Integrand size = 14, antiderivative size = 91 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right ) \] Output:
-3/8*b*cosh(a)*Chi(b*x^2)+3/8*b*cosh(3*a)*Chi(3*b*x^2)+3/8*sinh(b*x^2+a)/x ^2-1/8*sinh(3*b*x^2+3*a)/x^2-3/8*b*sinh(a)*Shi(b*x^2)+3/8*b*sinh(3*a)*Shi( 3*b*x^2)
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 b x^2 \cosh (a) \text {Chi}\left (b x^2\right )-3 b x^2 \cosh (3 a) \text {Chi}\left (3 b x^2\right )-3 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )+3 b x^2 \sinh (a) \text {Shi}\left (b x^2\right )-3 b x^2 \sinh (3 a) \text {Shi}\left (3 b x^2\right )}{8 x^2} \] Input:
Integrate[Sinh[a + b*x^2]^3/x^3,x]
Output:
-1/8*(3*b*x^2*Cosh[a]*CoshIntegral[b*x^2] - 3*b*x^2*Cosh[3*a]*CoshIntegral [3*b*x^2] - 3*Sinh[a + b*x^2] + Sinh[3*(a + b*x^2)] + 3*b*x^2*Sinh[a]*Sinh Integral[b*x^2] - 3*b*x^2*Sinh[3*a]*SinhIntegral[3*b*x^2])/x^2
Time = 0.46 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5863, 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.
\(\displaystyle \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 5863 |
\(\displaystyle \int \left (\frac {\sinh \left (3 a+3 b x^2\right )}{4 x^3}-\frac {3 \sinh \left (a+b x^2\right )}{4 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}\) |
Input:
Int[Sinh[a + b*x^2]^3/x^3,x]
Output:
(-3*b*Cosh[a]*CoshIntegral[b*x^2])/8 + (3*b*Cosh[3*a]*CoshIntegral[3*b*x^2 ])/8 + (3*Sinh[a + b*x^2])/(8*x^2) - Sinh[3*(a + b*x^2)]/(8*x^2) - (3*b*Si nh[a]*SinhIntegral[b*x^2])/8 + (3*b*Sinh[3*a]*SinhIntegral[3*b*x^2])/8
Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Sinh[c + d*x^n])^p, x], x ] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]
Time = 0.54 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.35
method | result | size |
risch | \(\frac {-3 \,\operatorname {expIntegral}_{1}\left (3 b \,x^{2}\right ) {\mathrm e}^{-3 a} b \,x^{2}+3 \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b \,x^{2}\right ) b \,x^{2}+3 \,\operatorname {expIntegral}_{1}\left (-b \,x^{2}\right ) {\mathrm e}^{a} b \,x^{2}-3 \,{\mathrm e}^{3 a} \operatorname {expIntegral}_{1}\left (-3 b \,x^{2}\right ) b \,x^{2}+{\mathrm e}^{-3 b \,x^{2}-3 a}-3 \,{\mathrm e}^{-b \,x^{2}-a}+3 \,{\mathrm e}^{b \,x^{2}+a}-{\mathrm e}^{3 b \,x^{2}+3 a}}{16 x^{2}}\) | \(123\) |
Input:
int(sinh(b*x^2+a)^3/x^3,x,method=_RETURNVERBOSE)
Output:
1/16*(-3*Ei(1,3*b*x^2)*exp(-3*a)*b*x^2+3*exp(-a)*Ei(1,b*x^2)*b*x^2+3*Ei(1, -b*x^2)*exp(a)*b*x^2-3*exp(3*a)*Ei(1,-3*b*x^2)*b*x^2+exp(-3*b*x^2-3*a)-3*e xp(-b*x^2-a)+3*exp(b*x^2+a)-exp(3*b*x^2+3*a))/x^2
Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.76 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {2 \, \sinh \left (b x^{2} + a\right )^{3} - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + 6 \, {\left (\cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right )}{16 \, x^{2}} \] Input:
