\(\int \cosh ^3(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [326]

Optimal result

Integrand size = 25, antiderivative size = 157 \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=-\frac {a^2 (a-6 b) \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac {a (a-6 b) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{16 b f}-\frac {(a-6 b) \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b f} \] Output:

-1/16*a^2*(a-6*b)*arctanh(b^(1/2)*sinh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2))/b 
^(3/2)/f-1/16*a*(a-6*b)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/b/f-1/24*(a- 
6*b)*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/b/f+1/6*sinh(f*x+e)*(a+b*sinh(f 
*x+e)^2)^(5/2)/b/f
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.95 \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {a+b \sinh ^2(e+f x)} \left (-3 a^{3/2} (a-6 b) \text {arcsinh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )+\sqrt {b} \sinh (e+f x) \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}} \left (3 a (a+10 b)+2 b (7 a+6 b) \sinh ^2(e+f x)+8 b^2 \sinh ^4(e+f x)\right )\right )}{48 b^{3/2} f \sqrt {1+\frac {b \sinh ^2(e+f x)}{a}}} \] Input:

Integrate[Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

(Sqrt[a + b*Sinh[e + f*x]^2]*(-3*a^(3/2)*(a - 6*b)*ArcSinh[(Sqrt[b]*Sinh[e 
 + f*x])/Sqrt[a]] + Sqrt[b]*Sinh[e + f*x]*Sqrt[1 + (b*Sinh[e + f*x]^2)/a]* 
(3*a*(a + 10*b) + 2*b*(7*a + 6*b)*Sinh[e + f*x]^2 + 8*b^2*Sinh[e + f*x]^4) 
))/(48*b^(3/2)*f*Sqrt[1 + (b*Sinh[e + f*x]^2)/a])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3669, 299, 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.

\(\displaystyle \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (i e+i f x)^3 \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \left (\sinh ^2(e+f x)+1\right ) \left (b \sinh ^2(e+f x)+a\right )^{3/2}d\sinh (e+f x)}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b}-\frac {(a-6 b) \int \left (b \sinh ^2(e+f x)+a\right )^{3/2}d\sinh (e+f x)}{6 b}}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b}-\frac {(a-6 b) \left (\frac {3}{4} a \int \sqrt {b \sinh ^2(e+f x)+a}d\sinh (e+f x)+\frac {1}{4} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{6 b}}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b}-\frac {(a-6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {1}{2} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{4} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{6 b}}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b}-\frac {(a-6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}+\frac {1}{2} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{4} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{6 b}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 b}-\frac {(a-6 b) \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 \sqrt {b}}+\frac {1}{2} \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )+\frac {1}{4} \sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{6 b}}{f}\)

Input:

Int[Cosh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

Output:

((Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(5/2))/(6*b) - ((a - 6*b)*((Sinh[e 
 + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/4 + (3*a*((a*ArcTanh[(Sqrt[b]*Sinh[ 
e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[b]) + (Sinh[e + f*x]*Sqrt[ 
a + b*Sinh[e + f*x]^2])/2))/4))/(6*b))/f
 

Defintions of rubi rules used

rule 211

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])
 

rule 219

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])
 

rule 224

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]
 

rule 299

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3669

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]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(\frac {\sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}+\frac {3 a \sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}+\frac {\sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 b f}-\frac {a \sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}\)	\(215\)
default	\(\frac {\sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{4 f}+\frac {3 a \sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{8 f \sqrt {b}}+\frac {\sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {5}{2}}}{6 b f}-\frac {a \sinh \left (f x +e \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}{24 f b}-\frac {a^{2} \sinh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}{16 f b}-\frac {a^{3} \ln \left (\sqrt {b}\, \sinh \left (f x +e \right )+\sqrt {a +b \sinh \left (f x +e \right )^{2}}\right )}{16 f \,b^{\frac {3}{2}}}\)	\(215\)

Input:

int(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/f+3/8*a*sinh(f*x+e)*(a+b*sinh(f* 
x+e)^2)^(1/2)/f+3/8/f*a^2/b^(1/2)*ln(b^(1/2)*sinh(f*x+e)+(a+b*sinh(f*x+e)^ 
2)^(1/2))+1/6*sinh(f*x+e)*(a+b*sinh(f*x+e)^2)^(5/2)/b/f-1/24/f/b*a*sinh(f* 
x+e)*(a+b*sinh(f*x+e)^2)^(3/2)-1/16/f/b*a^2*sinh(f*x+e)*(a+b*sinh(f*x+e)^2 
)^(1/2)-1/16/f/b^(3/2)*a^3*ln(b^(1/2)*sinh(f*x+e)+(a+b*sinh(f*x+e)^2)^(1/2 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1734 vs. \(2 (137) = 274\).

Time = 0.26 (sec) , antiderivative size = 4491, normalized size of antiderivative = 28.61 \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cosh(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cosh \left (f x + e\right )^{3} \,d x } \] Input:

integrate(cosh(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)*cosh(f*x + e)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1546 vs. \(2 (137) = 274\).

Time = 0.78 (sec) , antiderivative size = 1546, normalized size of antiderivative = 9.85 \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

1/384*(((b*e^(2*f*x + 10*e) + (7*a*b^2*e^(14*e) + b^3*e^(14*e))*e^(-6*e)/b 
^2)*e^(2*f*x) + (6*a^2*b*e^(12*e) + 39*a*b^2*e^(12*e) - 8*b^3*e^(12*e))*e^ 
(-6*e)/b^2)*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) - 24*(a^3*e^(6*e) - 6*a^2*b*e^(6*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) + 12*(a^3*sqrt(b)*e^(6*e) - 6*a^2*b^(3/2)*e^ 
(6*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 
+ 2*(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))^5*a^3*e^(6*e) + 72*(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))^5*a^2*b*e^(6*e) - 48*(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))^5*a*b^2*e^(6*e) + 9*(s 
qrt(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))^5*b^3*e^(6*e) + 48*(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))^4*a^2 
*b^(3/2)*e^(6*e) + 24*(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))^4*a*b^(5/2)*e^(6*e) - 9*(s 
qrt(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))^4*b^(7/2)*e^(6*e) + 32*(sqrt(b)*e^(2*f*x + 2*e)...
 

Mupad [F(-1)]

Timed out. \[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {cosh}\left (e+f\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \] Input:

int(cosh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2),x)
 

Output:

int(cosh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \cosh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \cosh \left (f x +e \right )^{3} \sinh \left (f x +e \right )^{2}d x \right ) b +\left (\int \sqrt {\sinh \left (f x +e \right )^{2} b +a}\, \cosh \left (f x +e \right )^{3}d x \right ) a \] Input:

int(cosh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x)
 

Output:

int(sqrt(sinh(e + f*x)**2*b + a)*cosh(e + f*x)**3*sinh(e + f*x)**2,x)*b + 
int(sqrt(sinh(e + f*x)**2*b + a)*cosh(e + f*x)**3,x)*a