Integrand size = 25, antiderivative size = 193 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {2 (a+b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-b) \text {sech}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]
2/3*(a+b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(si nh(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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*b*(1/(1+sin h(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)/f/(sec h(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(a-b)*sech(f*x+e)^2*(a+b*sinh( f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {4 i a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (2 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\frac {\left (8 a^2-3 a b+b^2+\left (4 a^2+6 a b-2 b^2\right ) \cosh (2 (e+f x))+b (a+b) \cosh (4 (e+f x))\right ) \text {sech}^2(e+f x) \tanh (e+f x)}{\sqrt {2}}}{6 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((4*I)*a*(a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*Ell ipticF[I*(e + f*x), b/a] + ((8*a^2 - 3*a*b + b^2 + (4*a^2 + 6*a*b - 2*b^2) *Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Sech[e + f*x]^2*Tanh[e + f*x])/Sqrt[2])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
Time = 0.39 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.23, 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.
\(\displaystyle \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\cos (i e+i f x)^4}dx\) |
\(\Big \downarrow \) 3671 |
\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^{5/2}}d\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \int \frac {b (a+2 b) \sinh ^2(e+f x)+a (2 a+b)}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
\(\Big \downarrow \) 400 |
\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a+b) \int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)-a b \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (\frac {2 (a+b) \sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-a b \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (\frac {2 (a+b) \sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {b \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 \left (\sinh ^2(e+f x)+1\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(((a - b)*Sinh[e + f*x]*Sqrt[a + b*Si nh[e + f*x]^2])/(3*(1 + Sinh[e + f*x]^2)^(3/2)) + ((2*(a + b)*EllipticE[Ar cTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sqrt[1 + Sinh[ e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) - (b* EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sq rt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x] ^2))]))/3))/f
3.4.72.3.1 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]
Time = 2.37 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {\left (2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4} \sinh \left (f x +e \right )+\left (2 \sqrt {-\frac {b}{a}}\, a^{2}+\sqrt {-\frac {b}{a}}\, a b -3 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, b \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a -2 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right ) \cosh \left (f x +e \right )^{2}+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \sinh \left (f x +e \right )}{3 \cosh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(324\) |
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)+(2* (-b/a)^(1/2)*a^2+(-b/a)^(1/2)*a*b-3*(-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*Ellipti cF(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 *EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)))*cosh(f*x+e)^2+((-b/a)^(1 /2)*a^2-2*(-b/a)^(1/2)*a*b+(-b/a)^(1/2)*b^2)*sinh(f*x+e))/cosh(f*x+e)^3/(- b/a)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 2071 vs. \(2 (205) = 410\).
Time = 0.12 (sec) , antiderivative size = 2071, normalized size of antiderivative = 10.73 \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2 + a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a^2 + a*b - b^2)*cosh(f* x + e)*sinh(f*x + e)^5 + (2*a^2 + a*b - b^2)*sinh(f*x + e)^6 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 2* a^2 + a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^ 3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 6*(2*a ^2 + a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b - b^2)*sinh(f*x + e)^2 + 2*a ^2 + a*b - b^2 + 6*((2*a^2 + a*b - b^2)*cosh(f*x + e)^5 + 2*(2*a^2 + a*b - b^2)*cosh(f*x + e)^3 + (2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b + b^2)*cosh(f*x + e)^6 + 6*(a*b + b^2)*cosh(f*x + e)*sinh(f*x + e )^5 + (a*b + b^2)*sinh(f*x + e)^6 + 3*(a*b + b^2)*cosh(f*x + e)^4 + 3*(5*( a*b + b^2)*cosh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(a*b + b^2) *cosh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e)^2 + 3*(5*(a*b + b^2)*cosh(f*x + e)^4 + 6*(a*b + b^2)*c osh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^2 + a*b + b^2 + 6*((a*b + b^2)*c osh(f*x + e)^5 + 2*(a*b + b^2)*cosh(f*x + e)^3 + (a*b + b^2)*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*a^2 - a*b)*cosh(f*x + e)^6 +...
Timed out. \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{4} \,d x } \]
Exception generated. \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage3:=type(sage2):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \text {sech}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {cosh}\left (e+f\,x\right )}^4} \,d x \]