Integrand size = 23, antiderivative size = 84 \[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{f}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{f} \]
-arctanh(cosh(f*x+e)*a^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))*a^(1/2)/f+arctan h(cosh(f*x+e)*b^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))*b^(1/2)/f
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.15 \[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \cosh (e+f x)}{\sqrt {2 a-b+b \cosh (2 (e+f x))}}\right )+\sqrt {b} \log \left (\sqrt {2} \sqrt {b} \cosh (e+f x)+\sqrt {2 a-b+b \cosh (2 (e+f x))}\right )}{f} \]
(-(Sqrt[a]*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2 *(e + f*x)]]]) + Sqrt[b]*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/f
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 26, 3665, 301, 224, 219, 291, 219}
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.
\(\displaystyle \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a-b \sin (i e+i f x)^2}}{\sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {a-b \sin (i e+i f x)^2}}{\sin (i e+i f x)}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {\sqrt {b \cosh ^2(e+f x)+a-b}}{1-\cosh ^2(e+f x)}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 301 |
\(\displaystyle -\frac {a \int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)-b \int \frac {1}{\sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {a \int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)-b \int \frac {1}{1-\frac {b \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a \int \frac {1}{\left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {a \int \frac {1}{1-\frac {a \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{f}\) |
-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2] ] - Sqrt[b]*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2 ]])/f)
3.1.68.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(72)=144\).
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.07
method | result | size |
default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (\sqrt {b}\, \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right )-\sqrt {a}\, \ln \left (\frac {\left (a +b \right ) \cosh \left (f x +e \right )^{2}+2 \sqrt {a}\, \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}+a -b}{\sinh \left (f x +e \right )^{2}}\right )\right )}{2 \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(174\) |
1/2*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(b^(1/2)*ln(1/2*(2*b*cosh(f* x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2)+a-b)/b^(1/2)) -a^(1/2)*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x +e)^2)^(1/2)+a-b)/sinh(f*x+e)^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. 719 vs. \(2 (72) = 144\).
Time = 0.38 (sec) , antiderivative size = 4423, normalized size of antiderivative = 52.65 \[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
[1/4*(sqrt(b)*log((a^2*b*cosh(f*x + e)^8 + 8*a^2*b*cosh(f*x + e)*sinh(f*x + e)^7 + a^2*b*sinh(f*x + e)^8 + 2*(a^3 + a^2*b)*cosh(f*x + e)^6 + 2*(14*a ^2*b*cosh(f*x + e)^2 + a^3 + a^2*b)*sinh(f*x + e)^6 + 4*(14*a^2*b*cosh(f*x + e)^3 + 3*(a^3 + a^2*b)*cosh(f*x + e))*sinh(f*x + e)^5 + (9*a^2*b - 4*a* b^2 + b^3)*cosh(f*x + e)^4 + (70*a^2*b*cosh(f*x + e)^4 + 9*a^2*b - 4*a*b^2 + b^3 + 30*(a^3 + a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(14*a^2*b*c osh(f*x + e)^5 + 10*(a^3 + a^2*b)*cosh(f*x + e)^3 + (9*a^2*b - 4*a*b^2 + b ^3)*cosh(f*x + e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - b^3)*cosh(f*x + e) ^2 + 2*(14*a^2*b*cosh(f*x + e)^6 + 15*(a^3 + a^2*b)*cosh(f*x + e)^4 + 3*a* b^2 - b^3 + 3*(9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + sqrt(2)*(a^2*cosh(f*x + e)^6 + 6*a^2*cosh(f*x + e)*sinh(f*x + e)^5 + a^2* sinh(f*x + e)^6 + 3*a^2*cosh(f*x + e)^4 + 3*(5*a^2*cosh(f*x + e)^2 + a^2)* sinh(f*x + e)^4 + 4*(5*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + e))*sinh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e)^2 + (15*a^2*cosh(f*x + e)^4 + 18*a^2 *cosh(f*x + e)^2 + 4*a*b - b^2)*sinh(f*x + e)^2 + b^2 + 2*(3*a^2*cosh(f*x + e)^5 + 6*a^2*cosh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e ))*sqrt(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*(2*a^2*b* cosh(f*x + e)^7 + 3*(a^3 + a^2*b)*cosh(f*x + e)^5 + (9*a^2*b - 4*a*b^2 + b ^3)*cosh(f*x + e)^3 + (3*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))/(co...
\[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname {csch}{\left (e + f x \right )}\, dx \]
\[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int { \sqrt {b \sinh \left (f x + e\right )^{2} + a} \operatorname {csch}\left (f x + e\right ) \,d x } \]
Exception generated. \[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \frac {\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}}{\mathrm {sinh}\left (e+f\,x\right )} \,d x \]