\(\int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx\) [22]

Optimal result

Integrand size = 16, antiderivative size = 184 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}+\frac {3 b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {3 b \cosh ^2(a+b x) \sinh (a+b x)}{2 d^2 (c+d x)}+\frac {3 b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{8 d^3}+\frac {9 b^2 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \] Output:

-1/2*cosh(b*x+a)^3/d/(d*x+c)^2+3/8*b^2*cosh(a-b*c/d)*Chi(b*c/d+b*x)/d^3+9/ 
8*b^2*cosh(3*a-3*b*c/d)*Chi(3*b*c/d+3*b*x)/d^3-3/2*b*cosh(b*x+a)^2*sinh(b* 
x+a)/d^2/(d*x+c)+3/8*b^2*sinh(a-b*c/d)*Shi(b*c/d+b*x)/d^3+9/8*b^2*sinh(3*a 
-3*b*c/d)*Shi(3*b*c/d+3*b*x)/d^3
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.18 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {6 d \cosh (b x) (d \cosh (a)+b (c+d x) \sinh (a))+2 d \cosh (3 b x) (d \cosh (3 a)+3 b (c+d x) \sinh (3 a))+6 d (b (c+d x) \cosh (a)+d \sinh (a)) \sinh (b x)+2 d (3 b (c+d x) \cosh (3 a)+d \sinh (3 a)) \sinh (3 b x)-6 b^2 (c+d x)^2 \left (\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b (c+d x)}{d}\right )+\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+3 \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )\right )}{16 d^3 (c+d x)^2} \] Input:

Integrate[Cosh[a + b*x]^3/(c + d*x)^3,x]
 

Output:

-1/16*(6*d*Cosh[b*x]*(d*Cosh[a] + b*(c + d*x)*Sinh[a]) + 2*d*Cosh[3*b*x]*( 
d*Cosh[3*a] + 3*b*(c + d*x)*Sinh[3*a]) + 6*d*(b*(c + d*x)*Cosh[a] + d*Sinh 
[a])*Sinh[b*x] + 2*d*(3*b*(c + d*x)*Cosh[3*a] + d*Sinh[3*a])*Sinh[3*b*x] - 
 6*b^2*(c + d*x)^2*(Cosh[a - (b*c)/d]*CoshIntegral[b*(c/d + x)] + 3*Cosh[3 
*a - (3*b*c)/d]*CoshIntegral[(3*b*(c + d*x))/d] + Sinh[a - (b*c)/d]*SinhIn 
tegral[b*(c/d + x)] + 3*Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x)) 
/d]))/(d^3*(c + d*x)^2)
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 3795, 3042, 3784, 26, 3042, 26, 3779, 3782, 3793, 2009}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3795

\(\displaystyle \frac {9 b^2 \int \frac {\cosh ^3(a+b x)}{c+d x}dx}{2 d^2}-\frac {3 b^2 \int \frac {\cosh (a+b x)}{c+d x}dx}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{c+d x}dx}{d^2}+\frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx-i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {i \sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\sinh \left (a-\frac {b c}{d}\right ) \int -\frac {i \sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx-i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3779

\(\displaystyle -\frac {3 b^2 \left (\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx\right )}{d^2}+\frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3782

\(\displaystyle \frac {9 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{c+d x}dx}{2 d^2}-\frac {3 b^2 \left (\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {9 b^2 \int \left (\frac {3 \cosh (a+b x)}{4 (c+d x)}+\frac {\cosh (3 a+3 b x)}{4 (c+d x)}\right )dx}{2 d^2}-\frac {3 b^2 \left (\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 b^2 \left (\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d^2}+\frac {9 b^2 \left (\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}+\frac {3 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\right )}{2 d^2}-\frac {3 b \sinh (a+b x) \cosh ^2(a+b x)}{2 d^2 (c+d x)}-\frac {\cosh ^3(a+b x)}{2 d (c+d x)^2}\)

Input:

Int[Cosh[a + b*x]^3/(c + d*x)^3,x]
 

Output:

-1/2*Cosh[a + b*x]^3/(d*(c + d*x)^2) - (3*b*Cosh[a + b*x]^2*Sinh[a + b*x]) 
/(2*d^2*(c + d*x)) - (3*b^2*((Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x 
])/d + (Sinh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d))/d^2 + (9*b^2*(( 
3*Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/(4*d) + (Cosh[3*a - (3*b* 
c)/d]*CoshIntegral[(3*b*c)/d + 3*b*x])/(4*d) + (3*Sinh[a - (b*c)/d]*SinhIn 
tegral[(b*c)/d + b*x])/(4*d) + (Sinh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*c) 
/d + 3*b*x])/(4*d)))/(2*d^2)
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(561\) vs. \(2(172)=344\).

