Integrand size = 25, antiderivative size = 126 \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {3 a^2 \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{8 \sqrt {a-b} f}+\frac {3 a \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{8 f}+\frac {\text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{4 f} \]
3/8*a^2*arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/f/(a-b)^ (1/2)+1/4*sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2)*tanh(f*x+e)/f+3/8*a*sech (f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52 \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {a^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},3,\frac {3}{2},-\frac {(a-b) \sinh ^2(e+f x)}{a+b \sinh ^2(e+f x)}\right ) \sinh (e+f x)}{f \sqrt {a+b \sinh ^2(e+f x)}} \]
(a^2*Hypergeometric2F1[1/2, 3, 3/2, -(((a - b)*Sinh[e + f*x]^2)/(a + b*Sin h[e + f*x]^2))]*Sinh[e + f*x])/(f*Sqrt[a + b*Sinh[e + f*x]^2])
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3669, 292, 292, 291, 216}
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.
\(\displaystyle \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\cos (i e+i f x)^5}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^3}d\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 292 |
\(\displaystyle \frac {\frac {3}{4} a \int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^2}d\sinh (e+f x)+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}}{f}\) |
\(\Big \downarrow \) 292 |
\(\displaystyle \frac {\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {(b-a) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {3}{4} a \left (\frac {a \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 \sqrt {a-b}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}}{f}\) |
((Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4*(1 + Sinh[e + f*x]^2)^2) + (3*a*((a*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2] ])/(2*Sqrt[a - b]) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*(1 + S inh[e + f*x]^2))))/4)/f
3.4.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
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/f S ubst[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 - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {b^{2} \sinh \left (f x +e \right )^{4}+2 \sinh \left (f x +e \right )^{2} a b +a^{2}}{\cosh \left (f x +e \right )^{6} \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(63\) |
`int/indef0`((b^2*sinh(f*x+e)^4+2*sinh(f*x+e)^2*a*b+a^2)/cosh(f*x+e)^6/(a+ b*sinh(f*x+e)^2)^(1/2),sinh(f*x+e))/f
Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (110) = 220\).
Time = 0.42 (sec) , antiderivative size = 3089, normalized size of antiderivative = 24.52 \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
[-1/16*(3*(a^2*cosh(f*x + e)^8 + 8*a^2*cosh(f*x + e)*sinh(f*x + e)^7 + a^2 *sinh(f*x + e)^8 + 4*a^2*cosh(f*x + e)^6 + 4*(7*a^2*cosh(f*x + e)^2 + a^2) *sinh(f*x + e)^6 + 6*a^2*cosh(f*x + e)^4 + 8*(7*a^2*cosh(f*x + e)^3 + 3*a^ 2*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(35*a^2*cosh(f*x + e)^4 + 30*a^2*cosh (f*x + e)^2 + 3*a^2)*sinh(f*x + e)^4 + 4*a^2*cosh(f*x + e)^2 + 8*(7*a^2*co sh(f*x + e)^5 + 10*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x + e )^3 + 4*(7*a^2*cosh(f*x + e)^6 + 15*a^2*cosh(f*x + e)^4 + 9*a^2*cosh(f*x + e)^2 + a^2)*sinh(f*x + e)^2 + a^2 + 8*(a^2*cosh(f*x + e)^7 + 3*a^2*cosh(f *x + e)^5 + 3*a^2*cosh(f*x + e)^3 + a^2*cosh(f*x + e))*sinh(f*x + e))*sqrt (-a + b)*log(((a - 2*b)*cosh(f*x + e)^4 + 4*(a - 2*b)*cosh(f*x + e)*sinh(f *x + e)^3 + (a - 2*b)*sinh(f*x + e)^4 - 2*(3*a - 2*b)*cosh(f*x + e)^2 + 2* (3*(a - 2*b)*cosh(f*x + e)^2 - 3*a + 2*b)*sinh(f*x + e)^2 - 2*sqrt(2)*(cos h(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(- a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a - 2*b)*co sh(f*x + e)^3 - (3*a - 2*b)*cosh(f*x + e))*sinh(f*x + e) + a - 2*b)/(cosh( f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh (f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) - 2*sqrt(2)*((3*a^2 - a*b - 2*b^2)*co sh(f*x + e)^6 + 6*(3*a^2 - a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 +...
Timed out. \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{5} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2017 vs. \(2 (110) = 220\).
Time = 0.86 (sec) , antiderivative size = 2017, normalized size of antiderivative = 16.01 \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
1/4*(3*a^2*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) + sqrt(b))/sqrt(a - b))/sq rt(a - b) - 2*(3*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e ^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a^2 - 8*(sqrt(b)*e^(2*f*x + 2 *e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*b^2 + 21*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^ (2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a^2*sqrt(b) - 64*(sqrt(b)*e^(2 *f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a*b^(3/2) + 8*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*b^(5/2) + 44*(sqr t(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b* e^(2*f*x + 2*e) + b))^5*a^3 - 237*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f *x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b + 96*( sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2 *b*e^(2*f*x + 2*e) + b))^5*a*b^2 - 8*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^( 4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b^3 - 292 *(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^3*sqrt(b) + 141*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4 *a^2*b^(3/2) - 96*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4...
Timed out. \[ \int \text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {cosh}\left (e+f\,x\right )}^5} \,d x \]