\(\int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx\) [12]

Optimal result

Integrand size = 17, antiderivative size = 195 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx=\frac {3 \sinh (a-3 c+(b-3 d) x)}{32 (b-3 d)}-\frac {9 \sinh (a-c+(b-d) x)}{32 (b-d)}-\frac {\sinh (3 (a-c)+3 (b-d) x)}{96 (b-d)}+\frac {3 \sinh (3 a-c+(3 b-d) x)}{32 (3 b-d)}+\frac {9 \sinh (a+c+(b+d) x)}{32 (b+d)}+\frac {\sinh (3 (a+c)+3 (b+d) x)}{96 (b+d)}-\frac {3 \sinh (3 a+c+(3 b+d) x)}{32 (3 b+d)}-\frac {3 \sinh (a+3 c+(b+3 d) x)}{32 (b+3 d)} \] Output:

3*sinh(a-3*c+(b-3*d)*x)/(32*b-96*d)-9*sinh(a-c+(b-d)*x)/(32*b-32*d)-sinh(3 
*a-3*c+3*(b-d)*x)/(96*b-96*d)+3*sinh(3*a-c+(3*b-d)*x)/(96*b-32*d)+9*sinh(a 
+c+(b+d)*x)/(32*b+32*d)+sinh(3*a+3*c+3*(b+d)*x)/(96*b+96*d)-3*sinh(3*a+c+( 
3*b+d)*x)/(96*b+32*d)-3*sinh(a+3*c+(b+3*d)*x)/(32*b+96*d)
 

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.91 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx=\frac {1}{96} \left (\frac {9 \sinh (a-3 c+b x-3 d x)}{b-3 d}-\frac {27 \sinh (a-c+b x-d x)}{b-d}-\frac {\sinh (3 (a-c+b x-d x))}{b-d}+\frac {9 \sinh (3 a-c+3 b x-d x)}{3 b-d}-\frac {9 \sinh (3 a+c+3 b x+d x)}{3 b+d}-\frac {9 \sinh (a+3 c+b x+3 d x)}{b+3 d}+\frac {27 \sinh (a+c+(b+d) x)}{b+d}+\frac {\sinh (3 (a+c+(b+d) x))}{b+d}\right ) \] Input:

Integrate[Sinh[a + b*x]^3*Sinh[c + d*x]^3,x]
 

Output:

((9*Sinh[a - 3*c + b*x - 3*d*x])/(b - 3*d) - (27*Sinh[a - c + b*x - d*x])/ 
(b - d) - Sinh[3*(a - c + b*x - d*x)]/(b - d) + (9*Sinh[3*a - c + 3*b*x - 
d*x])/(3*b - d) - (9*Sinh[3*a + c + 3*b*x + d*x])/(3*b + d) - (9*Sinh[a + 
3*c + b*x + 3*d*x])/(b + 3*d) + (27*Sinh[a + c + (b + d)*x])/(b + d) + Sin 
h[3*(a + c + (b + d)*x)]/(b + d))/96
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6147, 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.

\(\displaystyle \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx\)

\(\Big \downarrow \) 6147

\(\displaystyle \int \left (\frac {3}{32} \cosh (a+x (b-3 d)-3 c)-\frac {9}{32} \cosh (a+x (b-d)-c)-\frac {1}{32} \cosh (3 (a-c)+3 x (b-d))+\frac {3}{32} \cosh (3 a+x (3 b-d)-c)+\frac {9}{32} \cosh (a+x (b+d)+c)+\frac {1}{32} \cosh (3 (a+c)+3 x (b+d))-\frac {3}{32} \cosh (3 a+x (3 b+d)+c)-\frac {3}{32} \cosh (a+x (b+3 d)+3 c)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 \sinh (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac {9 \sinh (a+x (b-d)-c)}{32 (b-d)}-\frac {\sinh (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac {3 \sinh (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac {9 \sinh (a+x (b+d)+c)}{32 (b+d)}+\frac {\sinh (3 (a+c)+3 x (b+d))}{96 (b+d)}-\frac {3 \sinh (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac {3 \sinh (a+x (b+3 d)+3 c)}{32 (b+3 d)}\)

