Integrand size = 10, antiderivative size = 150 \[ \int (a+b \sinh (x))^{3/2} \, dx=\frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {8 i a E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{3 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 \sqrt {a+b \sinh (x)}} \] Output:
2/3*b*cosh(x)*(a+b*sinh(x))^(1/2)+8/3*I*a*EllipticE(cos(1/4*Pi+1/2*I*x),2^ (1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^(1/2)/((a+b*sinh(x))/(a-I*b))^(1/2) +2/3*I*(a^2+b^2)*InverseJacobiAM(-1/4*Pi+1/2*I*x,2^(1/2)*(b/(I*a+b))^(1/2) )*((a+b*sinh(x))/(a-I*b))^(1/2)/(a+b*sinh(x))^(1/2)
Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int (a+b \sinh (x))^{3/2} \, dx=\frac {2 b \cosh (x) (a+b \sinh (x))+8 a (i a+b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}-2 i \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{3 \sqrt {a+b \sinh (x)}} \] Input:
Integrate[(a + b*Sinh[x])^(3/2),x]
Output:
(2*b*Cosh[x]*(a + b*Sinh[x]) + 8*a*(I*a + b)*EllipticE[(Pi - (2*I)*x)/4, ( (-2*I)*b)/(a - I*b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)] - (2*I)*(a^2 + b^2)*E llipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*Sqrt[a + b*Sinh[x]])
Time = 0.87 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3135, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \sinh (x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a-i b \sin (i x))^{3/2}dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {2}{3} \int \frac {3 a^2+4 b \sinh (x) a-b^2}{2 \sqrt {a+b \sinh (x)}}dx+\frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {3 a^2+4 b \sinh (x) a-b^2}{\sqrt {a+b \sinh (x)}}dx+\frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \int \frac {3 a^2-4 i b \sin (i x) a-b^2}{\sqrt {a-i b \sin (i x)}}dx\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{3} \left (4 a \int \sqrt {a+b \sinh (x)}dx-\left (a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx\right )+\frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (4 a \int \sqrt {a-i b \sin (i x)}dx-\left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {4 a \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {4 a \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {8 i a \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {8 i a \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{\sqrt {a+b \sinh (x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {8 i a \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{\sqrt {a+b \sinh (x)}}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{3} b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {8 i a \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}}\right )\) |
Input:
Int[(a + b*Sinh[x])^(3/2),x]
Output:
(2*b*Cosh[x]*Sqrt[a + b*Sinh[x]])/3 + (((8*I)*a*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/Sqrt[(a + b*Sinh[x])/(a - I*b)] - (( 2*I)*(a^2 + b^2)*EllipticF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Si nh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]])/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[1/n Int[(a + b* Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x] , x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(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]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (129 ) = 258\).
Time = 0.89 (sec) , antiderivative size = 676, normalized size of antiderivative = 4.51
| method | result | size |
| default | \(\frac {\frac {2 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b}{3}+\frac {2 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{3}}{3}+2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{3}+2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \,b^{2}-\frac {8 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{3}}{3}-\frac {8 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \,b^{2}}{3}+\frac {2 b^{3} \sinh \left (x \right )^{3}}{3}+\frac {2 a \,b^{2} \sinh \left (x \right )^{2}}{3}+\frac {2 \sinh \left (x \right ) b^{3}}{3}+\frac {2 a \,b^{2}}{3}}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(676\) |
Input:
int((a+b*sinh(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
2/3*(I*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+ sinh(x))/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a) /(I*b+a))^(1/2))*a^2*b+I*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I* b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a ))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*b^3+3*(-(a+b*sinh(x))/(I*b-a))^(1/2)*(( I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticF((-(a+b *sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^3+3*(-(a+b*sinh(x))/( I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+sinh(x))/(I*b-a))^(1/2)* EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a*b^2-4 *(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*(b*(I+sinh(x ))/(I*b-a))^(1/2)*EllipticE((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+ a))^(1/2))*a^3-4*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1 /2)*(b*(I+sinh(x))/(I*b-a))^(1/2)*EllipticE((-(a+b*sinh(x))/(I*b-a))^(1/2) ,(-(I*b-a)/(I*b+a))^(1/2))*a*b^2+b^3*sinh(x)^3+a*b^2*sinh(x)^2+sinh(x)*b^3 +a*b^2)/b/cosh(x)/(a+b*sinh(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (125) = 250\).
Time = 0.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.71 \[ \int (a+b \sinh (x))^{3/2} \, dx=\frac {4 \, \sqrt {\frac {1}{2}} {\left ({\left (a^{2} - 3 \, b^{2}\right )} \cosh \left (x\right ) + {\left (a^{2} - 3 \, b^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 48 \, \sqrt {\frac {1}{2}} {\left (a b \cosh \left (x\right ) + a b \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} - 8 \, a b \cosh \left (x\right ) + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) - 4 \, a b\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{9 \, {\left (b \cosh \left (x\right ) + b \sinh \left (x\right )\right )}} \] Input:
integrate((a+b*sinh(x))^(3/2),x, algorithm="fricas")
Output:
1/9*(4*sqrt(1/2)*((a^2 - 3*b^2)*cosh(x) + (a^2 - 3*b^2)*sinh(x))*sqrt(b)*w eierstrassPInverse(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, 1 /3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b) - 48*sqrt(1/2)*(a*b*cosh(x) + a*b* sinh(x))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9 *a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9 *a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) + 3*(b^2*cosh(x)^2 + b^2*sinh(x)^2 - 8*a*b*cosh(x) + b^2 + 2*(b^2*cosh(x) - 4*a*b)*sinh(x))*s qrt(b*sinh(x) + a))/(b*cosh(x) + b*sinh(x))
\[ \int (a+b \sinh (x))^{3/2} \, dx=\int \left (a + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a+b*sinh(x))**(3/2),x)
Output:
Integral((a + b*sinh(x))**(3/2), x)
\[ \int (a+b \sinh (x))^{3/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*sinh(x))^(3/2),x, algorithm="maxima")
Output:
integrate((b*sinh(x) + a)^(3/2), x)
\[ \int (a+b \sinh (x))^{3/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*sinh(x))^(3/2),x, algorithm="giac")
Output:
integrate((b*sinh(x) + a)^(3/2), x)
Timed out. \[ \int (a+b \sinh (x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \] Input:
int((a + b*sinh(x))^(3/2),x)
Output:
int((a + b*sinh(x))^(3/2), x)
\[ \int (a+b \sinh (x))^{3/2} \, dx=\left (\int \sqrt {\sinh \left (x \right ) b +a}d x \right ) a +\left (\int \sqrt {\sinh \left (x \right ) b +a}\, \sinh \left (x \right )d x \right ) b \] Input:
int((a+b*sinh(x))^(3/2),x)
Output:
int(sqrt(sinh(x)*b + a),x)*a + int(sqrt(sinh(x)*b + a)*sinh(x),x)*b