Integrand size = 16, antiderivative size = 115 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {b \cosh (e+f x) \sinh (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {i E\left (i e+i f x\left |\frac {b}{a}\right .\right ) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b) f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \]
-b*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)-I*(cos(I*e+ I*f*x)^2)^(1/2)/cos(I*e+I*f*x)*EllipticE(sin(I*e+I*f*x),(b/a)^(1/2))*(a+b* sinh(f*x+e)^2)^(1/2)/a/(a-b)/f/(1+b*sinh(f*x+e)^2/a)^(1/2)
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-\sqrt {2} b \sinh (2 (e+f x))}{2 a (a-b) f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((-2*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b /a] - Sqrt[2]*b*Sinh[2*(e + f*x)])/(2*a*(a - b)*f*Sqrt[2*a - b + b*Cosh[2* (e + f*x)]])
Time = 0.46 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 3663, 25, 3042, 3657, 3042, 3656}
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.
\(\displaystyle \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3663 |
\(\displaystyle -\frac {\int -\sqrt {b \sinh ^2(e+f x)+a}dx}{a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \sqrt {b \sinh ^2(e+f x)+a}dx}{a (a-b)}-\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\int \sqrt {a-b \sin (i e+i f x)^2}dx}{a (a-b)}\) |
\(\Big \downarrow \) 3657 |
\(\displaystyle \frac {\sqrt {a+b \sinh ^2(e+f x)} \int \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}dx}{a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}-\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}+\frac {\sqrt {a+b \sinh ^2(e+f x)} \int \sqrt {1-\frac {b \sin (i e+i f x)^2}{a}}dx}{a (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}\) |
\(\Big \downarrow \) 3656 |
\(\displaystyle -\frac {b \sinh (e+f x) \cosh (e+f x)}{a f (a-b) \sqrt {a+b \sinh ^2(e+f x)}}-\frac {i \sqrt {a+b \sinh ^2(e+f x)} E\left (i e+i f x\left |\frac {b}{a}\right .\right )}{a f (a-b) \sqrt {\frac {b \sinh ^2(e+f x)}{a}+1}}\) |
-((b*Cosh[e + f*x]*Sinh[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Sinh[e + f*x]^2] )) - (I*EllipticE[I*e + I*f*x, b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b )*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])
3.4.89.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)] Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Time = 0.67 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.19
method | result | size |
default | \(\frac {-\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right ) b +a \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b +\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b}{\left (a -b \right ) a \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(252\) |
(-(-b/a)^(1/2)*cosh(f*x+e)^2*sinh(f*x+e)*b+a*(b/a*cosh(f*x+e)^2+(a-b)/a)^( 1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)) -(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f* x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f *x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b)/(a-b)/a/ (-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 1464 vs. \(2 (123) = 246\).
Time = 0.11 (sec) , antiderivative size = 1464, normalized size of antiderivative = 12.73 \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
(((2*a*b^2 - b^3)*cosh(f*x + e)^4 + 4*(2*a*b^2 - b^3)*cosh(f*x + e)*sinh(f *x + e)^3 + (2*a*b^2 - b^3)*sinh(f*x + e)^4 + 2*a*b^2 - b^3 + 2*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2 + 2*(4*a^2*b - 4*a*b^2 + b^3 + 3*(2*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((2*a*b^2 - b^3)*cosh(f*x + e) ^3 + (4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*(b^3*cosh( f*x + e)^4 + 4*b^3*cosh(f*x + e)*sinh(f*x + e)^3 + b^3*sinh(f*x + e)^4 + b ^3 + 2*(2*a*b^2 - b^3)*cosh(f*x + e)^2 + 2*(3*b^3*cosh(f*x + e)^2 + 2*a*b^ 2 - b^3)*sinh(f*x + e)^2 + 4*(b^3*cosh(f*x + e)^3 + (2*a*b^2 - b^3)*cosh(f *x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^ 2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - 2*((2*a^2*b - a*b^2)*cosh(f* x + e)^4 + 4*(2*a^2*b - a*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (2*a^2*b - a*b^2)*sinh(f*x + e)^4 + 2*a^2*b - a*b^2 + 2*(4*a^3 - 4*a^2*b + a*b^2)*cos h(f*x + e)^2 + 2*(4*a^3 - 4*a^2*b + a*b^2 + 3*(2*a^2*b - a*b^2)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 4*((2*a^2*b - a*b^2)*cosh(f*x + e)^3 + (4*a^3 - 4 *a^2*b + a*b^2)*cosh(f*x + e))*sinh(f*x + e) + 2*((a*b^2 - b^3)*cosh(f*x + e)^4 + 4*(a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^3 + (a*b^2 - b^3)*sinh (f*x + e)^4 + a*b^2 - b^3 + 2*(2*a^2*b - 3*a*b^2 + b^3)*cosh(f*x + e)^2 + 2*(2*a^2*b - 3*a*b^2 + b^3 + 3*(a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x ...
\[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {1}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]