3.8.62 \(\int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx\) [762]

Optimal. Leaf size=249 \[ \frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}-\frac {8 i a E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \left (a^2-b^2+c^2\right ) F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \sqrt {a+b \cosh (x)+c \sinh (x)}} \]

[Out]

2/3*(c*cosh(x)+b*sinh(x))*(a+b*cosh(x)+c*sinh(x))^(1/2)-8/3*I*a*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(
1/2*I*x-1/2*arctan(b,-I*c))*EllipticE(sin(1/2*I*x-1/2*arctan(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1
/2)))^(1/2))*(a+b*cosh(x)+c*sinh(x))^(1/2)/((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^(1/2)+2/3*I*(a^2-b^2+
c^2)*(cos(1/2*I*x-1/2*arctan(b,-I*c))^2)^(1/2)/cos(1/2*I*x-1/2*arctan(b,-I*c))*EllipticF(sin(1/2*I*x-1/2*arcta
n(b,-I*c)),2^(1/2)*((b^2-c^2)^(1/2)/(a+(b^2-c^2)^(1/2)))^(1/2))*((a+b*cosh(x)+c*sinh(x))/(a+(b^2-c^2)^(1/2)))^
(1/2)/(a+b*cosh(x)+c*sinh(x))^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3199, 3228, 3198, 2732, 3206, 2740} \begin {gather*} \frac {2 i \left (a^2-b^2+c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}} F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \sqrt {a+b \cosh (x)+c \sinh (x)}}-\frac {8 i a \sqrt {a+b \cosh (x)+c \sinh (x)} E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right )}{3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2}{3} (b \sinh (x)+c \cosh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c*Cosh[x] + b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/3 - (((8*I)/3)*a*EllipticE[(I*x - ArcTan[b, (-I)*c
])/2, (2*Sqrt[b^2 - c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/Sqrt[(a + b*Cosh[x] + c*Sinh
[x])/(a + Sqrt[b^2 - c^2])] + (((2*I)/3)*(a^2 - b^2 + c^2)*EllipticF[(I*x - ArcTan[b, (-I)*c])/2, (2*Sqrt[b^2
- c^2])/(a + Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a + Sqrt[b^2 - c^2])])/Sqrt[a + b*Cosh[x] + c
*Sinh[x]]

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3199

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(c*Cos[d
+ e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Dist[1/n, Int[Simp[n*a^2
+ (n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x], x]*(a + b*Cos[d + e*x] + c*S
in[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]

Rule 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3228

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.)
+ (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e
, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]

