Integrand size = 18, antiderivative size = 123 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac {b f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3} \] Output:
-1/2*a/d/(d*x+c)^2-1/2*b*f*cosh(f*x+e)/d^2/(d*x+c)-1/2*b*f^2*Chi(c*f/d+f*x )*sinh(-e+c*f/d)/d^3-1/2*b*sinh(f*x+e)/d/(d*x+c)^2+1/2*b*f^2*cosh(-e+c*f/d )*Shi(c*f/d+f*x)/d^3
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {b f^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-\frac {d (b f (c+d x) \cosh (e+f x)+d (a+b \sinh (e+f x)))}{(c+d x)^2}+b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )}{2 d^3} \] Input:
Integrate[(a + b*Sinh[e + f*x])/(c + d*x)^3,x]
Output:
(b*f^2*CoshIntegral[f*(c/d + x)]*Sinh[e - (c*f)/d] - (d*(b*f*(c + d*x)*Cos h[e + f*x] + d*(a + b*Sinh[e + f*x])))/(c + d*x)^2 + b*f^2*Cosh[e - (c*f)/ d]*SinhIntegral[f*(c/d + x)])/(2*d^3)
Time = 0.46 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3798, 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.
\(\displaystyle \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-i b \sin (i e+i f x)}{(c+d x)^3}dx\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \int \left (\frac {a}{(c+d x)^3}+\frac {b \sinh (e+f x)}{(c+d x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a}{2 d (c+d x)^2}+\frac {b f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}+\frac {b f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {b f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac {b \sinh (e+f x)}{2 d (c+d x)^2}\) |
Input:
Int[(a + b*Sinh[e + f*x])/(c + d*x)^3,x]
Output:
-1/2*a/(d*(c + d*x)^2) - (b*f*Cosh[e + f*x])/(2*d^2*(c + d*x)) + (b*f^2*Co shIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/(2*d^3) - (b*Sinh[e + f*x])/( 2*d*(c + d*x)^2) + (b*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/( 2*d^3)
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(295\) vs. \(2(115)=230\).
Time = 0.35 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.41
method | result | size |
risch | \(-\frac {a}{2 d \left (d x +c \right )^{2}}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{3} b \,{\mathrm e}^{-f x -e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{-f x -e}}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{2} b \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{4 d^{3}}-\frac {f^{2} b \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {f^{2} b \,{\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {f^{2} b \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{4 d^{3}}\) | \(296\) |
Input:
int((a+b*sinh(f*x+e))/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a/d/(d*x+c)^2-1/4*f^3*b*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^ 2)*x-1/4*f^3*b*exp(-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/4*f^2 *b*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/4*f^2*b/d^3*exp((c*f- d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-1/4*f^2*b/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/4 *f^2*b/d^3*exp(f*x+e)/(c*f/d+f*x)-1/4*f^2*b/d^3*exp(-(c*f-d*e)/d)*Ei(1,-f* x-e-(c*f-d*e)/d)
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (115) = 230\).
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {2 \, b d^{2} \sinh \left (f x + e\right ) + 2 \, a d^{2} + 2 \, {\left (b d^{2} f x + b c d f\right )} \cosh \left (f x + e\right ) - {\left ({\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+b*sinh(f*x+e))/(d*x+c)^3,x, algorithm="fricas")
Output:
-1/4*(2*b*d^2*sinh(f*x + e) + 2*a*d^2 + 2*(b*d^2*f*x + b*c*d*f)*cosh(f*x + e) - ((b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*Ei((d*f*x + c*f)/d) - ( b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*Ei(-(d*f*x + c*f)/d))*cosh(-(d* e - c*f)/d) + ((b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*Ei((d*f*x + c*f )/d) + (b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*Ei(-(d*f*x + c*f)/d))*s inh(-(d*e - c*f)/d))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*sinh(f*x+e))/(d*x+c)**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {1}{2} \, b {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:
integrate((a+b*sinh(f*x+e))/(d*x+c)^3,x, algorithm="maxima")
Output:
1/2*b*(e^(-e + c*f/d)*exp_integral_e(3, (d*x + c)*f/d)/((d*x + c)^2*d) - e ^(e - c*f/d)*exp_integral_e(3, -(d*x + c)*f/d)/((d*x + c)^2*d)) - 1/2*a/(d ^3*x^2 + 2*c*d^2*x + c^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (115) = 230\).
