Integrand size = 16, antiderivative size = 145 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 b \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}+\frac {3 b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{4 d^2}+\frac {3 b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d^2}+\frac {3 b \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]
-cosh(b*x+a)^3/d/(d*x+c)+3/4*b*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^2+3/4*b*cosh (3*a-3*b*c/d)*Shi(3*b*c/d+3*b*x)/d^2+3/4*b*Chi(3*b*c/d+3*b*x)*sinh(3*a-3*b *c/d)/d^2+3/4*b*Chi(b*c/d+b*x)*sinh(a-b*c/d)/d^2
Time = 0.47 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \cosh (a) \cosh (b x)}{4 d (c+d x)}-\frac {\cosh (3 a) \cosh (3 b x)}{4 d (c+d x)}-\frac {3 \sinh (a) \sinh (b x)}{4 d (c+d x)}-\frac {\sinh (3 a) \sinh (3 b x)}{4 d (c+d x)}-\frac {3 b \left (-2 \text {Chi}\left (\frac {3 b c}{d}+3 b x\right ) \sinh \left (3 a-\frac {3 b c}{d}\right )-2 \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )-2 \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )-2 \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )\right )}{8 d^2} \]
(-3*Cosh[a]*Cosh[b*x])/(4*d*(c + d*x)) - (Cosh[3*a]*Cosh[3*b*x])/(4*d*(c + d*x)) - (3*Sinh[a]*Sinh[b*x])/(4*d*(c + d*x)) - (Sinh[3*a]*Sinh[3*b*x])/( 4*d*(c + d*x)) - (3*b*(-2*CoshIntegral[(3*b*c)/d + 3*b*x]*Sinh[3*a - (3*b* c)/d] - 2*CoshIntegral[(b*c)/d + b*x]*Sinh[a - (b*c)/d] - 2*Cosh[a - (b*c) /d]*SinhIntegral[(b*c)/d + b*x] - 2*Cosh[3*a - (3*b*c)/d]*SinhIntegral[(3* b*c)/d + 3*b*x]))/(8*d^2)
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3794, 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.
\(\displaystyle \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^3}{(c+d x)^2}dx\) |
\(\Big \downarrow \) 3794 |
\(\displaystyle -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 i b \int \left (-\frac {i \sinh (a+b x)}{4 (c+d x)}-\frac {i \sinh (3 a+3 b x)}{4 (c+d x)}\right )dx}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\cosh ^3(a+b x)}{d (c+d x)}+\frac {3 i b \left (-\frac {i \sinh \left (3 a-\frac {3 b c}{d}\right ) \text {Chi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}-\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {i \cosh \left (3 a-\frac {3 b c}{d}\right ) \text {Shi}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\right )}{d}\) |
-(Cosh[a + b*x]^3/(d*(c + d*x))) + ((3*I)*b*(((-1/4*I)*CoshIntegral[(3*b*c )/d + 3*b*x]*Sinh[3*a - (3*b*c)/d])/d - ((I/4)*CoshIntegral[(b*c)/d + b*x] *Sinh[a - (b*c)/d])/d - ((I/4)*Cosh[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))/ d
3.1.21.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Time = 0.42 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {b \,{\mathrm e}^{-3 b x -3 a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {3 \left (d a -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 b x +3 a -\frac {3 \left (d a -c b \right )}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{-b x -a}}{8 d \left (d x b +c b \right )}+\frac {3 b \,{\mathrm e}^{-\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {d a -c b}{d}\right )}{8 d^{2}}-\frac {3 b \,{\mathrm e}^{b x +a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-d a +c b}{d}\right )}{8 d^{2}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{8 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {3 b \,{\mathrm e}^{\frac {3 d a -3 c b}{d}} \operatorname {Ei}_{1}\left (-3 b x -3 a -\frac {3 \left (-d a +c b \right )}{d}\right )}{8 d^{2}}\) | \(271\) |
-1/8*b*exp(-3*b*x-3*a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-3*(a*d-b*c)/d)*Ei(1,3* b*x+3*a-3*(a*d-b*c)/d)-3/8*b*exp(-b*x-a)/d/(b*d*x+b*c)+3/8*b/d^2*exp(-(a*d -b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-3/8*b/d^2*exp(b*x+a)/(b*c/d+b*x)-3/8*b/d^ 2*exp((a*d-b*c)/d)*Ei(1,-b*x-a-(-a*d+b*c)/d)-1/8*b/d^2*exp(3*b*x+3*a)/(b*c /d+b*x)-3/8*b/d^2*exp(3*(a*d-b*c)/d)*Ei(1,-3*b*x-3*a-3*(-a*d+b*c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (137) = 274\).
Time = 0.25 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.10 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \cosh \left (b x + a\right )^{3} + 6 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 6 \, d \cosh \left (b x + a\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right ) - 3 \, {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \]
-1/8*(2*d*cosh(b*x + a)^3 + 6*d*cosh(b*x + a)*sinh(b*x + a)^2 + 6*d*cosh(b *x + a) - 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) - (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) - (b*d*x + b*c)*Ei(-3*(b*d*x + b*c)/d))*cosh(-3*(b*c - a*d)/d) - 3*((b*d*x + b*c)*Ei((b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*sinh(-(b* c - a*d)/d) - 3*((b*d*x + b*c)*Ei(3*(b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-3 *(b*d*x + b*c)/d))*sinh(-3*(b*c - a*d)/d))/(d^3*x + c*d^2)
\[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{2}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {3 \, e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} - \frac {e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{2}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, {\left (d x + c\right )} d} \]
-1/8*e^(-3*a + 3*b*c/d)*exp_integral_e(2, 3*(d*x + c)*b/d)/((d*x + c)*d) - 3/8*e^(-a + b*c/d)*exp_integral_e(2, (d*x + c)*b/d)/((d*x + c)*d) - 3/8*e ^(a - b*c/d)*exp_integral_e(2, -(d*x + c)*b/d)/((d*x + c)*d) - 1/8*e^(3*a - 3*b*c/d)*exp_integral_e(2, -3*(d*x + c)*b/d)/((d*x + c)*d)
Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (137) = 274\).
Time = 0.32 (sec) , antiderivative size = 1075, normalized size of antiderivative = 7.41 \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]
-1/8*(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-3*((d*x + c) *(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(3*(b*c - a*d)/d) + 3*b^3*c*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d) /d)*e^(3*(b*c - a*d)/d) - 3*a*b^2*d*Ei(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(3*(b*c - a*d)/d) + 3*(d*x + c)*(b - b*c/ (d*x + c) + a*d/(d*x + c))*b^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d* x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 3*b^3*c*Ei(-((d*x + c)*(b - b* c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 3*a*b^2*d *Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b* c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(-(b*c - a*d)/ d) - 3*b^3*c*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d )/d)*e^(-(b*c - a*d)/d) + 3*a*b^2*d*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d /(d*x + c)) + b*c - a*d)/d)*e^(-(b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) - 3*b^3*c*Ei(3*((d*x + c)*(b - b *c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(-3*(b*c - a*d)/d) + 3*a*b ^2*d*Ei(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e ^(-3*(b*c - a*d)/d) + b^2*d*e^(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) + 3*b^2*d*e^((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d) +...
Timed out. \[ \int \frac {\cosh ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]