Rubi steps

\begin {align*} \int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {2}{3} \int \frac {\frac {1}{2} \left (3 a^2+b^2-c^2\right )+2 a b \cosh (x)+2 a c \sinh (x)}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx\\ &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {1}{3} (4 a) \int \sqrt {a+b \cosh (x)+c \sinh (x)} \, dx+\frac {1}{3} \left (-a^2+b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cosh (x)+c \sinh (x)}} \, dx\\ &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}+\frac {\left (4 a \sqrt {a+b \cosh (x)+c \sinh (x)}\right ) \int \sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}} \, dx}{3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {\left (\left (-a^2+b^2-c^2\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2-c^2}}+\frac {\sqrt {b^2-c^2} \cosh \left (x+i \tan ^{-1}(b,-i c)\right )}{a+\sqrt {b^2-c^2}}}} \, dx}{3 \sqrt {a+b \cosh (x)+c \sinh (x)}}\\ &=\frac {2}{3} (c \cosh (x)+b \sinh (x)) \sqrt {a+b \cosh (x)+c \sinh (x)}-\frac {8 i a E\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {a+b \cosh (x)+c \sinh (x)}}{3 \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}+\frac {2 i \left (a^2-b^2+c^2\right ) F\left (\frac {1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac {2 \sqrt {b^2-c^2}}{a+\sqrt {b^2-c^2}}\right ) \sqrt {\frac {a+b \cosh (x)+c \sinh (x)}{a+\sqrt {b^2-c^2}}}}{3 \sqrt {a+b \cosh (x)+c \sinh (x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 6.11, size = 2292, normalized size = 9.20 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((8*a*b)/(3*c) + (2*c*Cosh[x])/3 + (2*b*Sinh[x])/3)*Sqrt[a + b*Cosh[x] + c*Sinh[x]] + (2*a^2*AppellF1[1/2, 1/2
, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^
2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1
- b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] -
I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2
)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sq
rt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(Sqrt[1 - b^2/c^2]*c*Sqrt[I*(I + Sinh[x + ArcTanh[b/c]])]) + (2*
b^2*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]*(1
- (I*a)/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/(Sqrt[1 - b^2/c^2]
*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTanh[b/c]]]*Sqrt[(c*Sqr
t[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-b^2 + c^2)/c^2])]*Sqr
t[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/((-I)*a + c*Sqrt[(-b^2 + c^2)
/c^2])]*Sqrt[a + c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(3*Sqrt[1 - b^2/c^2]*c*Sqrt[I*(I + Sinh[x +
ArcTanh[b/c]])]) - (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]))/
(Sqrt[1 - b^2/c^2]*(1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c), ((-I)*(a + Sqrt[1 - b^2/c^2]*c*Sinh[x + ArcTanh[b/c]]
))/(Sqrt[1 - b^2/c^2]*(-1 - (I*a)/(Sqrt[1 - b^2/c^2]*c))*c)]*Sech[x + ArcTanh[b/c]]*Sqrt[-1 + I*Sinh[x + ArcTa
nh[b/c]]]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] - I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/(I*a + c*Sqrt[(-
b^2 + c^2)/c^2])]*Sqrt[(c*Sqrt[(-b^2 + c^2)/c^2] + I*c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]])/((-I)*a
+ c*Sqrt[(-b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(-b^2 + c^2)/c^2]*Sinh[x + ArcTanh[b/c]]])/(3*Sqrt[1 - b^2/c^2]*S
qrt[I*(I + Sinh[x + ArcTanh[b/c]])]) - (4*a*b^2*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*C
osh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x +
ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b
^2]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^
2])]*Sqrt[a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^
2)/b^2]*Cosh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + Ar
cTanh[c/b]]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cos
h[x + ArcTanh[c/b]]]))/(3*c) + (4*a*c*((c*AppellF1[-1/2, -1/2, -1/2, 1/2, (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + Ar
cTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*(1 + a/(b*Sqrt[1 - c^2/b^2]))), (a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/
b]])/(b*Sqrt[1 - c^2/b^2]*(-1 + a/(b*Sqrt[1 - c^2/b^2])))]*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]*Sqrt[(
b*Sqrt[(b^2 - c^2)/b^2] - b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]])/(a + b*Sqrt[(b^2 - c^2)/b^2])]*Sqrt[
a + b*Sqrt[(b^2 - c^2)/b^2]*Cosh[x + ArcTanh[c/b]]]*Sqrt[(b*Sqrt[(b^2 - c^2)/b^2] + b*Sqrt[(b^2 - c^2)/b^2]*Co
sh[x + ArcTanh[c/b]])/(-a + b*Sqrt[(b^2 - c^2)/b^2])]) - ((-2*b*(a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcTanh[c/b]
]))/(b^2 - c^2) + (c*Sinh[x + ArcTanh[c/b]])/(b*Sqrt[1 - c^2/b^2]))/Sqrt[a + b*Sqrt[1 - c^2/b^2]*Cosh[x + ArcT
anh[c/b]]]))/3

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Maple [A]
time = 2.48, size = 321, normalized size = 1.29