Time = 2.00 (sec) , antiderivative size = 562, normalized size of antiderivative = 3.05

method	result	size
risch	\(\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} x}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-3 b x -3 a} c}{16 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{2} {\mathrm e}^{-3 b x -3 a}}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {9 b^{2} {\mathrm e}^{-\frac {3 \left (a d -c b \right )}{d}} \operatorname {expIntegral}_{1}\left (3 b x +3 a -\frac {3 \left (a d -c b \right )}{d}\right )}{16 d^{3}}+\frac {3 b^{3} {\mathrm e}^{-b x -a} x}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {3 b^{3} {\mathrm e}^{-b x -a} c}{16 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-b x -a}}{16 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {3 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{16 d^{3}}-\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{b x +a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {3 b^{2} {\mathrm e}^{\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-a d +c b}{d}\right )}{16 d^{3}}-\frac {b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{16 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {9 b^{2} {\mathrm e}^{\frac {3 a d -3 c b}{d}} \operatorname {expIntegral}_{1}\left (-3 b x -3 a -\frac {3 \left (-a d +c b \right )}{d}\right )}{16 d^{3}}\)	\(562\)

Input:

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

Output:

3/16*b^3*exp(-3*b*x-3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*x+3/16*b^3*ex 
p(-3*b*x-3*a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c-1/16*b^2*exp(-3*b*x- 
3*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-9/16*b^2/d^3*exp(-3*(a*d-b*c)/d)* 
Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)+3/16*b^3*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c* 
d*x+b^2*c^2)*x+3/16*b^3*exp(-b*x-a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)* 
c-3/16*b^2*exp(-b*x-a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-3/16*b^2/d^3*ex 
p(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b*x 
)^2-3/16*b^2/d^3*exp(b*x+a)/(b*c/d+b*x)-3/16*b^2/d^3*exp((a*d-b*c)/d)*Ei(1 
,-b*x-a-(-a*d+b*c)/d)-1/16*b^2/d^3*exp(3*b*x+3*a)/(b*c/d+b*x)^2-3/16*b^2/d 
^3*exp(3*b*x+3*a)/(b*c/d+b*x)-9/16*b^2/d^3*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x- 
3*a-3*(-a*d+b*c)/d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (172) = 344\).

Time = 0.14 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.86 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {2 \, d^{2} \cosh \left (b x + a\right )^{3} + 6 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right )^{3} + 6 \, d^{2} \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 6 \, {\left (b d^{2} x + b c d + 3 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right ) - 3 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 9 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

-1/16*(2*d^2*cosh(b*x + a)^3 + 6*d^2*cosh(b*x + a)*sinh(b*x + a)^2 + 6*(b* 
d^2*x + b*c*d)*sinh(b*x + a)^3 + 6*d^2*cosh(b*x + a) - 3*((b^2*d^2*x^2 + 2 
*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b 
^2*c^2)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - 9*((b^2*d^2*x^2 + 2*b 
^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b 
^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) + 6*(b*d^2*x + b*c* 
d + 3*(b*d^2*x + b*c*d)*cosh(b*x + a)^2)*sinh(b*x + a) - 3*((b^2*d^2*x^2 + 
 2*b^2*c*d*x + b^2*c^2)*Ei((b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + 
 b^2*c^2)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - 9*((b^2*d^2*x^2 + 2 
*b^2*c*d*x + b^2*c^2)*Ei(3*(b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + 
 b^2*c^2)*Ei(-3*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^5*x^2 + 2*c*d 
^4*x + c^2*d^3)
 

Sympy [F]

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

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

Output:

Integral(cosh(a + b*x)**3/(c + d*x)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.79 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{3}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{3}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )}^{2} d} \] Input:

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

Output:

-1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(3, 3*(d*x + c)*b/d)/((d*x + c)^2*d) 
 - 3/8*e^(-a + b*c/d)*exp_integral_e(3, (d*x + c)*b/d)/((d*x + c)^2*d) - 3 
/8*e^(a - b*c/d)*exp_integral_e(3, -(d*x + c)*b/d)/((d*x + c)^2*d) - 1/8*e 
^(3*a - 3*b*c/d)*exp_integral_e(3, -3*(d*x + c)*b/d)/((d*x + c)^2*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (172) = 344\).