integrate(sinh(b*x^2+a)^3/x^3,x, algorithm="fricas")
Output:
-1/16*(2*sinh(b*x^2 + a)^3 - 3*(b*x^2*Ei(3*b*x^2) + b*x^2*Ei(-3*b*x^2))*co sh(3*a) + 3*(b*x^2*Ei(b*x^2) + b*x^2*Ei(-b*x^2))*cosh(a) + 6*(cosh(b*x^2 + a)^2 - 1)*sinh(b*x^2 + a) - 3*(b*x^2*Ei(3*b*x^2) - b*x^2*Ei(-3*b*x^2))*si nh(3*a) + 3*(b*x^2*Ei(b*x^2) - b*x^2*Ei(-b*x^2))*sinh(a))/x^2
\[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {\sinh ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \] Input:
integrate(sinh(b*x**2+a)**3/x**3,x)
Output:
Integral(sinh(a + b*x**2)**3/x**3, x)
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac {3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) - \frac {3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac {3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \] Input:
integrate(sinh(b*x^2+a)^3/x^3,x, algorithm="maxima")
Output:
3/16*b*e^(-3*a)*gamma(-1, 3*b*x^2) - 3/16*b*e^(-a)*gamma(-1, b*x^2) - 3/16 *b*e^a*gamma(-1, -b*x^2) + 3/16*b*e^(3*a)*gamma(-1, -3*b*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (80) = 160\).
Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.45 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} + 3 \, a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} + 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} + b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \] Input:
integrate(sinh(b*x^2+a)^3/x^3,x, algorithm="giac")
Output:
1/16*(3*(b*x^2 + a)*b^2*Ei(3*b*x^2)*e^(3*a) - 3*a*b^2*Ei(3*b*x^2)*e^(3*a) - 3*(b*x^2 + a)*b^2*Ei(-b*x^2)*e^(-a) + 3*a*b^2*Ei(-b*x^2)*e^(-a) + 3*(b*x ^2 + a)*b^2*Ei(-3*b*x^2)*e^(-3*a) - 3*a*b^2*Ei(-3*b*x^2)*e^(-3*a) - 3*(b*x ^2 + a)*b^2*Ei(b*x^2)*e^a + 3*a*b^2*Ei(b*x^2)*e^a - b^2*e^(3*b*x^2 + 3*a) + 3*b^2*e^(b*x^2 + a) - 3*b^2*e^(-b*x^2 - a) + b^2*e^(-3*b*x^2 - 3*a))/(b^ 2*x^2)
Timed out. \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^3}{x^3} \,d x \] Input:
int(sinh(a + b*x^2)^3/x^3,x)
Output:
int(sinh(a + b*x^2)^3/x^3, x)
Time = 0.16 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.74 \[ \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {-3 e^{3 b \,x^{2}+2 a} \mathit {ei} \left (-b \,x^{2}\right ) b \,x^{2}+3 e^{3 b \,x^{2}} \mathit {ei} \left (-3 b \,x^{2}\right ) b \,x^{2}-3 e^{3 b \,x^{2}+4 a} \mathit {ei} \left (b \,x^{2}\right ) b \,x^{2}+3 e^{3 b \,x^{2}+6 a} \mathit {ei} \left (3 b \,x^{2}\right ) b \,x^{2}-e^{6 b \,x^{2}+6 a}+3 e^{4 b \,x^{2}+4 a}-3 e^{2 b \,x^{2}+2 a}+1}{16 e^{3 b \,x^{2}+3 a} x^{2}} \] Input:
int(sinh(b*x^2+a)^3/x^3,x)
Output:
( - 3*e**(2*a + 3*b*x**2)*ei( - b*x**2)*b*x**2 + 3*e**(3*b*x**2)*ei( - 3*b *x**2)*b*x**2 - 3*e**(4*a + 3*b*x**2)*ei(b*x**2)*b*x**2 + 3*e**(6*a + 3*b* x**2)*ei(3*b*x**2)*b*x**2 - e**(6*a + 6*b*x**2) + 3*e**(4*a + 4*b*x**2) - 3*e**(2*a + 2*b*x**2) + 1)/(16*e**(3*a + 3*b*x**2)*x**2)