Integrand size = 23, antiderivative size = 121 \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {3 (a-b)^2 \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{8 \sqrt {b} f}+\frac {3 (a-b) \cosh (e+f x) \sqrt {a-b+b \cosh ^2(e+f x)}}{8 f}+\frac {\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{4 f} \]
1/4*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(3/2)/f+3/8*(a-b)^2*arctanh(cosh(f*x +e)*b^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1/2))/f/b^(1/2)+3/8*(a-b)*cosh(f*x+e)*( a-b+b*cosh(f*x+e)^2)^(1/2)/f
Time = 0.36 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\frac {1}{2} \cosh (e+f x) (5 a-4 b+b \cosh (2 (e+f x))) \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}+\frac {3 (a-b)^2 \log \left (\sqrt {2} \sqrt {b} \cosh (e+f x)+\sqrt {2 a-b+b \cosh (2 (e+f x))}\right )}{\sqrt {b}}}{8 f} \]
((Cosh[e + f*x]*(5*a - 4*b + b*Cosh[2*(e + f*x)])*Sqrt[4*a - 2*b + 2*b*Cos h[2*(e + f*x)]])/2 + (3*(a - b)^2*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt [2*a - b + b*Cosh[2*(e + f*x)]]])/Sqrt[b])/(8*f)
Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 3665, 211, 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.
\(\displaystyle \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \sin (i e+i f x) \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sin (i e+i f x) \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle \frac {\int \left (b \cosh ^2(e+f x)+a-b\right )^{3/2}d\cosh (e+f x)}{f}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {3}{4} (a-b) \int \sqrt {b \cosh ^2(e+f x)+a-b}d\cosh (e+f x)+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {3}{4} (a-b) \left (\frac {1}{2} (a-b) \int \frac {1}{\sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\frac {3}{4} (a-b) \left (\frac {1}{2} (a-b) \int \frac {1}{1-\frac {b \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {3}{4} (a-b) \left (\frac {(a-b) \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 \sqrt {b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{f}\) |
((Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/4 + (3*(a - b)*(((a - b )*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*Sqr t[b]) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/2))/4)/f
3.1.77.3.1 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[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. \(335\) vs. \(2(105)=210\).
Time = 0.06 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.78
method | result | size |
default | \(\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (4 b^{\frac {3}{2}} \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \cosh \left (f x +e \right )^{2}-10 b^{\frac {3}{2}} \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}+3 \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a^{2}-6 b \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a +3 b^{2} \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right )+10 a \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}\right )}{16 \sqrt {b}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(336\) |
1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(4*b^(3/2)*(b*cosh(f*x+e)^4 +(a-b)*cosh(f*x+e)^2)^(1/2)*cosh(f*x+e)^2-10*b^(3/2)*(b*cosh(f*x+e)^4+(a-b )*cosh(f*x+e)^2)^(1/2)+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))*a^2-6*b*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 +3*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))+10*a*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2 )*b^(1/2))/b^(1/2)/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. 1151 vs. \(2 (105) = 210\).
Time = 0.38 (sec) , antiderivative size = 2977, normalized size of antiderivative = 24.60 \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/64*(6*((a^2 - 2*a*b + b^2)*cosh(f*x + e)^4 + 4*(a^2 - 2*a*b + b^2)*cosh (f*x + e)^3*sinh(f*x + e) + 6*(a^2 - 2*a*b + b^2)*cosh(f*x + e)^2*sinh(f*x + e)^2 + 4*(a^2 - 2*a*b + b^2)*cosh(f*x + e)*sinh(f*x + e)^3 + (a^2 - 2*a *b + b^2)*sinh(f*x + e)^4)*sqrt(b)*log((a^2*b*cosh(f*x + e)^8 + 8*a^2*b*co sh(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*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)*c osh(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)*s inh(f*x + e)^5 + a^2*sinh(f*x + e)^6 + 3*a^2*cosh(f*x + e)^4 + 3*(5*a^2*co sh(f*x + e)^2 + a^2)*sinh(f*x + e)^4 + 4*(5*a^2*cosh(f*x + e)^3 + 3*a^2*co sh(f*x + e))*sinh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e)^2 + (15*a^2*cos h(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 + ...
Timed out. \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sinh \left (f x + e\right ) \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (105) = 210\).
Time = 0.73 (sec) , antiderivative size = 897, normalized size of antiderivative = 7.41 \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
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)*(b*e^(2*f*x + 6*e) + (10*a*b*e^(6*e) - 7*b^2*e^(6*e))*e^(-2*e)/b) - 24 *(a^2*e^(4*e) - 2*a*b*e^(4*e) + 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))/sqrt(-b))/sqrt(-b) - 12*(a^2*sqrt(b)*e^(4*e) - 2*a*b^(3/2)*e^(4*e) + 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 - 4*(10*(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) - 12*(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) + 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))^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) - 6*(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) + 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))*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. \[ \int \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \mathrm {sinh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]