Integrand size = 25, antiderivative size = 79 \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {(a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 b^{3/2} f}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 b f} \] Output:
-1/2*(a-2*b)*arctanh(b^(1/2)*sinh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2))/b^(3/2 )/f+1/2*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/b/f
Time = 0.07 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {-\frac {(a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 b^{3/2}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 b}}{f} \] Input:
Integrate[Cosh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]
Output:
(-1/2*((a - 2*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^ 2]])/b^(3/2) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*b))/f
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 3669, 299, 224, 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.
| \(\displaystyle \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i e+i f x)^3}{\sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\sinh ^2(e+f x)+1}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 b}-\frac {(a-2 b) \int \frac {1}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{2 b}}{f}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 b}-\frac {(a-2 b) \int \frac {1}{1-\frac {b \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}}{2 b}}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 b}-\frac {(a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 b^{3/2}}}{f}\) |
Input:
Int[Cosh[e + f*x]^3/Sqrt[a + b*Sinh[e + f*x]^2],x]
Output:
(-1/2*((a - 2*b)*ArcTanh[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^ 2]])/b^(3/2) + (Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(2*b))/f
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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]
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{\sqrt {b}}+\frac {\sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{2 b^{\frac {3}{2}}}}{f}\) | \(93\) |
| default | \(\frac {\frac {\ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{\sqrt {b}}+\frac {\sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{2 b}-\frac {a \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{2 b^{\frac {3}{2}}}}{f}\) | \(93\) |
Input:
int(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/f*(ln(b^(1/2)*sinh(f*x+e)+(a+b*sinh(f*x+e)^2)^(1/2))/b^(1/2)+1/2*sinh(f* x+e)/b*(a+b*sinh(f*x+e)^2)^(1/2)-1/2*a/b^(3/2)*ln(b^(1/2)*sinh(f*x+e)+(a+b *sinh(f*x+e)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 673 vs. \(2 (67) = 134\).
Time = 0.15 (sec) , antiderivative size = 2367, normalized size of antiderivative = 29.96 \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \] Input:
integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")
Output:
[-1/8*(((a - 2*b)*cosh(f*x + e)^2 + 2*(a - 2*b)*cosh(f*x + e)*sinh(f*x + e ) + (a - 2*b)*sinh(f*x + e)^2)*sqrt(b)*log(-((a^2*b - 2*a*b^2 + b^3)*cosh( f*x + e)^8 + 8*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^ 2*b - 2*a*b^2 + b^3)*sinh(f*x + e)^8 + 2*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3) *cosh(f*x + e)^6 + 2*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3 + 14*(a^2*b - 2*a*b^ 2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 4*(14*(a^2*b - 2*a*b^2 + b^3)* cosh(f*x + e)^3 + 3*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh( f*x + e)^5 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^4 + (70*(a^2*b - 2 *a*b^2 + b^3)*cosh(f*x + e)^4 + 9*a^2*b - 14*a*b^2 + 6*b^3 + 30*(a^3 - 4*a ^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(14*(a^2*b - 2*a*b^2 + b^3)*cosh(f*x + e)^5 + 10*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh (f*x + e)^3 + (9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2 + 2*(14*(a^2*b - 2*a*b^2 + b^3 )*cosh(f*x + e)^6 + 15*(a^3 - 4*a^2*b + 5*a*b^2 - 2*b^3)*cosh(f*x + e)^4 + 3*a*b^2 - 2*b^3 + 3*(9*a^2*b - 14*a*b^2 + 6*b^3)*cosh(f*x + e)^2)*sinh(f* x + e)^2 + sqrt(2)*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^6 + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^2 - 2*a*b + b^2)*sinh(f*x + e)^6 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 3*(5*(a^2 - 2*a*b + b^2)*cosh(f* x + e)^2 - a^2 + 2*a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(a^2 - 2*a*b + b^2)*c osh(f*x + e)^3 - 3*(a^2 - 2*a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 -...
Timed out. \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Timed out} \] Input:
integrate(cosh(f*x+e)**3/(a+b*sinh(f*x+e)**2)**(1/2),x)
Output:
Timed out
\[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\cosh \left (f x + e\right )^{3}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \] Input:
integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")
Output:
integrate(cosh(f*x + e)^3/sqrt(b*sinh(f*x + e)^2 + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (67) = 134\).
Time = 0.22 (sec) , antiderivative size = 433, normalized size of antiderivative = 5.48 \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {{\left (\frac {4 \, {\left (a e^{\left (2 \, e\right )} - 2 \, b e^{\left (2 \, e\right )}\right )} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} e^{\left (2 \, e\right )}}{b} - \frac {2 \, {\left (a e^{\left (2 \, e\right )} - 2 \, b e^{\left (2 \, e\right )}\right )} \log \left ({\left | {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} \sqrt {b} + 2 \, a - b \right |}\right )}{b^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} a e^{\left (2 \, e\right )} - {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} b e^{\left (2 \, e\right )} + b^{\frac {3}{2}} e^{\left (2 \, e\right )}\right )}}{{\left ({\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} - b\right )} b}\right )} e^{\left (-2 \, e\right )}}{8 \, f} \] Input:
integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
Output:
-1/8*(4*(a*e^(2*e) - 2*b*e^(2*e))*arctan(-(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(-b)*b) - 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)*e^(2*e)/b - 2*(a*e^(2*e) - 2*b*e^(2*e))*log(abs((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) + 2*a - b))/b^(3/2) - 2*(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))*a*e^(2*e) - (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))*b*e^(2*e) + b^(3/2)*e^(2*e))/((( 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))^2 - b)*b))*e^(-2*e)/f
Timed out. \[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {{\mathrm {cosh}\left (e+f\,x\right )}^3}{\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \] Input:
int(cosh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(1/2),x)
Output:
int(cosh(e + f*x)^3/(a + b*sinh(e + f*x)^2)^(1/2), x)
\[ \int \frac {\cosh ^3(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \cosh \left (f x +e \right )^{3}}{\sinh \left (f x +e \right )^{2} b +a}d x \] Input:
int(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(1/2),x)
Output:
int((sqrt(sinh(e + f*x)**2*b + a)*cosh(e + f*x)**3)/(sinh(e + f*x)**2*b + a),x)