\(\int x^3 \sqrt {a+a \cosh (c+d x)} \, dx\) [121]

Optimal result

Integrand size = 18, antiderivative size = 110 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {96 \sqrt {a+a \cosh (c+d x)}}{d^4}-\frac {12 x^2 \sqrt {a+a \cosh (c+d x)}}{d^2}+\frac {48 x \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^3}+\frac {2 x^3 \sqrt {a+a \cosh (c+d x)} \tanh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d} \] Output:

-96*(a+a*cosh(d*x+c))^(1/2)/d^4-12*x^2*(a+a*cosh(d*x+c))^(1/2)/d^2+48*x*(a 
+a*cosh(d*x+c))^(1/2)*tanh(1/2*d*x+1/2*c)/d^3+2*x^3*(a+a*cosh(d*x+c))^(1/2 
)*tanh(1/2*d*x+1/2*c)/d
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.48 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {2 \sqrt {a (1+\cosh (c+d x))} \left (-6 \left (8+d^2 x^2\right )+d x \left (24+d^2 x^2\right ) \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d^4} \] Input:

Integrate[x^3*Sqrt[a + a*Cosh[c + d*x]],x]
 

Output:

(2*Sqrt[a*(1 + Cosh[c + d*x])]*(-6*(8 + d^2*x^2) + d*x*(24 + d^2*x^2)*Tanh 
[(c + d*x)/2]))/d^4
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {3042, 3800, 3042, 3777, 26, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118}

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.

\(\displaystyle \int x^3 \sqrt {a \cosh (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 \sqrt {a+a \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3800

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int x^3 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \int x^3 \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 3777

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {6 i \int -i x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {6 \int x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {6 \int -i x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \int x^2 \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \int x \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \int x \sin \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{2}\right )dx}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 3777

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {2 i \int -i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {2 \int \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )dx}{d}\right )}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {2 \int -i \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 26

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {2 i \int \sin \left (\frac {i c}{2}+\frac {i d x}{2}\right )dx}{d}\right )}{d}\right )}{d}\right )\)

\(\Big \downarrow \) 3118

\(\displaystyle \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {a \cosh (c+d x)+a} \left (\frac {2 x^3 \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}+\frac {6 i \left (\frac {2 i x^2 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 i \left (\frac {2 x \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d}-\frac {4 \cosh \left (\frac {c}{2}+\frac {d x}{2}\right )}{d^2}\right )}{d}\right )}{d}\right )\)

Input:

Int[x^3*Sqrt[a + a*Cosh[c + d*x]],x]
 

Output:

Sqrt[a + a*Cosh[c + d*x]]*Sech[c/2 + (d*x)/2]*((2*x^3*Sinh[c/2 + (d*x)/2]) 
/d + ((6*I)*(((2*I)*x^2*Cosh[c/2 + (d*x)/2])/d - ((4*I)*((-4*Cosh[c/2 + (d 
*x)/2])/d^2 + (2*x*Sinh[c/2 + (d*x)/2])/d))/d))/d)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), 
 x_Symbol] :> Simp[(2*a)^IntPart[n]*((a + b*Sin[e + f*x])^FracPart[n]/Sin[e 
/2 + a*(Pi/(4*b)) + f*(x/2)]^(2*FracPart[n]))   Int[(c + d*x)^m*Sin[e/2 + a 
*(Pi/(4*b)) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && 
EqQ[a^2 - b^2, 0] && IntegerQ[n + 1/2] && (GtQ[n, 0] || IGtQ[m, 0])
 

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98

method	result	size
risch	\(\frac {\sqrt {2}\, \sqrt {a \left ({\mathrm e}^{d x +c}+1\right )^{2} {\mathrm e}^{-d x -c}}\, \left (d^{3} x^{3} {\mathrm e}^{d x +c}-d^{3} x^{3}-6 d^{2} x^{2} {\mathrm e}^{d x +c}-6 x^{2} d^{2}+24 d x \,{\mathrm e}^{d x +c}-24 d x -48 \,{\mathrm e}^{d x +c}-48\right )}{\left ({\mathrm e}^{d x +c}+1\right ) d^{4}}\)	\(108\)

Input:

int(x^3*(a+a*cosh(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2^(1/2)*(a*(exp(d*x+c)+1)^2*exp(-d*x-c))^(1/2)/(exp(d*x+c)+1)*(d^3*x^3*exp 
(d*x+c)-d^3*x^3-6*d^2*x^2*exp(d*x+c)-6*x^2*d^2+24*d*x*exp(d*x+c)-24*d*x-48 
*exp(d*x+c)-48)/d^4
 

Fricas [F(-2)]

