Integrand size = 25, antiderivative size = 223 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(a-b) \cosh (e+f x) \sinh (e+f x)}{3 a b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 (a+b) \cosh (e+f x) E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} b^{3/2} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
-1/3*(a-b)*cosh(f*x+e)*sinh(f*x+e)/a/b/f/(a+b*sinh(f*x+e)^2)^(3/2)+2/3*(a+ b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1/2) *EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-a/b) ^(1/2))/a^(3/2)/b^(3/2)/f/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2))^(1/2)/(a+b *sinh(f*x+e)^2)^(1/2)-1/3*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1 /2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+ e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/b/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a) ^(1/2)
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.80 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 i a^2 (a+b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i a^2 (2 a+b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\sqrt {2} b \left (a^2+2 a b-b^2+b (a+b) \cosh (2 (e+f x))\right ) \sinh (2 (e+f x))}{3 a^2 b^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]
((2*I)*a^2*(a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I*( e + f*x), b/a] - I*a^2*(2*a + b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2) *EllipticF[I*(e + f*x), b/a] + Sqrt[2]*b*(a^2 + 2*a*b - b^2 + b*(a + b)*Co sh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*a^2*b^2*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))
Time = 0.41 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3671, 315, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i e+i f x)^4}{\left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\left (\sinh ^2(e+f x)+1\right )^{3/2}}{\left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {(2 a+b) \sinh ^2(e+f x)+a+2 b}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {2 (a+b) \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)-\int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{3 a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a b \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-1/3*((a - b)*Sinh[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2])/(a*b*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((2*(a + b)*Ellip ticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b]*Sqrt[1 + Sinh[e + f *x]^2])/(Sqrt[a]*Sqrt[b]*Sqrt[(a*(1 + Sinh[e + f*x]^2))/(a + b*Sinh[e + f* x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - (EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]))/(3*a*b)))/f
3.4.96.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(596\) vs. \(2(293)=586\).
Time = 2.54 (sec) , antiderivative size = 597, normalized size of antiderivative = 2.68
| method | result | size |
| default | \(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (\sqrt {-\frac {b}{a}}\, a^{2}+\sqrt {-\frac {b}{a}}\, a b -2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, b \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a -2 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \cosh \left (f x +e \right )^{2}+a^{2} \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 )+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 ) b -2 \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^{2}-2 \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 ) a^{2}+2 \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^{2}}{3 a^{2} \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \sqrt {-\frac {b}{a}}\, b \cosh \left (f x +e \right ) f}\) | \(597\) |
1/3*((2*(-b/a)^(1/2)*a*b+2*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^4*sinh(f*x+e)+((- b/a)^(1/2)*a^2+(-b/a)^(1/2)*a*b-2*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^2*sinh(f*x +e)+(cosh(f*x+e)^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*b*(a*EllipticF (sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))+2*b*EllipticF(sinh(f*x+e)*(-b/a)^(1 /2),(a/b)^(1/2))-2*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a-2*b*E llipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)))*cosh(f*x+e)^2+a^2*(b/a*cos h(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))+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-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^2-2*(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))*a^2+2*(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^2)/a^2/(a+b*sinh(f*x+e)^2)^(3/2)/(-b/a)^(1/2)/b/cosh(f*x+e)/ f
Leaf count of result is larger than twice the leaf count of optimal. 4355 vs. \(2 (231) = 462\).
Time = 0.19 (sec) , antiderivative size = 4355, normalized size of antiderivative = 19.53 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^8 + 8*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a^2*b^2 + a*b^3 - b^4)*sinh(f*x + e)^8 + 4*(4*a^3*b - 3*a*b^3 + b^4)*cosh(f*x + e)^6 + 4*(4*a^3*b - 3*a*b^3 + b^4 + 7*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*( 7*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^3 + 3*(4*a^3*b - 3*a*b^3 + b^4)* cosh(f*x + e))*sinh(f*x + e)^5 + 2*(16*a^4 - 8*a^3*b - 10*a^2*b^2 + 11*a*b ^3 - 3*b^4)*cosh(f*x + e)^4 + 2*(35*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e )^4 + 16*a^4 - 8*a^3*b - 10*a^2*b^2 + 11*a*b^3 - 3*b^4 + 30*(4*a^3*b - 3*a *b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 2*a^2*b^2 + a*b^3 - b^4 + 8 *(7*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^5 + 10*(4*a^3*b - 3*a*b^3 + b^ 4)*cosh(f*x + e)^3 + (16*a^4 - 8*a^3*b - 10*a^2*b^2 + 11*a*b^3 - 3*b^4)*co sh(f*x + e))*sinh(f*x + e)^3 + 4*(4*a^3*b - 3*a*b^3 + b^4)*cosh(f*x + e)^2 + 4*(7*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^6 + 15*(4*a^3*b - 3*a*b^3 + b^4)*cosh(f*x + e)^4 + 4*a^3*b - 3*a*b^3 + b^4 + 3*(16*a^4 - 8*a^3*b - 1 0*a^2*b^2 + 11*a*b^3 - 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((2*a^2 *b^2 + a*b^3 - b^4)*cosh(f*x + e)^7 + 3*(4*a^3*b - 3*a*b^3 + b^4)*cosh(f*x + e)^5 + (16*a^4 - 8*a^3*b - 10*a^2*b^2 + 11*a*b^3 - 3*b^4)*cosh(f*x + e) ^3 + (4*a^3*b - 3*a*b^3 + b^4)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a*b^3 + b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a* b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^2*b^2 + a*b^3 - b^4)*cosh(f*x + e)^...
Timed out. \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\cosh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {cosh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]