Input:

Int[Sinh[a + b*x]^3*Sinh[c + d*x]^3,x]
 

Output:

(3*Sinh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Sinh[a - c + (b - d)*x 
])/(32*(b - d)) - Sinh[3*(a - c) + 3*(b - d)*x]/(96*(b - d)) + (3*Sinh[3*a 
 - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sinh[a + c + (b + d)*x])/(32*(b + 
 d)) + Sinh[3*(a + c) + 3*(b + d)*x]/(96*(b + d)) - (3*Sinh[3*a + c + (3*b 
 + d)*x])/(32*(3*b + d)) - (3*Sinh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))
 

Defintions of rubi rules used

rule 2009

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

rule 6147

Int[Sinh[v_]^(p_.)*Sinh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v 
]^p*Sinh[w]^q, x], x] /; IGtQ[p, 0] && IGtQ[q, 0] && ((PolynomialQ[v, x] && 
 PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x 
]))
 

Maple [A] (verified)

Time = 26.38 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {3 \sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{32 \left (b -3 d \right )}-\frac {9 \sinh \left (a -c +\left (b -d \right ) x \right )}{32 \left (b -d \right )}+\frac {9 \sinh \left (a +c +\left (b +d \right ) x \right )}{32 \left (b +d \right )}-\frac {3 \sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{32 \left (b +3 d \right )}-\frac {\sinh \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{32 \left (3 b -3 d \right )}+\frac {3 \sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{32 \left (3 b -d \right )}-\frac {3 \sinh \left (3 a +c +\left (3 b +d \right ) x \right )}{32 \left (3 b +d \right )}+\frac {\sinh \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{96 b +96 d}\)	\(190\)
parallelrisch	\(\frac {\frac {9 \left (b -d \right ) \left (b +3 d \right ) \left (b -3 d \right ) \left (b +d \right ) \left (b +\frac {d}{3}\right ) \sinh \left (3 a -c +\left (3 b -d \right ) x \right )}{32}-\frac {9 \left (\frac {\left (b +3 d \right ) \left (b -3 d \right ) \left (b +d \right ) \left (b +\frac {d}{3}\right ) \sinh \left (\left (3 b -3 d \right ) x +3 a -3 c \right )}{3}-\frac {\left (b -d \right ) \left (b +3 d \right ) \left (b -3 d \right ) \left (b +\frac {d}{3}\right ) \sinh \left (\left (3 b +3 d \right ) x +3 a +3 c \right )}{3}+\left (-3 b^{4}-10 b^{3} d +10 b \,d^{3}+3 d^{4}\right ) \sinh \left (a -3 c +\left (b -3 d \right ) x \right )+\left (9 \left (b +3 d \right ) \left (b +d \right ) \left (b +\frac {d}{3}\right ) \sinh \left (a -c +\left (b -d \right ) x \right )+\left (b -d \right ) \left (\left (3 b^{2}+4 b d +d^{2}\right ) \sinh \left (a +3 c +\left (b +3 d \right ) x \right )+\left (b +3 d \right ) \left (\left (b +d \right ) \sinh \left (3 a +c +\left (3 b +d \right ) x \right )-9 \left (b +\frac {d}{3}\right ) \sinh \left (a +c +\left (b +d \right ) x \right )\right )\right )\right ) \left (b -3 d \right )\right ) \left (b -\frac {d}{3}\right )}{32}}{9 b^{6}-91 b^{4} d^{2}+91 b^{2} d^{4}-9 d^{6}}\)	\(304\)
risch	\(\frac {\left (b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-9 b \,d^{2} {\mathrm e}^{6 b x +6 a}+9 d^{3} {\mathrm e}^{6 b x +6 a}-9 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+9 b \,d^{2} {\mathrm e}^{4 b x +4 a}-27 d^{3} {\mathrm e}^{4 b x +4 a}-9 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+9 b \,d^{2} {\mathrm e}^{2 b x +2 a}+27 d^{3} {\mathrm e}^{2 b x +2 a}+b^{3}+b^{2} d -9 b \,d^{2}-9 d^{3}\right ) {\mathrm e}^{-3 b x +3 d x -3 a +3 c}}{192 \left (b +d \right ) \left (b +3 d \right ) \left (b -d \right ) \left (b -3 d \right )}-\frac {3 \left (3 b^{3} {\mathrm e}^{6 b x +6 a}-b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}+d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}+27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}-3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}+3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+b^{2} d -3 b \,d^{2}-d^{3}\right ) {\mathrm e}^{-3 b x +d x -3 a +c}}{64 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}+\frac {3 \left (3 b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-3 b \,d^{2} {\mathrm e}^{6 b x +6 a}-d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+3 b \,d^{2} {\mathrm e}^{4 b x +4 a}+3 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+3 b \,d^{2} {\mathrm e}^{2 b x +2 a}-3 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}-b^{2} d -3 b \,d^{2}+d^{3}\right ) {\mathrm e}^{-3 b x -d x -3 a -c}}{64 \left (3 b +d \right ) \left (b +d \right ) \left (3 b -d \right ) \left (b -d \right )}-\frac {\left (b^{3} {\mathrm e}^{6 b x +6 a}+b^{2} d \,{\mathrm e}^{6 b x +6 a}-9 b \,d^{2} {\mathrm e}^{6 b x +6 a}-9 d^{3} {\mathrm e}^{6 b x +6 a}-9 b^{3} {\mathrm e}^{4 b x +4 a}-27 b^{2} d \,{\mathrm e}^{4 b x +4 a}+9 b \,d^{2} {\mathrm e}^{4 b x +4 a}+27 d^{3} {\mathrm e}^{4 b x +4 a}-9 b^{3} {\mathrm e}^{2 b x +2 a}+27 b^{2} d \,{\mathrm e}^{2 b x +2 a}+9 b \,d^{2} {\mathrm e}^{2 b x +2 a}-27 d^{3} {\mathrm e}^{2 b x +2 a}+b^{3}-b^{2} d -9 b \,d^{2}+9 d^{3}\right ) {\mathrm e}^{-3 b x -3 d x -3 a -3 c}}{192 \left (b +d \right ) \left (b +3 d \right ) \left (b -d \right ) \left (b -3 d \right )}\)	\(954\)
orering	\(\text {Expression too large to display}\)	\(7417\)

