\(\int (a \cosh (x)+b \sinh (x))^{3/2} \, dx\) [591]

Optimal result

Integrand size = 13, antiderivative size = 103 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3152, 3157, 2720} \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} (a \sinh (x)+b \cosh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right )}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \]

[In]

Int[(a*Cosh[x] + b*Sinh[x])^(3/2),x]

[Out]

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

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3152

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

Rule 3157

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[c + d*x] +
b*Sin[c + d*x])^n/((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n, Int[Cos[c + d*x - ArcTan[a, b]]^n, x]
, x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) &&  !(GtQ[a^2 + b^2, 0] || EqQ[a^2 + b^2, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}+\frac {1}{3} \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a \cosh (x)+b \sinh (x)}} \, dx \\ & = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}+\frac {\left (\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}\right ) \int \frac {1}{\sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )}} \, dx}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \\ & = \frac {2}{3} (b \cosh (x)+a \sinh (x)) \sqrt {a \cosh (x)+b \sinh (x)}-\frac {2 i \left (a^2-b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right ),2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}{3 \sqrt {a \cosh (x)+b \sinh (x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.89 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2}{3} \left (b \cosh (x)-\sqrt {1-\frac {a^2}{b^2}} b \sqrt {\cosh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\sinh ^2\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )\right ) \text {sech}\left (x+\text {arctanh}\left (\frac {a}{b}\right )\right )+a \sinh (x)\right ) \sqrt {a \cosh (x)+b \sinh (x)} \]

[In]

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

[Out]

(2*(b*Cosh[x] - Sqrt[1 - a^2/b^2]*b*Sqrt[Cosh[x + ArcTanh[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, -Sinh[
x + ArcTanh[a/b]]^2]*Sech[x + ArcTanh[a/b]] + a*Sinh[x])*Sqrt[a*Cosh[x] + b*Sinh[x]])/3

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.66

method	result	size
default	\(-\frac {\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}\, \left (\cosh \left (x \right ) \sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}\, \sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \left (a^{2}-b^{2}\right )+\sinh \left (x \right ) \left (a^{2}-b^{2}\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \cosh \left (x \right )}{\sqrt {-\sqrt {a^{2}-b^{2}}\, \sinh \left (x \right )^{3}}}\right )\right )}{2 \sqrt {\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}\, \sinh \left (x \right )^{2} \sqrt {a^{2}-b^{2}}\, \sqrt {-\sinh \left (x \right ) \sqrt {a^{2}-b^{2}}}}\)	\(171\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.98 \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (a - b\right )} \cosh \left (x\right ) + \sqrt {2} {\left (a - b\right )} \sinh \left (x\right )\right )} \sqrt {a + b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left ({\left (a + b\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a + b\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a + b\right )} \sinh \left (x\right )^{2} - a + b\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}}{3 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

1/3*(2*(sqrt(2)*(a - b)*cosh(x) + sqrt(2)*(a - b)*sinh(x))*sqrt(a + b)*weierstrassPInverse(-4*(a - b)/(a + b),
0, cosh(x) + sinh(x)) + ((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - a + b)*sqrt(a*co
sh(x) + b*sinh(x)))/(cosh(x) + sinh(x))

Sympy [F]

\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int \left (a \cosh {\left (x \right )} + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int { {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (a \cosh (x)+b \sinh (x))^{3/2} \, dx=\int {\left (a\,\mathrm {cosh}\left (x\right )+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]

[In]

int((a*cosh(x) + b*sinh(x))^(3/2),x)

[Out]

int((a*cosh(x) + b*sinh(x))^(3/2), x)