Integrand size = 16, antiderivative size = 121 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d}-\frac {3 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d}-\frac {3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d} \] Output:
1/4*Chi(3*b*c/d+3*b*x)*sinh(3*a-3*b*c/d)/d-3/4*Chi(b*c/d+b*x)*sinh(a-b*c/d )/d-3/4*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d+1/4*cosh(3*a-3*b*c/d)*Shi(3*b*c/d+3 *b*x)/d
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.84 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {\text {Chi}\left (\frac {3 b (c+d x)}{d}\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-3 \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )-3 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )+\cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b (c+d x)}{d}\right )}{4 d} \] Input:
Integrate[Sinh[a + b*x]^3/(c + d*x),x]
Output:
(CoshIntegral[(3*b*(c + d*x))/d]*Sinh[3*a - (3*b*c)/d] - 3*CoshIntegral[b* (c/d + x)]*Sinh[a - (b*c)/d] - 3*Cosh[a - (b*c)/d]*SinhIntegral[b*(c/d + x )] + Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3*b*(c + d*x))/d])/(4*d)
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 26, 3793, 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.
\(\displaystyle \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sin (i a+i b x)^3}{c+d x}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sin (i a+i b x)^3}{c+d x}dx\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle i \int \left (\frac {3 i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {i \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}+\frac {3 i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}+\frac {3 i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {i \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\right )\) |
Input:
Int[Sinh[a + b*x]^3/(c + d*x),x]
Output:
I*(((-1/4*I)*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/d + (( (3*I)/4)*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d])/d + (((3*I)/4)*Cos h[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)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Time = 0.68 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {{\mathrm e}^{-\frac {3 \left (d a -b c \right )}{d}} \operatorname {expIntegral}_{1}\left (3 b x +3 a -\frac {3 \left (d a -b c \right )}{d}\right )}{8 d}-\frac {3 \,{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{8 d}+\frac {3 \,{\mathrm e}^{\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-d a +b c}{d}\right )}{8 d}-\frac {{\mathrm e}^{\frac {3 d a -3 b c}{d}} \operatorname {expIntegral}_{1}\left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right )}{8 d}\) | \(166\) |
Input:
int(sinh(b*x+a)^3/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/8/d*exp(-3*(a*d-b*c)/d)*Ei(1,3*b*x+3*a-3*(a*d-b*c)/d)-3/8/d*exp(-(a*d-b* c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+3/8/d*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c )/d)-1/8/d*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x-3*a-3*(-a*d+b*c)/d)
Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.55 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=-\frac {3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - {\left ({\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, d} \] Input:
integrate(sinh(b*x+a)^3/(d*x+c),x, algorithm="fricas")
Output:
-1/8*(3*(Ei((b*d*x + b*c)/d) - Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - (Ei(3*(b*d*x + b*c)/d) - Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) + 3*(Ei((b*d*x + b*c)/d) + Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d) - (E i(3*(b*d*x + b*c)/d) + Ei(-3*(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/d
\[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )}}{c + d x}\, dx \] Input:
integrate(sinh(b*x+a)**3/(d*x+c),x)
Output:
Integral(sinh(a + b*x)**3/(c + d*x), x)
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.97 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{1}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{1}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \] Input:
integrate(sinh(b*x+a)^3/(d*x+c),x, algorithm="maxima")
Output:
1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(1, 3*(d*x + c)*b/d)/d - 3/8*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d + 3/8*e^(a - b*c/d)*exp_integral _e(1, -(d*x + c)*b/d)/d - 1/8*e^(3*a - 3*b*c/d)*exp_integral_e(1, -3*(d*x + c)*b/d)/d
Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93 \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\frac {{\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} - 3 \, {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 3 \, {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )}}{8 \, d} \] Input:
integrate(sinh(b*x+a)^3/(d*x+c),x, algorithm="giac")
Output:
1/8*(Ei(3*(b*d*x + b*c)/d)*e^(3*a - 3*b*c/d) - 3*Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + 3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - Ei(-3*(b*d*x + b*c)/d)* e^(-3*a + 3*b*c/d))/d
Timed out. \[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{c+d\,x} \,d x \] Input:
int(sinh(a + b*x)^3/(c + d*x),x)
Output:
int(sinh(a + b*x)^3/(c + d*x), x)
\[ \int \frac {\sinh ^3(a+b x)}{c+d x} \, dx=\int \frac {\sinh \left (b x +a \right )^{3}}{d x +c}d x \] Input:
int(sinh(b*x+a)^3/(d*x+c),x)
Output:
int(sinh(a + b*x)**3/(c + d*x),x)