Integrand size = 21, antiderivative size = 131 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=-\frac {a}{2 d (c+d x)^2}-\frac {i a f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac {i a f^2 \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}-\frac {i a \sinh (e+f x)}{2 d (c+d x)^2}+\frac {i a f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{2 d^3} \] Output:
-1/2*a/d/(d*x+c)^2-1/2*I*a*f*cosh(f*x+e)/d^2/(d*x+c)-1/2*I*a*f^2*Chi(c*f/d +f*x)*sinh(-e+c*f/d)/d^3-1/2*I*a*sinh(f*x+e)/d/(d*x+c)^2+1/2*I*a*f^2*cosh( -e+c*f/d)*Shi(c*f/d+f*x)/d^3
Time = 0.55 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.83 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {i a \left (f^2 (c+d x)^2 \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )-d (f (c+d x) \cosh (e+f x)+d (-i+\sinh (e+f x)))+f^2 (c+d x)^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{2 d^3 (c+d x)^2} \] Input:
Integrate[(a + I*a*Sinh[e + f*x])/(c + d*x)^3,x]
Output:
((I/2)*a*(f^2*(c + d*x)^2*CoshIntegral[f*(c/d + x)]*Sinh[e - (c*f)/d] - d* (f*(c + d*x)*Cosh[e + f*x] + d*(-I + Sinh[e + f*x])) + f^2*(c + d*x)^2*Cos h[e - (c*f)/d]*SinhIntegral[f*(c/d + x)]))/(d^3*(c + d*x)^2)
Time = 0.46 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.
\(\displaystyle \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+a \sin (i e+i f x)}{(c+d x)^3}dx\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \int \left (\frac {a}{(c+d x)^3}+\frac {i a \sinh (e+f x)}{(c+d x)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i a f^2 \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{2 d^3}+\frac {i a f^2 \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{2 d^3}-\frac {i a f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac {i a \sinh (e+f x)}{2 d (c+d x)^2}-\frac {a}{2 d (c+d x)^2}\) |
Input:
Int[(a + I*a*Sinh[e + f*x])/(c + d*x)^3,x]
Output:
-1/2*a/(d*(c + d*x)^2) - ((I/2)*a*f*Cosh[e + f*x])/(d^2*(c + d*x)) + ((I/2 )*a*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - ((I/2)*a*Sinh [e + f*x])/(d*(c + d*x)^2) + ((I/2)*a*f^2*Cosh[e - (c*f)/d]*SinhIntegral[( c*f)/d + f*x])/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])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (119 ) = 238\).
Time = 0.48 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.31
method | result | size |
risch | \(-\frac {a}{2 d \left (d x +c \right )^{2}}-\frac {i a \,f^{3} {\mathrm e}^{-f x -e} x}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {i a \,f^{3} {\mathrm e}^{-f x -e} c}{4 d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a \,f^{2} {\mathrm e}^{-f x -e}}{4 d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {i a \,f^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{4 d^{3}}-\frac {i a \,f^{2} {\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )^{2}}-\frac {i a \,f^{2} {\mathrm e}^{f x +e}}{4 d^{3} \left (\frac {c f}{d}+f x \right )}-\frac {i a \,f^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{4 d^{3}}\) | \(303\) |
Input:
int((a+I*a*sinh(f*x+e))/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a/d/(d*x+c)^2-1/4*I*a*f^3*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2* f^2)*x-1/4*I*a*f^3*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 *I*a*f^2*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)+1/4*I*a*f^2/d^3*e xp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-1/4*I*a*f^2/d^3*exp(f*x+e)/(c*f/d+ f*x)^2-1/4*I*a*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)-1/4*I*a*f^2/d^3*exp(-(c*f-d* e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.69 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {{\left (-i \, a d^{2} f x - i \, a c d f + i \, a d^{2} + {\left (-i \, a d^{2} f x - i \, a c d f - i \, a d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} - {\left (2 \, a d^{2} - {\left (i \, a d^{2} f^{2} x^{2} + 2 i \, a c d f^{2} x + i \, a c^{2} f^{2}\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - {\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a c d f^{2} x - i \, a c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )}\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^3,x, algorithm="fricas")
Output:
1/4*(-I*a*d^2*f*x - I*a*c*d*f + I*a*d^2 + (-I*a*d^2*f*x - I*a*c*d*f - I*a* d^2)*e^(2*f*x + 2*e) - (2*a*d^2 - (I*a*d^2*f^2*x^2 + 2*I*a*c*d*f^2*x + I*a *c^2*f^2)*Ei((d*f*x + c*f)/d)*e^((d*e - c*f)/d) - (-I*a*d^2*f^2*x^2 - 2*I* a*c*d*f^2*x - I*a*c^2*f^2)*Ei(-(d*f*x + c*f)/d)*e^(-(d*e - c*f)/d))*e^(f*x + e))*e^(-f*x - e)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)**3,x)
Output:
Timed out
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.76 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {1}{2} i \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{3}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{3}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac {a}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^3,x, algorithm="maxima")
Output:
1/2*I*a*(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)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (115) = 230\).