Input:

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

Output:

3/32*sinh(a-3*c+(b-3*d)*x)/(b-3*d)-9/32*sinh(a-c+(b-d)*x)/(b-d)+9/32/(b+d) 
*sinh(a+c+(b+d)*x)-3/32/(b+3*d)*sinh(a+3*c+(b+3*d)*x)-1/32/(3*b-3*d)*sinh( 
(3*b-3*d)*x+3*a-3*c)+3/32/(3*b-d)*sinh(3*a-c+(3*b-d)*x)-3/32/(3*b+d)*sinh( 
3*a+c+(3*b+d)*x)+1/32/(3*b+3*d)*sinh((3*b+3*d)*x+3*a+3*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (179) = 358\).

Time = 0.10 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.75 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/48*(((9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(d*x + c)^3 - 9*(b^4*d - 10*b^2 
*d^3 + 9*d^5)*cosh(d*x + c))*sinh(b*x + a)^3 - ((9*b^5 - 82*b^3*d^2 + 9*b* 
d^4)*cosh(b*x + a)^3 + 3*(9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)*sinh 
(b*x + a)^2 - 9*(9*b^5 - 10*b^3*d^2 + b*d^4)*cosh(b*x + a))*sinh(d*x + c)^ 
3 + 3*((9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(d*x + c)*sinh(b*x + a)^3 - 3*(8 
1*b^4*d - 90*b^2*d^3 + 9*d^5 - (9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(b*x + a 
)^2)*cosh(d*x + c)*sinh(b*x + a))*sinh(d*x + c)^2 - 3*((81*b^4*d - 90*b^2* 
d^3 + 9*d^5 - (9*b^4*d - 82*b^2*d^3 + 9*d^5)*cosh(b*x + a)^2)*cosh(d*x + c 
)^3 - 9*(9*b^4*d - 82*b^2*d^3 + 9*d^5 - (b^4*d - 10*b^2*d^3 + 9*d^5)*cosh( 
b*x + a)^2)*cosh(d*x + c))*sinh(b*x + a) + 3*(9*(b^5 - 10*b^3*d^2 + 9*b*d^ 
4)*cosh(b*x + a)^3 - ((9*b^5 - 82*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)^3 - 9*( 
9*b^5 - 10*b^3*d^2 + b*d^4)*cosh(b*x + a))*cosh(d*x + c)^2 - 3*((9*b^5 - 8 
2*b^3*d^2 + 9*b*d^4)*cosh(b*x + a)*cosh(d*x + c)^2 - 9*(b^5 - 10*b^3*d^2 + 
 9*b*d^4)*cosh(b*x + a))*sinh(b*x + a)^2 - 9*(9*b^5 - 82*b^3*d^2 + 9*b*d^4 
)*cosh(b*x + a))*sinh(d*x + c))/((9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6) 
*cosh(b*x + a)^4 - 2*(9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)*cosh(b*x + 
a)^2*sinh(b*x + a)^2 + (9*b^6 - 91*b^4*d^2 + 91*b^2*d^4 - 9*d^6)*sinh(b*x 
