### 3.762 $$\int (a+b \cosh (x)+c \sinh (x))^{3/2} \, dx$$

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

[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]]

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Rubi [A]  time = 0.270464, antiderivative size = 249, normalized size of antiderivative = 1., 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 = {3120, 3149, 3119, 2653, 3127, 2661} $\frac{2 i \left (a^2-b^2+c^2\right ) \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}} F\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{3 \sqrt{a+b \cosh (x)+c \sinh (x)}}-\frac{8 i a \sqrt{a+b \cosh (x)+c \sinh (x)} E\left (\frac{1}{2} \left (i x-\tan ^{-1}(b,-i c)\right )|\frac{2 \sqrt{b^2-c^2}}{a+\sqrt{b^2-c^2}}\right )}{3 \sqrt{\frac{a+b \cosh (x)+c \sinh (x)}{a+\sqrt{b^2-c^2}}}}+\frac{2}{3} (b \sinh (x)+c \cosh (x)) \sqrt{a+b \cosh (x)+c \sinh (x)}$

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 3120

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*Sin
[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 3149

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]

Rule 3119

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2653

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

Rule 3127

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]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], 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 2661

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

Mathematica [C]  time = 6.14134, size = 2292, normalized size = 9.2 $\text{Result too large to show}$

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 = 0.362, size = 318, normalized size = 1.3 \begin{align*} 2\,{\frac{a \left ( -{b}^{2}+{c}^{2} \right ) \cosh \left ( x \right ) }{\sqrt{{b}^{2}-{c}^{2}}}{\frac{1}{\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{{a}^{2}}{\sinh \left ( x \right ) }\sqrt{{ \left ( \sinh \left ( x \right ) \right ) ^{2} \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}}\ln \left ({ \left ( \cosh \left ( x \right ) \sinh \left ( x \right ) \left ( -{b}^{2}+{c}^{2} \right ) +\cosh \left ( x \right ) \sqrt{{b}^{2}-{c}^{2}}a+\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{3}{\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{{b}^{2}-{c}^{2}}\sqrt{{ \left ( -{b}^{2}+{c}^{2} \right ) \sinh \left ( x \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}}+a} \right ){\frac{1}{\sqrt{{b}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{{ \left ( -\sinh \left ( x \right ){b}^{2}+\sinh \left ( x \right ){c}^{2}+a\sqrt{{b}^{2}-{c}^{2}} \right ){\frac{1}{\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 ) ^{-1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*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)*cosh(x)+((
-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+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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}, x\right ) \end{align*}

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]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \cosh \left (x\right ) + c \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)