\(\int (-\frac {3 d^2 e^{a+b x}}{4 (b^2-\frac {9 d^2}{4}) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)) \, dx\) [17]

Optimal result

Integrand size = 56, antiderivative size = 73 \[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=-\frac {6 d e^{a+b x} \cosh (c+d x) \sqrt {\sinh (c+d x)}}{4 b^2-9 d^2}+\frac {4 b e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)}{4 b^2-9 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 7.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\frac {2 e^{a+b x} \sqrt {\sinh (c+d x)} (-3 d \cosh (c+d x)+2 b \sinh (c+d x))}{4 b^2-9 d^2} \] Input:

Integrate[(-3*d^2*E^(a + b*x))/(4*(b^2 - (9*d^2)/4)*Sqrt[Sinh[c + d*x]]) + 
 E^(a + b*x)*Sinh[c + d*x]^(3/2),x]
 

Output:

(2*E^(a + b*x)*Sqrt[Sinh[c + d*x]]*(-3*d*Cosh[c + d*x] + 2*b*Sinh[c + d*x] 
))/(4*b^2 - 9*d^2)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {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.

\(\displaystyle \int \left (e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 b e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)}{4 b^2-9 d^2}-\frac {6 d e^{a+b x} \sqrt {\sinh (c+d x)} \cosh (c+d x)}{4 b^2-9 d^2}\)

Input:

Int[(-3*d^2*E^(a + b*x))/(4*(b^2 - (9*d^2)/4)*Sqrt[Sinh[c + d*x]]) + E^(a 
+ b*x)*Sinh[c + d*x]^(3/2),x]
 

Output:

(-6*d*E^(a + b*x)*Cosh[c + d*x]*Sqrt[Sinh[c + d*x]])/(4*b^2 - 9*d^2) + (4* 
b*E^(a + b*x)*Sinh[c + d*x]^(3/2))/(4*b^2 - 9*d^2)
 

Defintions of rubi rules used

rule 2009

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

Maple [F]

\[\int \left (-\frac {3 d^{2} {\mathrm e}^{b x +a}}{4 \left (b^{2}-\frac {9 d^{2}}{4}\right ) \sqrt {\sinh \left (d x +c \right )}}+{\mathrm e}^{b x +a} \sinh \left (d x +c \right )^{\frac {3}{2}}\right )d x\]

Input:

int(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*sinh(d* 
x+c)^(3/2),x)
 

Output:

int(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*sinh(d* 
x+c)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*s 
inh(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\frac {\left (\int 4 b^{2} e^{b x} \sinh ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- \frac {3 d^{2} e^{b x}}{\sqrt {\sinh {\left (c + d x \right )}}}\right )\, dx + \int \left (- 9 d^{2} e^{b x} \sinh ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx\right ) e^{a}}{\left (2 b - 3 d\right ) \left (2 b + 3 d\right )} \] Input:

integrate(-3/4*d**2*exp(b*x+a)/(b**2-9/4*d**2)/sinh(d*x+c)**(1/2)+exp(b*x+ 
a)*sinh(d*x+c)**(3/2),x)
 

Output:

(Integral(4*b**2*exp(b*x)*sinh(c + d*x)**(3/2), x) + Integral(-3*d**2*exp( 
b*x)/sqrt(sinh(c + d*x)), x) + Integral(-9*d**2*exp(b*x)*sinh(c + d*x)**(3 
/2), x))*exp(a)/((2*b - 3*d)*(2*b + 3*d))
 

Maxima [F]

\[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\int { e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac {3}{2}} - \frac {3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt {\sinh \left (d x + c\right )}} \,d x } \] Input:

integrate(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*s 
inh(d*x+c)^(3/2),x, algorithm="maxima")
 

Output:

integrate(e^(b*x + a)*sinh(d*x + c)^(3/2) - 3*d^2*e^(b*x + a)/((4*b^2 - 9* 
d^2)*sqrt(sinh(d*x + c))), x)
 

Giac [F]

\[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\int { e^{\left (b x + a\right )} \sinh \left (d x + c\right )^{\frac {3}{2}} - \frac {3 \, d^{2} e^{\left (b x + a\right )}}{{\left (4 \, b^{2} - 9 \, d^{2}\right )} \sqrt {\sinh \left (d x + c\right )}} \,d x } \] Input:

integrate(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*s 
inh(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

integrate(e^(b*x + a)*sinh(d*x + c)^(3/2) - 3*d^2*e^(b*x + a)/((4*b^2 - 9* 
d^2)*sqrt(sinh(d*x + c))), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\int {\mathrm {e}}^{a+b\,x}\,{\mathrm {sinh}\left (c+d\,x\right )}^{3/2}-\frac {3\,d^2\,{\mathrm {e}}^{a+b\,x}}{4\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\,\left (b^2-\frac {9\,d^2}{4}\right )} \,d x \] Input:

int(exp(a + b*x)*sinh(c + d*x)^(3/2) - (3*d^2*exp(a + b*x))/(4*sinh(c + d* 
x)^(1/2)*(b^2 - (9*d^2)/4)),x)
 

Output:

int(exp(a + b*x)*sinh(c + d*x)^(3/2) - (3*d^2*exp(a + b*x))/(4*sinh(c + d* 
x)^(1/2)*(b^2 - (9*d^2)/4)), x)
 

Reduce [F]

\[ \int \left (-\frac {3 d^2 e^{a+b x}}{4 \left (b^2-\frac {9 d^2}{4}\right ) \sqrt {\sinh (c+d x)}}+e^{a+b x} \sinh ^{\frac {3}{2}}(c+d x)\right ) \, dx=\frac {e^{a} \left (8 e^{b x} \sqrt {\sinh \left (d x +c \right )}\, \sinh \left (d x +c \right ) b^{2}-18 e^{b x} \sqrt {\sinh \left (d x +c \right )}\, \sinh \left (d x +c \right ) d^{2}-6 \left (\int \frac {e^{b x} \sqrt {\sinh \left (d x +c \right )}}{\sinh \left (d x +c \right )}d x \right ) b \,d^{2}-12 \left (\int e^{b x} \sqrt {\sinh \left (d x +c \right )}\, \cosh \left (d x +c \right )d x \right ) b^{2} d +27 \left (\int e^{b x} \sqrt {\sinh \left (d x +c \right )}\, \cosh \left (d x +c \right )d x \right ) d^{3}\right )}{2 b \left (4 b^{2}-9 d^{2}\right )} \] Input:

int(-3/4*d^2*exp(b*x+a)/(b^2-9/4*d^2)/sinh(d*x+c)^(1/2)+exp(b*x+a)*sinh(d* 
x+c)^(3/2),x)
 

Output:

(e**a*(8*e**(b*x)*sqrt(sinh(c + d*x))*sinh(c + d*x)*b**2 - 18*e**(b*x)*sqr 
t(sinh(c + d*x))*sinh(c + d*x)*d**2 - 6*int((e**(b*x)*sqrt(sinh(c + d*x))) 
/sinh(c + d*x),x)*b*d**2 - 12*int(e**(b*x)*sqrt(sinh(c + d*x))*cosh(c + d* 
x),x)*b**2*d + 27*int(e**(b*x)*sqrt(sinh(c + d*x))*cosh(c + d*x),x)*d**3)) 
/(2*b*(4*b**2 - 9*d**2))