+ a)^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3577 vs. \(2 (172) = 344\).

Time = 16.62 (sec) , antiderivative size = 3577, normalized size of antiderivative = 18.34 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((x*sinh(a)**3*sinh(c)**3, Eq(b, 0) & Eq(d, 0)), (3*x*sinh(a - 3* 
d*x)**3*sinh(c + d*x)**3/32 + 9*x*sinh(a - 3*d*x)**3*sinh(c + d*x)*cosh(c 
+ d*x)**2/32 + 9*x*sinh(a - 3*d*x)**2*sinh(c + d*x)**2*cosh(a - 3*d*x)*cos 
h(c + d*x)/32 + 3*x*sinh(a - 3*d*x)**2*cosh(a - 3*d*x)*cosh(c + d*x)**3/32 
 - 3*x*sinh(a - 3*d*x)*sinh(c + d*x)**3*cosh(a - 3*d*x)**2/32 - 9*x*sinh(a 
 - 3*d*x)*sinh(c + d*x)*cosh(a - 3*d*x)**2*cosh(c + d*x)**2/32 - 9*x*sinh( 
c + d*x)**2*cosh(a - 3*d*x)**3*cosh(c + d*x)/32 - 3*x*cosh(a - 3*d*x)**3*c 
osh(c + d*x)**3/32 - 61*sinh(a - 3*d*x)**3*sinh(c + d*x)**2*cosh(c + d*x)/ 
(320*d) + sinh(a - 3*d*x)**3*cosh(c + d*x)**3/(30*d) - 117*sinh(a - 3*d*x) 
**2*sinh(c + d*x)**3*cosh(a - 3*d*x)/(320*d) + 3*sinh(a - 3*d*x)*sinh(c + 
d*x)**2*cosh(a - 3*d*x)**2*cosh(c + d*x)/(20*d) - 11*sinh(a - 3*d*x)*cosh( 
a - 3*d*x)**2*cosh(c + d*x)**3/(320*d) + sinh(c + d*x)**3*cosh(a - 3*d*x)* 
*3/(4*d) - 3*sinh(c + d*x)*cosh(a - 3*d*x)**3*cosh(c + d*x)**2/(320*d), Eq 
(b, -3*d)), (5*x*sinh(a - d*x)**3*sinh(c + d*x)**3/16 - 3*x*sinh(a - d*x)* 
*3*sinh(c + d*x)*cosh(c + d*x)**2/16 + 9*x*sinh(a - d*x)**2*sinh(c + d*x)* 
*2*cosh(a - d*x)*cosh(c + d*x)/16 - 3*x*sinh(a - d*x)**2*cosh(a - d*x)*cos 
h(c + d*x)**3/16 - 3*x*sinh(a - d*x)*sinh(c + d*x)**3*cosh(a - d*x)**2/16 
+ 9*x*sinh(a - d*x)*sinh(c + d*x)*cosh(a - d*x)**2*cosh(c + d*x)**2/16 - 3 
*x*sinh(c + d*x)**2*cosh(a - d*x)**3*cosh(c + d*x)/16 + 5*x*cosh(a - d*x)* 
*3*cosh(c + d*x)**3/16 + 3*sinh(a - d*x)**3*sinh(c + d*x)**2*cosh(c + d...
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(-(3*d)/b>0)', see `assume?` for 
more detai
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (179) = 358\).

