Integrand size = 25, antiderivative size = 133 \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {a-b} (a+2 b) \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 f}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}+\frac {(a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{2 f} \] Output:
1/2*(a-b)^(1/2)*(a+2*b)*arctan((a-b)^(1/2)*sinh(f*x+e)/(a+b*sinh(f*x+e)^2) ^(1/2))/f+b^(3/2)*arctanh(b^(1/2)*sinh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2))/f +1/2*(a-b)*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Time = 0.52 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {2 \sqrt {a-b} (a+2 b) \arctan \left (\frac {\sqrt {2 a-2 b} \sinh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )+4 b^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {b} \sinh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )+(a-b) \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))} \text {sech}(e+f x) \tanh (e+f x)}{4 f} \] Input:
Integrate[Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
Output:
(2*Sqrt[a - b]*(a + 2*b)*ArcTan[(Sqrt[2*a - 2*b]*Sinh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + 4*b^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[b]*Sinh[e + f *x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] + (a - b)*Sqrt[4*a - 2*b + 2*b*C osh[2*(e + f*x)]]*Sech[e + f*x]*Tanh[e + f*x])/(4*f)
Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3669, 315, 398, 224, 219, 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.
\(\displaystyle \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\cos (i e+i f x)^3}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^2}d\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 315 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {2 b^2 \sinh ^2(e+f x)+a (a+b)}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {\frac {1}{2} \left (2 b^2 \int \frac {1}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\left (2 b^2-a (a+b)\right ) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {1}{2} \left (2 b^2 \int \frac {1}{1-\frac {b \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}-\left (2 b^2-a (a+b)\right ) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {1}{2} \left (2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )-\left (2 b^2-a (a+b)\right ) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {1}{2} \left (2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )-\left (2 b^2-a (a+b)\right ) \int \frac {1}{1-\frac {(b-a) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{2} \left (2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )-\frac {\left (2 b^2-a (a+b)\right ) \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{\sqrt {a-b}}\right )+\frac {(a-b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}}{f}\) |
Input:
Int[Sech[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
Output:
((-(((2*b^2 - a*(a + b))*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sin h[e + f*x]^2]])/Sqrt[a - b]) + 2*b^(3/2)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/S qrt[a + b*Sinh[e + f*x]^2]])/2 + ((a - b)*Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*(1 + Sinh[e + f*x]^2)))/f
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 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] :> 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[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
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 = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.47
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {b^{2} \sinh \left (f x +e \right )^{4}+2 \sinh \left (f x +e \right )^{2} a b +a^{2}}{\cosh \left (f x +e \right )^{4} \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(63\) |
Input:
int(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
`int/indef0`((b^2*sinh(f*x+e)^4+2*sinh(f*x+e)^2*a*b+a^2)/cosh(f*x+e)^4/(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. 1213 vs. \(2 (115) = 230\).
Time = 0.33 (sec) , antiderivative size = 7126, normalized size of antiderivative = 53.58 \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate(sech(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(3/2),x)
Output:
Timed out
\[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{3} \,d x } \] Input:
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
Output:
integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sech(f*x + e)^3, x)
Exception generated. \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {cosh}\left (e+f\,x\right )}^3} \,d x \] Input:
int((a + b*sinh(e + f*x)^2)^(3/2)/cosh(e + f*x)^3,x)
Output:
int((a + b*sinh(e + f*x)^2)^(3/2)/cosh(e + f*x)^3, x)
\[ \int \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \mathrm {sech}\left (f x +e \right )^{3} \sinh \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \mathrm {sech}\left (f x +e \right )^{3}d x \right ) a \] Input:
int(sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x)
Output:
int(sqrt(sinh(e + f*x)**2*b + a)*sech(e + f*x)**3*sinh(e + f*x)**2,x)*b + int(sqrt(sinh(e + f*x)**2*b + a)*sech(e + f*x)**3,x)*a