Integrand size = 25, antiderivative size = 130 \[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=-\frac {(a-b) (a+3 b) \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 b^{3/2} f}-\frac {(a+3 b) \cosh (e+f x) \sqrt {a-b+b \cosh ^2(e+f x)}}{8 b f}+\frac {\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 b f} \]
-1/8*(a-b)*(a+3*b)*arctanh(cosh(f*x+e)*b^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2) )/b^(3/2)/f+1/4*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(3/2)/b/f-1/8*(a+3*b)*co sh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(1/2)/b/f
Time = 0.72 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {\frac {\cosh (e+f x) (a-4 b+b \cosh (2 (e+f x))) \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}}{2 b}+\frac {(-a+b) (a+3 b) \log \left (\sqrt {2} \sqrt {b} \cosh (e+f x)+\sqrt {2 a-b+b \cosh (2 (e+f x))}\right )}{b^{3/2}}}{8 f} \]
((Cosh[e + f*x]*(a - 4*b + b*Cosh[2*(e + f*x)])*Sqrt[4*a - 2*b + 2*b*Cosh[ 2*(e + f*x)]])/(2*b) + ((-a + b)*(a + 3*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f* x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]])/b^(3/2))/(8*f)
Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 26, 3665, 299, 211, 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.
\(\displaystyle \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i e+i f x)^3 \sqrt {a-b \sin (i e+i f x)^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin (i e+i f x)^3 \sqrt {a-b \sin (i e+i f x)^2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \left (1-\cosh ^2(e+f x)\right ) \sqrt {b \cosh ^2(e+f x)+a-b}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle -\frac {\frac {(a+3 b) \int \sqrt {b \cosh ^2(e+f x)+a-b}d\cosh (e+f x)}{4 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b}}{f}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle -\frac {\frac {(a+3 b) \left (\frac {1}{2} (a-b) \int \frac {1}{\sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )}{4 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b}}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {\frac {(a+3 b) \left (\frac {1}{2} (a-b) \int \frac {1}{1-\frac {b \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )}{4 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {(a+3 b) \left (\frac {(a-b) \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 \sqrt {b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )}{4 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 b}}{f}\) |
-((-1/4*(Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/b + ((a + 3*b)*( ((a - b)*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]]) /(2*Sqrt[b]) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/2))/(4*b))/ f)
3.1.66.3.1 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)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[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. \(338\) vs. \(2(114)=228\).
Time = 0.23 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.61
method | result | size |
default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (4 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {5}{2}} \cosh \left (f x +e \right )^{2}-10 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {5}{2}}+2 a \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {3}{2}}-\ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a^{2} b -2 \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a \,b^{2}+3 b^{3} \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right )\right )}{16 b^{\frac {5}{2}} \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(339\) |
1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(4*(b*cosh(f*x+e)^4+(a-b)*c osh(f*x+e)^2)^(1/2)*b^(5/2)*cosh(f*x+e)^2-10*(b*cosh(f*x+e)^4+(a-b)*cosh(f *x+e)^2)^(1/2)*b^(5/2)+2*a*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^( 3/2)-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^2*b-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*b^2+3*b^3*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)))/b^(5/2)/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. 1180 vs. \(2 (114) = 228\).
Time = 0.37 (sec) , antiderivative size = 3037, normalized size of antiderivative = 23.36 \[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
[-1/64*(2*((a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^4 + 4*(a^2 + 2*a*b - 3*b^2) *cosh(f*x + e)^3*sinh(f*x + e) + 6*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)^2*s inh(f*x + e)^2 + 4*(a^2 + 2*a*b - 3*b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + ( a^2 + 2*a*b - 3*b^2)*sinh(f*x + e)^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))*s inh(f*x + e)^5 + (9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^4 + (70*a^2*b*cos h(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*cosh(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)*co sh(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*...
\[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \sinh ^{3}{\left (e + f x \right )}\, dx \]
\[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int { \sqrt {b \sinh \left (f x + e\right )^{2} + a} \sinh \left (f x + e\right )^{3} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 890 vs. \(2 (114) = 228\).
Time = 0.64 (sec) , antiderivative size = 890, normalized size of antiderivative = 6.85 \[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
1/64*(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*a*e^(6*e) - 7*b*e^(6*e))*e^(-2*e)/b + e^(2*f*x + 6*e)) + 8*(a^2*e^ (4*e) + 2*a*b*e^(4*e) - 3*b^2*e^(4*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))/s qrt(-b))/(sqrt(-b)*b) + 4*(a^2*sqrt(b)*e^(4*e) + 2*a*b^(3/2)*e^(4*e) - 3*b ^(5/2)*e^(4*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))*b - 2*a*sqrt(b) + b^(3/ 2)))/b^2 - 4*(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))^3*a^2*e^(4*e) - 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))^3*a*b*e^(4*e) + 4*(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))^3*b^2*e^(4*e) + 4* (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*a*b^(3/2)*e^(4*e) - 5*(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^(5/2)*e^(4*e) + 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^2*b*e^(4*e) + 4*(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*b^2*e^(4*e) - 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))*b^3*e^(...
Timed out. \[ \int \sinh ^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int {\mathrm {sinh}\left (e+f\,x\right )}^3\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]