Time = 0.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.91 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx=\frac {e^{\left (3 \, b x + 3 \, d x + 3 \, a + 3 \, c\right )}}{192 \, {\left (b + d\right )}} - \frac {3 \, e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{64 \, {\left (3 \, b + d\right )}} + \frac {3 \, e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{64 \, {\left (3 \, b - d\right )}} - \frac {e^{\left (3 \, b x - 3 \, d x + 3 \, a - 3 \, c\right )}}{192 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} + \frac {9 \, e^{\left (b x + d x + a + c\right )}}{64 \, {\left (b + d\right )}} - \frac {9 \, e^{\left (b x - d x + a - c\right )}}{64 \, {\left (b - d\right )}} + \frac {3 \, e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} - \frac {3 \, e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{64 \, {\left (b - 3 \, d\right )}} + \frac {9 \, e^{\left (-b x + d x - a + c\right )}}{64 \, {\left (b - d\right )}} - \frac {9 \, e^{\left (-b x - d x - a - c\right )}}{64 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{64 \, {\left (b + 3 \, d\right )}} + \frac {e^{\left (-3 \, b x + 3 \, d x - 3 \, a + 3 \, c\right )}}{192 \, {\left (b - d\right )}} - \frac {3 \, e^{\left (-3 \, b x + d x - 3 \, a + c\right )}}{64 \, {\left (3 \, b - d\right )}} + \frac {3 \, e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{64 \, {\left (3 \, b + d\right )}} - \frac {e^{\left (-3 \, b x - 3 \, d x - 3 \, a - 3 \, c\right )}}{192 \, {\left (b + d\right )}} \] Input:

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

Output:

1/192*e^(3*b*x + 3*d*x + 3*a + 3*c)/(b + d) - 3/64*e^(3*b*x + d*x + 3*a + 
c)/(3*b + d) + 3/64*e^(3*b*x - d*x + 3*a - c)/(3*b - d) - 1/192*e^(3*b*x - 
 3*d*x + 3*a - 3*c)/(b - d) - 3/64*e^(b*x + 3*d*x + a + 3*c)/(b + 3*d) + 9 
/64*e^(b*x + d*x + a + c)/(b + d) - 9/64*e^(b*x - d*x + a - c)/(b - d) + 3 
/64*e^(b*x - 3*d*x + a - 3*c)/(b - 3*d) - 3/64*e^(-b*x + 3*d*x - a + 3*c)/ 
(b - 3*d) + 9/64*e^(-b*x + d*x - a + c)/(b - d) - 9/64*e^(-b*x - d*x - a - 
 c)/(b + d) + 3/64*e^(-b*x - 3*d*x - a - 3*c)/(b + 3*d) + 1/192*e^(-3*b*x 
+ 3*d*x - 3*a + 3*c)/(b - d) - 3/64*e^(-3*b*x + d*x - 3*a + c)/(3*b - d) + 
 3/64*e^(-3*b*x - d*x - 3*a - c)/(3*b + d) - 1/192*e^(-3*b*x - 3*d*x - 3*a 
 - 3*c)/(b + d)
 

Mupad [B] (verification not implemented)