Exception generated. \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\int x^{3} \sqrt {a \left (\cosh {\left (c + d x \right )} + 1\right )}\, dx \] Input:

integrate(x**3*(a+a*cosh(d*x+c))**(1/2),x)
 

Output:

Integral(x**3*sqrt(a*(cosh(c + d*x) + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.09 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {{\left (\sqrt {2} \sqrt {a} d^{3} x^{3} + 6 \, \sqrt {2} \sqrt {a} d^{2} x^{2} + 24 \, \sqrt {2} \sqrt {a} d x - {\left (\sqrt {2} \sqrt {a} d^{3} x^{3} e^{c} - 6 \, \sqrt {2} \sqrt {a} d^{2} x^{2} e^{c} + 24 \, \sqrt {2} \sqrt {a} d x e^{c} - 48 \, \sqrt {2} \sqrt {a} e^{c}\right )} e^{\left (d x\right )} + 48 \, \sqrt {2} \sqrt {a}\right )} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{d^{4}} \] Input:

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

-(sqrt(2)*sqrt(a)*d^3*x^3 + 6*sqrt(2)*sqrt(a)*d^2*x^2 + 24*sqrt(2)*sqrt(a) 
*d*x - (sqrt(2)*sqrt(a)*d^3*x^3*e^c - 6*sqrt(2)*sqrt(a)*d^2*x^2*e^c + 24*s 
qrt(2)*sqrt(a)*d*x*e^c - 48*sqrt(2)*sqrt(a)*e^c)*e^(d*x) + 48*sqrt(2)*sqrt 
(a))*e^(-1/2*d*x - 1/2*c)/d^4
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\frac {\sqrt {2} {\left (\sqrt {a} d^{3} x^{3} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \sqrt {a} d^{3} x^{3} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 6 \, \sqrt {a} d^{2} x^{2} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 6 \, \sqrt {a} d^{2} x^{2} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + 24 \, \sqrt {a} d x e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 24 \, \sqrt {a} d x e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 48 \, \sqrt {a} e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 48 \, \sqrt {a} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}\right )}}{d^{4}} \] Input:

integrate(x^3*(a+a*cosh(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

sqrt(2)*(sqrt(a)*d^3*x^3*e^(1/2*d*x + 1/2*c) - sqrt(a)*d^3*x^3*e^(-1/2*d*x 
 - 1/2*c) - 6*sqrt(a)*d^2*x^2*e^(1/2*d*x + 1/2*c) - 6*sqrt(a)*d^2*x^2*e^(- 
1/2*d*x - 1/2*c) + 24*sqrt(a)*d*x*e^(1/2*d*x + 1/2*c) - 24*sqrt(a)*d*x*e^( 
-1/2*d*x - 1/2*c) - 48*sqrt(a)*e^(1/2*d*x + 1/2*c) - 48*sqrt(a)*e^(-1/2*d* 
x - 1/2*c))/d^4
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06 \[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=-\frac {\sqrt {a+a\,\left (\frac {{\mathrm {e}}^{c+d\,x}}{2}+\frac {{\mathrm {e}}^{-c-d\,x}}{2}\right )}\,\left (\frac {96\,{\mathrm {e}}^{c+d\,x}}{d^4}+\frac {48\,x}{d^3}+\frac {96}{d^4}+\frac {2\,x^3}{d}+\frac {12\,x^2}{d^2}-\frac {2\,x^3\,{\mathrm {e}}^{c+d\,x}}{d}+\frac {12\,x^2\,{\mathrm {e}}^{c+d\,x}}{d^2}-\frac {48\,x\,{\mathrm {e}}^{c+d\,x}}{d^3}\right )}{{\mathrm {e}}^{c+d\,x}+1} \] Input:

int(x^3*(a + a*cosh(c + d*x))^(1/2),x)
 

Output:

-((a + a*(exp(c + d*x)/2 + exp(- c - d*x)/2))^(1/2)*((96*exp(c + d*x))/d^4 
 + (48*x)/d^3 + 96/d^4 + (2*x^3)/d + (12*x^2)/d^2 - (2*x^3*exp(c + d*x))/d 
 + (12*x^2*exp(c + d*x))/d^2 - (48*x*exp(c + d*x))/d^3))/(exp(c + d*x) + 1 
)
 

Reduce [F]

\[ \int x^3 \sqrt {a+a \cosh (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\cosh \left (d x +c \right )+1}\, x^{3}d x \right ) \] Input:

int(x^3*(a+a*cosh(d*x+c))^(1/2),x)
 

Output:

sqrt(a)*int(sqrt(cosh(c + d*x) + 1)*x**3,x)