Time = 0.13 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.46 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\frac {i \, a d^{2} f^{2} x^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - i \, a d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + 2 i \, a c d f^{2} x {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - 2 i \, a c d f^{2} x {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} + i \, a c^{2} f^{2} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} - i \, a c^{2} f^{2} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} - i \, a d^{2} f x e^{\left (f x + e\right )} - i \, a d^{2} f x e^{\left (-f x - e\right )} - i \, a c d f e^{\left (f x + e\right )} - i \, a c d f e^{\left (-f x - e\right )} - i \, a d^{2} e^{\left (f x + e\right )} + i \, a d^{2} e^{\left (-f x - e\right )} - 2 \, a d^{2}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^3,x, algorithm="giac")
Output:
1/4*(I*a*d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) - I*a*d^2*f^2*x^2*E i(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + 2*I*a*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e ^(e - c*f/d) - 2*I*a*c*d*f^2*x*Ei(-(d*f*x + c*f)/d)*e^(-e + c*f/d) + I*a*c ^2*f^2*Ei((d*f*x + c*f)/d)*e^(e - c*f/d) - I*a*c^2*f^2*Ei(-(d*f*x + c*f)/d )*e^(-e + c*f/d) - I*a*d^2*f*x*e^(f*x + e) - I*a*d^2*f*x*e^(-f*x - e) - I* a*c*d*f*e^(f*x + e) - I*a*c*d*f*e^(-f*x - e) - I*a*d^2*e^(f*x + e) + I*a*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. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\int \frac {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int((a + a*sinh(e + f*x)*1i)/(c + d*x)^3,x)
Output:
int((a + a*sinh(e + f*x)*1i)/(c + d*x)^3, x)
\[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^3} \, dx=\text {too large to display} \] Input:
int((a+I*a*sinh(f*x+e))/(d*x+c)^3,x)
Output:
(a*(2*e**(e + f*x)*cosh(e + f*x)*c**3*d*f**3*i + 2*e**(e + f*x)*cosh(e + f *x)*c**2*d**2*f**3*i*x - 8*e**(e + f*x)*cosh(e + f*x)*c*d**3*f*i - 8*e**(e + f*x)*cosh(e + f*x)*d**4*f*i*x - e**(2*e + 2*f*x)*c**2*d**2*f**3*i*x + e **(2*e + 2*f*x)*c**2*d**2*f**2*i + 2*e**(2*e + 2*f*x)*c*d**3*f*i + 4*e**(2 *e + 2*f*x)*d**4*f*i*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)*c**7*d**2*f**6*i - 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)*c**6*d**3*f**6*i*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)*c**6 *d**3*f**5*i - 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)*c**5*d**4*f**6*i*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)*c**5*d**4*f**5*i*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)*c**5*d...