\(\int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx\) [21]

Optimal result

Integrand size = 16, antiderivative size = 145 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 b \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}+\frac {3 b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \] Output:

-cosh(b*x+a)^3/d/(d*x+c)+3/4*b*Chi(3*b*c/d+3*b*x)*sinh(3*a-3*b*c/d)/d^2+3/ 
4*b*Chi(b*c/d+b*x)*sinh(a-b*c/d)/d^2+3/4*b*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^ 
2+3/4*b*cosh(3*a-3*b*c/d)*Shi(3*b*c/d+3*b*x)/d^2
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \cosh (a) \cosh (b x)}{4 d (c+d x)}-\frac {\cosh (3 a) \cosh (3 b x)}{4 d (c+d x)}-\frac {3 \sinh (a) \sinh (b x)}{4 d (c+d x)}-\frac {\sinh (3 a) \sinh (3 b x)}{4 d (c+d x)}-\frac {3 b \left (-2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )-2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )-2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )\right )}{8 d^2} \] Input:

Integrate[Cosh[a + b*x]^3/(c + d*x)^2,x]
 

Output:

(-3*Cosh[a]*Cosh[b*x])/(4*d*(c + d*x)) - (Cosh[3*a]*Cosh[3*b*x])/(4*d*(c + 
 d*x)) - (3*Sinh[a]*Sinh[b*x])/(4*d*(c + d*x)) - (Sinh[3*a]*Sinh[3*b*x])/( 
4*d*(c + d*x)) - (3*b*(-2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b* 
c)/d] - 2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d] - 2*Cosh[a - (b*c) 
/d]*SinhIntegral[(b*c)/d + b*x] - 2*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3* 
b*c)/d + 3*b*x]))/(8*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3794, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3794

\(\displaystyle -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 i b \int \left (-\frac {i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right )dx}{d}\)

\(\Big \downarrow \) 2009

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

Input:

Int[Cosh[a + b*x]^3/(c + d*x)^2,x]
 

Output:

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

Defintions of rubi rules used

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 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

Maple [A] (verified)

Time = 1.79 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87

method	result	size
risch	\(-\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {3 \left (a d -c b \right )}{d}} \operatorname {expIntegral}_{1}\left (3 b x +3 a -\frac {3 \left (a d -c b \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-a d +c b}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 a d -3 c b}{d}} \operatorname {expIntegral}_{1}\left (-3 b x -3 a -\frac {3 \left (-a d +c b \right )}{d}\right )}{8 d^{2}}\)	\(271\)

Input:

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

Output:

-1/8*b*exp(-3*b*x-3*a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-3*(a*d-b*c)/d)*Ei(1,3* 
b*x+3*a-3*(a*d-b*c)/d)-3/8*b*exp(-b*x-a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-(a*d 
-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/8*b/d^2*exp(b*x+a)/(b*c/d+b*x)-3/8*b/d^ 
2*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/8*b/d^2*exp(3*b*x+3*a)/(b*c 
/d+b*x)-3/8*b/d^2*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. 305 vs. \(2 (137) = 274\).

Time = 0.11 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.10 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \cosh \left (b x + a\right )^{3} + 6 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, d \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:

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

Output:

-1/8*(2*d*cosh(b*x + a)^3 + 6*d*cosh(b*x + a)*sinh(b*x + a)^2 + 6*d*cosh(b 
*x + a) - 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) - (b*d*x + b*c)*Ei(-(b*d*x 
+ b*c)/d))*cosh(-(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) - 
 (b*d*x + b*c)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) - 3*((b*d*x 
+ b*c)*Ei((b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*sinh(-(b* 
c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-3 
*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^3*x + c*d^2)
 

Sympy [F]

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

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

Output:

Integral(cosh(a + b*x)**3/(c + d*x)**2, x)
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (137) = 274\).

Time = 0.17 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.41 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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