Time = 0.13 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.27 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} + 3 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 9 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 6 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 18 \, b^{2} c d x {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} + 9 \, b^{2} c^{2} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} + 3 \, b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 9 \, b^{2} c^{2} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} - 3 \, b d^{2} x e^{\left (3 \, b x + 3 \, a\right )} - 3 \, b d^{2} x e^{\left (b x + a\right )} + 3 \, b d^{2} x e^{\left (-b x - a\right )} + 3 \, b d^{2} x e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b c d e^{\left (3 \, b x + 3 \, a\right )} - 3 \, b c d e^{\left (b x + a\right )} + 3 \, b c d e^{\left (-b x - a\right )} + 3 \, b c d e^{\left (-3 \, b x - 3 \, a\right )} - d^{2} e^{\left (3 \, b x + 3 \, a\right )} - 3 \, d^{2} e^{\left (b x + a\right )} - 3 \, d^{2} e^{\left (-b x - a\right )} - d^{2} e^{\left (-3 \, b x - 3 \, a\right )}}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

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

Output:

1/16*(9*b^2*d^2*x^2*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 3*b^2*d^2*x^ 
2*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*b^2*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e 
^(-a + b*c/d) + 9*b^2*d^2*x^2*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 
18*b^2*c*d*x*Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 6*b^2*c*d*x*Ei((b*d 
*x + b*c)/d)*e^(a - b*c/d) + 6*b^2*c*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/ 
d) + 18*b^2*c*d*x*Ei(-3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) + 9*b^2*c^2*Ei 
(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) + 3*b^2*c^2*Ei((b*d*x + b*c)/d)*e^(a 
 - b*c/d) + 3*b^2*c^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 9*b^2*c^2*Ei(- 
3*(b*d*x + b*c)/d)*e^(-3*a + 3*b*c/d) - 3*b*d^2*x*e^(3*b*x + 3*a) - 3*b*d^ 
2*x*e^(b*x + a) + 3*b*d^2*x*e^(-b*x - a) + 3*b*d^2*x*e^(-3*b*x - 3*a) - 3* 
b*c*d*e^(3*b*x + 3*a) - 3*b*c*d*e^(b*x + a) + 3*b*c*d*e^(-b*x - a) + 3*b*c 
*d*e^(-3*b*x - 3*a) - d^2*e^(3*b*x + 3*a) - 3*d^2*e^(b*x + a) - 3*d^2*e^(- 
b*x - a) - d^2*e^(-3*b*x - 3*a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^3} \, dx=\frac {e^{4 a} \left (\int \frac {e^{3 b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right )+3 e^{2 a} \left (\int \frac {e^{b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right )+e^{a} \left (\int \frac {1}{e^{3 b x +3 a} c^{3}+3 e^{3 b x +3 a} c^{2} d x +3 e^{3 b x +3 a} c \,d^{2} x^{2}+e^{3 b x +3 a} d^{3} x^{3}}d x \right )+3 \left (\int \frac {1}{e^{b x} c^{3}+3 e^{b x} c^{2} d x +3 e^{b x} c \,d^{2} x^{2}+e^{b x} d^{3} x^{3}}d x \right )}{8 e^{a}} \] Input:

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

Output:

(e**(4*a)*int(e**(3*b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x 
) + 3*e**(2*a)*int(e**(b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 
),x) + e**a*int(1/(e**(3*a + 3*b*x)*c**3 + 3*e**(3*a + 3*b*x)*c**2*d*x + 3 
*e**(3*a + 3*b*x)*c*d**2*x**2 + e**(3*a + 3*b*x)*d**3*x**3),x) + 3*int(1/( 
e**(b*x)*c**3 + 3*e**(b*x)*c**2*d*x + 3*e**(b*x)*c*d**2*x**2 + e**(b*x)*d* 
*3*x**3),x))/(8*e**a)