Time = 0.11 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.59 \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {b d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - b d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 2 \, b c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 2 \, b c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + b c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - b c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} - b d^{2} f x e^{\left (f x + e\right )} - b d^{2} f x e^{\left (-f x - e\right )} - b c d f e^{\left (f x + e\right )} - b c d f e^{\left (-f x - e\right )} - b d^{2} e^{\left (f x + e\right )} + b d^{2} e^{\left (-f x - e\right )} - 2 \, a d^{2}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+b*sinh(f*x+e))/(d*x+c)^3,x, algorithm="giac")
Output:
1/4*(b*d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) - b*d^2*f^2*x^2*Ei(-( d*f*x + c*f)/d)*e^(-e + c*f/d) + 2*b*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) - 2*b*c*d*f^2*x*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + b*c^2*f^2*Ei( (d*f*x + c*f)/d)*e^(e - c*f/d) - b*c^2*f^2*Ei(-(d*f*x + c*f)/d)*e^(-e + c* f/d) - b*d^2*f*x*e^(f*x + e) - b*d^2*f*x*e^(-f*x - e) - b*c*d*f*e^(f*x + e ) - b*c*d*f*e^(-f*x - e) - b*d^2*e^(f*x + e) + b*d^2*e^(-f*x - e) - 2*a*d^ 2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\int \frac {a+b\,\mathrm {sinh}\left (e+f\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + b*sinh(e + f*x))/(c + d*x)^3,x)
Output:
int((a + b*sinh(e + f*x))/(c + d*x)^3, x)
\[ \int \frac {a+b \sinh (e+f x)}{(c+d x)^3} \, dx=\text {too large to display} \] Input:
int((a+b*sinh(f*x+e))/(d*x+c)^3,x)
Output:
(2*e**(e + f*x)*cosh(e + f*x)*b*c**3*d*f**3 + 2*e**(e + f*x)*cosh(e + f*x) *b*c**2*d**2*f**3*x - 8*e**(e + f*x)*cosh(e + f*x)*b*c*d**3*f - 8*e**(e + f*x)*cosh(e + f*x)*b*d**4*f*x - e**(2*e + 2*f*x)*b*c**2*d**2*f**3*x + e**( 2*e + 2*f*x)*b*c**2*d**2*f**2 + 2*e**(2*e + 2*f*x)*b*c*d**3*f + 4*e**(2*e + 2*f*x)*b*d**4*f*x - e**(2*e + f*x)*int((e**(f*x)*x)/(c**5*f**2 + 3*c**4* d*f**2*x + 3*c**3*d**2*f**2*x**2 - 4*c**3*d**2 + c**2*d**3*f**2*x**3 - 12* c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b*c**7*d**2*f**6 - 2*e**(2* e + f*x)*int((e**(f*x)*x)/(c**5*f**2 + 3*c**4*d*f**2*x + 3*c**3*d**2*f**2* x**2 - 4*c**3*d**2 + c**2*d**3*f**2*x**3 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b*c**6*d**3*f**6*x - 2*e**(2*e + f*x)*int((e**(f*x)*x)/ (c**5*f**2 + 3*c**4*d*f**2*x + 3*c**3*d**2*f**2*x**2 - 4*c**3*d**2 + c**2* d**3*f**2*x**3 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b*c**6* d**3*f**5 - e**(2*e + f*x)*int((e**(f*x)*x)/(c**5*f**2 + 3*c**4*d*f**2*x + 3*c**3*d**2*f**2*x**2 - 4*c**3*d**2 + c**2*d**3*f**2*x**3 - 12*c**2*d**3* x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b*c**5*d**4*f**6*x**2 - 4*e**(2*e + f *x)*int((e**(f*x)*x)/(c**5*f**2 + 3*c**4*d*f**2*x + 3*c**3*d**2*f**2*x**2 - 4*c**3*d**2 + c**2*d**3*f**2*x**3 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4* d**5*x**3),x)*b*c**5*d**4*f**5*x + 6*e**(2*e + f*x)*int((e**(f*x)*x)/(c**5 *f**2 + 3*c**4*d*f**2*x + 3*c**3*d**2*f**2*x**2 - 4*c**3*d**2 + c**2*d**3* f**2*x**3 - 12*c**2*d**3*x - 12*c*d**4*x**2 - 4*d**5*x**3),x)*b*c**5*d*...