Time = 1.62 (sec) , antiderivative size = 906, normalized size of antiderivative = 4.65 \[ \int \sinh ^3(a+b x) \sinh ^3(c+d x) \, dx={\mathrm {e}}^{3\,a+c+3\,b\,x+d\,x}\,\left (\frac {-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )-{\mathrm {e}}^{3\,a-c+3\,b\,x-d\,x}\,\left (\frac {-9\,b^3-3\,b^2\,d+9\,b\,d^2+3\,d^3}{576\,b^4-640\,b^2\,d^2+64\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-9\,b^3+3\,b^2\,d+9\,b\,d^2-3\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-81\,b^3-81\,b^2\,d+9\,b\,d^2+9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-81\,b^3+81\,b^2\,d+9\,b\,d^2-9\,d^3\right )}{576\,b^4-640\,b^2\,d^2+64\,d^4}\right )+{\mathrm {e}}^{3\,a-3\,c+3\,b\,x-3\,d\,x}\,\left (\frac {-b^3-b^2\,d+9\,b\,d^2+9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3+b^2\,d+9\,b\,d^2-9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right )-{\mathrm {e}}^{3\,a+3\,c+3\,b\,x+3\,d\,x}\,\left (\frac {-b^3+b^2\,d+9\,b\,d^2-9\,d^3}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}+\frac {{\mathrm {e}}^{-6\,a-6\,b\,x}\,\left (-b^3-b^2\,d+9\,b\,d^2+9\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-2\,a-2\,b\,x}\,\left (-9\,b^3+27\,b^2\,d+9\,b\,d^2-27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}-\frac {{\mathrm {e}}^{-4\,a-4\,b\,x}\,\left (-9\,b^3-27\,b^2\,d+9\,b\,d^2+27\,d^3\right )}{192\,b^4-1920\,b^2\,d^2+1728\,d^4}\right ) \] Input:

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

Output:

exp(3*a + c + 3*b*x + d*x)*((9*b*d^2 + 3*b^2*d - 9*b^3 - 3*d^3)/(576*b^4 + 
 64*d^4 - 640*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d^2 - 3*b^2*d - 9*b^3 + 
3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 2*a - 2*b*x)*(9*b*d^2 + 
81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 4*a 
- 4*b*x)*(9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^ 
2*d^2)) - exp(3*a - c + 3*b*x - d*x)*((9*b*d^2 - 3*b^2*d - 9*b^3 + 3*d^3)/ 
(576*b^4 + 64*d^4 - 640*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d^2 + 3*b^2*d 
- 9*b^3 - 3*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - (exp(- 2*a - 2*b*x)*( 
9*b*d^2 - 81*b^2*d - 81*b^3 + 9*d^3))/(576*b^4 + 64*d^4 - 640*b^2*d^2) - ( 
exp(- 4*a - 4*b*x)*(9*b*d^2 + 81*b^2*d - 81*b^3 - 9*d^3))/(576*b^4 + 64*d^ 
4 - 640*b^2*d^2)) + exp(3*a - 3*c + 3*b*x - 3*d*x)*((9*b*d^2 - b^2*d - b^3 
 + 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- 6*a - 6*b*x)*(9*b*d 
^2 + b^2*d - b^3 - 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- 2* 
a - 2*b*x)*(9*b*d^2 - 27*b^2*d - 9*b^3 + 27*d^3))/(192*b^4 + 1728*d^4 - 19 
20*b^2*d^2) - (exp(- 4*a - 4*b*x)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/( 
192*b^4 + 1728*d^4 - 1920*b^2*d^2)) - exp(3*a + 3*c + 3*b*x + 3*d*x)*((9*b 
*d^2 + b^2*d - b^3 - 9*d^3)/(192*b^4 + 1728*d^4 - 1920*b^2*d^2) + (exp(- 6 
*a - 6*b*x)*(9*b*d^2 - b^2*d - b^3 + 9*d^3))/(192*b^4 + 1728*d^4 - 1920*b^ 
2*d^2) - (exp(- 2*a - 2*b*x)*(9*b*d^2 + 27*b^2*d - 9*b^3 - 27*d^3))/(192*b 
^4 + 1728*d^4 - 1920*b^2*d^2) - (exp(- 4*a - 4*b*x)*(9*b*d^2 - 27*b^2*d...
 

Reduce [F]

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

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

Output:

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