^2*d^4 - a*b*c*e^4 - (b^2*c + 2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^
2)*c)*d*e^3)*sinh(x) - 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)*d^2 + (a
^2*b^3 + b^5 - 4*a*b*c^3 + (8*a^2*b + b^3)*c^2 - 2*(2*a^3*b + 3*a*b^3)*c)*d*e - (a^3*b^2 + a*b^4 - 4*a^2*c^3 +
(8*a^3 + a*b^2)*c^2 - 2*(2*a^4 + 3*a^2*b^2)*c)*e^2)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2
- b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 -
4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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(-2*b^2*c*d^4 + 2*a*b^2*e^4 + 2*(b^3 + 2*a*b*c - 2*b*c^
2)*d^3*e - 6*(a*b^2 - b^2*c)*d^2*e^2 + 2*(2*a^2*b - b^3 - 2*a*b*c)*d*e^3 - sqrt(2)*((b^4 - 4*a*b^2*c)*d^3 - 3*
(a*b^3 + 4*a*b*c^2 - (4*a^2*b + b^3)*c)*d^2*e + (2*a^2*b^2 - b^4 - 8*a^3*c - 8*a*c^3 + 2*(8*a^2 + b^2)*c^2)*d*
e^2 + (a*b^3 + 4*a*b*c^2 - (4*a^2*b + b^3)*c)*e^3 - ((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)*d - (a^3*b^3 + a*b^5 - 4*a*b*
c^4 + (4*a^2*b + b^3)*c^3 + (4*a^3*b - 5*a*b^3)*c^2 - (4*a^4*b + 5*a^2*b^3 - b^5)*c)*e)*sqrt((b^2*d^4 + b^2*e^
4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*
e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)) - 4*(b*c^2*d^4 - a*b*c*e^4 - (b^2*c +
2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*cosh(x) - 4*(b*c^2*d^
4 - a*b*c*e^4 - (b^2*c + 2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^
3)*sinh(x) - 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)*d^2 + (a^2*b^3 + b
^5 - 4*a*b*c^3 + (8*a^2*b + b^3)*c^2 - 2*(2*a^3*b + 3*a*b^3)*c)*d*e - (a^3*b^2 + a*b^4 - 4*a^2*c^3 + (8*a^3 +
a*b^2)*c^2 - 2*(2*a^4 + 3*a^2*b^2)*c)*e^2)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*
a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*
c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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(-2*b^2*c*d^4 + 2*a*b^2*e^4 + 2*(b^3 + 2*a*b*c - 2*b*c^2)*d^3*e -
6*(a*b^2 - b^2*c)*d^2*e^2 + 2*(2*a^2*b - b^3 - 2*a*b*c)*d*e^3 + sqrt(2)*((b^4 - 4*a*b^2*c)*d^3 - 3*(a*b^3 + 4
*a*b*c^2 - (4*a^2*b + b^3)*c)*d^2*e + (2*a^2*b^2 - b^4 - 8*a^3*c - 8*a*c^3 + 2*(8*a^2 + b^2)*c^2)*d*e^2 + (a*b
^3 + 4*a*b*c^2 - (4*a^2*b + b^3)*c)*e^3 + ((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)*d - (a^3*b^3 + a*b^5 - 4*a*b*c^4 + (4*a
^2*b + b^3)*c^3 + (4*a^3*b - 5*a*b^3)*c^2 - (4*a^4*b + 5*a^2*b^3 - b^5)*c)*e)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b
- b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a
^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)) - 4*(b*c^2*d^4 - a*b*c*e^4 - (b^2*c + 2*a*c^2 -
2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*cosh(x) - 4*(b*c^2*d^4 - a*b*c*
e^4 - (b^2*c + 2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*sinh(x)
+ 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)*d^2 + (a^2*b^3 + b^5 - 4*a*b
*c^3 + (8*a^2*b + b^3)*c^2 - 2*(2*a^3*b + 3*a*b^3)*c)*d*e - (a^3*b^2 + a*b^4 - 4*a^2*c^3 + (8*a^3 + a*b^2)*c^2
- 2*(2*a^4 + 3*a^2*b^2)*c)*e^2)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^
2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e