method	result	size
default	\(\frac {2 \cosh \left (x \right ) a \left (-b^{2}+c^{2}\right )}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}+\frac {\sqrt {\frac {\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}}\, a^{2} \ln \left (\frac {-\cosh \left (x \right ) \sinh \left (x \right ) b^{2}+\cosh \left (x \right ) \sinh \left (x \right ) c^{2}+\cosh \left (x \right ) \sqrt {b^{2}-c^{2}}\, a +\sqrt {\frac {\left (-b^{2}+c^{2}\right ) \left (\sinh ^{3}\left (x \right )\right )}{\sqrt {b^{2}-c^{2}}}+a \left (\sinh ^{2}\left (x \right )\right )}\, \sqrt {b^{2}-c^{2}}\, \sqrt {\frac {\left (-b^{2}+c^{2}\right ) \sinh \left (x \right )}{\sqrt {b^{2}-c^{2}}}+a}}{\sqrt {b^{2}-c^{2}}\, \sqrt {\frac {-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}}{\sqrt {b^{2}-c^{2}}}}}\right ) \sqrt {b^{2}-c^{2}}}{\left (-\sinh \left (x \right ) b^{2}+\sinh \left (x \right ) c^{2}+a \sqrt {b^{2}-c^{2}}\right ) \sinh \left (x \right )}\)	\(321\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*cosh(x)*a/(b^2-c^2)^(1/2)*(-b^2+c^2)/((-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2))^(1/2)+((
-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))/(b^2-c^2)^(1/2)*sinh(x)^2)^(1/2)*a^2*ln((-cosh(x)*sinh(x)*b^2+cosh
(x)*sinh(x)*c^2+cosh(x)*(b^2-c^2)^(1/2)*a+((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)^3+a*sinh(x)^2)^(1/2)*(b^2-c^2)^(
1/2)*((-b^2+c^2)/(b^2-c^2)^(1/2)*sinh(x)+a)^(1/2))/(b^2-c^2)^(1/2)/((-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2
))/(b^2-c^2)^(1/2))^(1/2))/(-sinh(x)*b^2+sinh(x)*c^2+a*(b^2-c^2)^(1/2))*(b^2-c^2)^(1/2)/sinh(x)

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 463, normalized size = 1.86 \begin {gather*} \frac {2 \, {\left (\sqrt {2} {\left (a^{2} + 3 \, b^{2} - 3 \, c^{2}\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a^{2} + 3 \, b^{2} - 3 \, c^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b + c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right ) - 24 \, {\left (\sqrt {2} {\left (a b + a c\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a b + a c\right )} \sinh \left (x\right )\right )} \sqrt {b + c} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2} + 3 \, c^{2}\right )}}{3 \, {\left (b^{2} + 2 \, b c + c^{2}\right )}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2} + 9 \, a c^{2}\right )}}{27 \, {\left (b^{3} + 3 \, b^{2} c + 3 \, b c^{2} + c^{3}\right )}}, \frac {3 \, {\left (b + c\right )} \cosh \left (x\right ) + 3 \, {\left (b + c\right )} \sinh \left (x\right ) + 2 \, a}{3 \, {\left (b + c\right )}}\right )\right ) + 3 \, {\left ({\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )^{2} + {\left (b^{2} + 2 \, b c + c^{2}\right )} \sinh \left (x\right )^{2} - b^{2} + c^{2} - 8 \, {\left (a b + a c\right )} \cosh \left (x\right ) - 2 \, {\left (4 \, a b + 4 \, a c - {\left (b^{2} + 2 \, b c + c^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + c \sinh \left (x\right ) + a}}{9 \, {\left ({\left (b + c\right )} \cosh \left (x\right ) + {\left (b + c\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(2*(sqrt(2)*(a^2 + 3*b^2 - 3*c^2)*cosh(x) + sqrt(2)*(a^2 + 3*b^2 - 3*c^2)*sinh(x))*sqrt(b + c)*weierstrass
PInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3
*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x) + 3*(b + c)*sinh(x) + 2*a)/(b + c)) - 24*(sqrt(2)*(a*b + a*c)*cosh(x) +
sqrt(2)*(a*b + a*c)*sinh(x))*sqrt(b + c)*weierstrassZeta(4/3*(4*a^2 - 3*b^2 + 3*c^2)/(b^2 + 2*b*c + c^2), -8/2
7*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), weierstrassPInverse(4/3*(4*a^2 - 3*b^2 + 3*c^2)
/(b^2 + 2*b*c + c^2), -8/27*(8*a^3 - 9*a*b^2 + 9*a*c^2)/(b^3 + 3*b^2*c + 3*b*c^2 + c^3), 1/3*(3*(b + c)*cosh(x
) + 3*(b + c)*sinh(x) + 2*a)/(b + c))) + 3*((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 + 2*b*c + c^2)*sinh(x)^2 - b^
2 + c^2 - 8*(a*b + a*c)*cosh(x) - 2*(4*a*b + 4*a*c - (b^2 + 2*b*c + c^2)*cosh(x))*sinh(x))*sqrt(b*cosh(x) + c*
sinh(x) + a))/((b + c)*cosh(x) + (b + c)*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \cosh {\left (x \right )} + c \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cosh(x)+c*sinh(x))**(3/2),x)

[Out]

Integral((a + b*cosh(x) + c*sinh(x))**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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