^2 + 4*(a*b - b*c)*d*e^3)/(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(-2*b^2*c*d^4 + 2*a*b^2*e^4 + 2*(b^3 + 2*a*b*c - 2*b*c^2)*d^3*e - 6*(a*b^2
- b^2*c)*d^2*e^2 + 2*(2*a^2*b - b^3 - 2*a*b*c)*d*e^3 - sqrt(2)*((b^4 - 4*a*b^2*c)*d^3 - 3*(a*b^3 + 4*a*b*c^2 -
(4*a^2*b + b^3)*c)*d^2*e + (2*a^2*b^2 - b^4 - 8*a^3*c - 8*a*c^3 + 2*(8*a^2 + b^2)*c^2)*d*e^2 + (a*b^3 + 4*a*b
*c^2 - (4*a^2*b + b^3)*c)*e^3 + ((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)*d - (a^3*b^3 + a*b^5 - 4*a*b*c^4 + (4*a^2*b + b^3
)*c^3 + (4*a^3*b - 5*a*b^3)*c^2 - (4*a^4*b + 5*a^2*b^3 - b^5)*c)*e)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^
3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(a^4*b^2 + 2*a^2*b^4 + b^6 - 4*a*c^5 + (1
6*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)*d^2 - 2*(a*b + b*c)*d*e + (2*a^2 + b^2 - 2*a*c)*e^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*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 -
4*a*c + 2*c^2)*d^2*e^2 + 4*(a*b - b*c)*d*e^3)/(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)) - 4*(b*c^2*d^4 - a*b*c*e^4 - (b^2*c + 2*a*c^2 - 2*c^3)*d^
3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*cosh(x) - 4*(b*c^2*d^4 - a*b*c*e^4 - (b^2
*c + 2*a*c^2 - 2*c^3)*d^3*e + 3*(a*b*c - b*c^2)*d^2*e^2 + (2*a*c^2 - (2*a^2 - b^2)*c)*d*e^3)*sinh(x) + 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)*d^2 + (a^2*b^3 + b^5 - 4*a*b*c^3 + (8*
a^2*b + b^3)*c^2 - 2*(2*a^3*b + 3*a*b^3)*c)*d*e - (a^3*b^2 + a*b^4 - 4*a^2*c^3 + (8*a^3 + a*b^2)*c^2 - 2*(2*a^
4 + 3*a^2*b^2)*c)*e^2)*sqrt((b^2*d^4 + b^2*e^4 - 4*(a*b - b*c)*d^3*e + 2*(2*a^2 - b^2 - 4*a*c + 2*c^2)*d^2*e^2
+ 4*(a*b - b*c)*d*e^3)/(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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giac [A] time = 4.28, size = 1, normalized size = 0.00 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*sinh(x))/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="giac")
[Out]
0
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maple [C] time = 0.26, size = 79, normalized size = 0.26 \[ \munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-2 b \,\textit {\_Z}^{3}+\left (-2 a +4 c \right ) \textit {\_Z}^{2}+2 b \textit {\_Z} +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} d +2 \textit {\_R} e +d \right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{2 \textit {\_R}^{3} a -3 b \,\textit {\_R}^{2}-2 \textit {\_R} a +4 \textit {\_R} c +b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d+e*sinh(x))/(a+b*sinh(x)+c*sinh(x)^2),x)
[Out]
sum((-_R^2*d+2*_R*e+d)/(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 \[ \int \frac {e \sinh \relax (x) + d}{c \sinh \relax (x)^{2} + b \sinh \relax (x) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d+e*sinh(x))/(a+b*sinh(x)+c*sinh(x)^2),x, algorithm="maxima")
[Out]
integrate((e*sinh(x) + d)/(c*sinh(x)^2 + b*sinh(x) + a), 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((d + e*sinh(x))/(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((d+e*sinh(x))/(a+b*sinh(x)+c*sinh(x)**2),x)
[Out]
Timed out
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