\(\int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx\) [58]

Optimal result

Integrand size = 19, antiderivative size = 209 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {\cosh (c+d x)}{b d^2}-\frac {a \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {x \sinh (c+d x)}{b d}+\frac {a \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2} \]

[Out]

-cosh(d*x+c)/b/d^2-1/2*a*Chi(d*x+d*(-a)^(1/2)/b^(1/2))*cosh(c-d*(-a)^(1/2)/b^(1/2))/b^2-1/2*a*Chi(-d*x+d*(-a)^
(1/2)/b^(1/2))*cosh(c+d*(-a)^(1/2)/b^(1/2))/b^2+x*sinh(d*x+c)/b/d-1/2*a*Shi(d*x+d*(-a)^(1/2)/b^(1/2))*sinh(c-d
*(-a)^(1/2)/b^(1/2))/b^2-1/2*a*Shi(d*x-d*(-a)^(1/2)/b^(1/2))*sinh(c+d*(-a)^(1/2)/b^(1/2))/b^2

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5401, 3377, 2718, 3384, 3379, 3382} \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {a \cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}+\frac {a \sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b^2}-\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d} \]

[In]

Int[(x^3*Cosh[c + d*x])/(a + b*x^2),x]

[Out]

-(Cosh[c + d*x]/(b*d^2)) - (a*Cosh[c + (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2)
- (a*Cosh[c - (Sqrt[-a]*d)/Sqrt[b]]*CoshIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2) + (x*Sinh[c + d*x])/(b*
d) + (a*Sinh[c + (Sqrt[-a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b^2) - (a*Sinh[c - (Sqrt[-
a]*d)/Sqrt[b]]*SinhIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b^2)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5401

Int[Cosh[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[c
+ d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && IGtQ[n, 0] && (Eq
Q[n, 2] || EqQ[p, -1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x \cosh (c+d x)}{b}-\frac {a x \cosh (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {\int x \cosh (c+d x) \, dx}{b}-\frac {a \int \frac {x \cosh (c+d x)}{a+b x^2} \, dx}{b} \\ & = \frac {x \sinh (c+d x)}{b d}-\frac {a \int \left (-\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\cosh (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{b}-\frac {\int \sinh (c+d x) \, dx}{b d} \\ & = -\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d}+\frac {a \int \frac {\cosh (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}-\frac {a \int \frac {\cosh (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}} \\ & = -\frac {\cosh (c+d x)}{b d^2}+\frac {x \sinh (c+d x)}{b d}-\frac {\left (a \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}+\frac {\left (a \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cosh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}}-\frac {\left (a \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{2 b^{3/2}}-\frac {\left (a \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sinh \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{2 b^{3/2}} \\ & = -\frac {\cosh (c+d x)}{b d^2}-\frac {a \cosh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \cosh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Chi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2}+\frac {x \sinh (c+d x)}{b d}+\frac {a \sinh \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b^2}-\frac {a \sinh \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Shi}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=-\frac {a e^{c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (d \left (-\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )+\operatorname {ExpIntegralEi}\left (d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )\right )\right )+a e^{-c-\frac {i \sqrt {a} d}{\sqrt {b}}} \left (e^{\frac {2 i \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+\operatorname {ExpIntegralEi}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )+\frac {4 b \cosh (d x) (\cosh (c)-d x \sinh (c))}{d^2}-\frac {4 b (d x \cosh (c)-\sinh (c)) \sinh (d x)}{d^2}}{4 b^2} \]

[In]

Integrate[(x^3*Cosh[c + d*x])/(a + b*x^2),x]

[Out]

-1/4*(a*E^(c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sqrt[b])*ExpIntegralEi[d*(((-I)*Sqrt[a])/Sqrt[b] +
x)] + ExpIntegralEi[d*((I*Sqrt[a])/Sqrt[b] + x)]) + a*E^(-c - (I*Sqrt[a]*d)/Sqrt[b])*(E^(((2*I)*Sqrt[a]*d)/Sq
rt[b])*ExpIntegralEi[((-I)*Sqrt[a]*d)/Sqrt[b] - d*x] + ExpIntegralEi[(I*Sqrt[a]*d)/Sqrt[b] - d*x]) + (4*b*Cosh
[d*x]*(Cosh[c] - d*x*Sinh[c]))/d^2 - (4*b*(d*x*Cosh[c] - Sinh[c])*Sinh[d*x])/d^2)/b^2

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.28

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a}{4 b^{2}}+\frac {{\mathrm e}^{\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a}{4 b^{2}}+\frac {{\mathrm e}^{-\frac {d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (-\frac {d \sqrt {-a b}-\left (d x +c \right ) b +c b}{b}\right ) a}{4 b^{2}}+\frac {{\mathrm e}^{-\frac {-d \sqrt {-a b}+c b}{b}} \operatorname {Ei}_{1}\left (\frac {d \sqrt {-a b}+\left (d x +c \right ) b -c b}{b}\right ) a}{4 b^{2}}-\frac {{\mathrm e}^{-d x -c} x}{2 d b}+\frac {{\mathrm e}^{d x +c} x}{2 d b}-\frac {{\mathrm e}^{-d x -c}}{2 d^{2} b}-\frac {{\mathrm e}^{d x +c}}{2 d^{2} b}\)	\(268\)

[In]

int(x^3*cosh(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

1/4/b^2*exp((d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)-(d*x+c)*b+c*b)/b)*a+1/4/b^2*exp((-d*(-a*b)^(1/2)+c*b)
/b)*Ei(1,-(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a+1/4/b^2*exp(-(d*(-a*b)^(1/2)+c*b)/b)*Ei(1,-(d*(-a*b)^(1/2)-(d*x+
c)*b+c*b)/b)*a+1/4/b^2*exp(-(-d*(-a*b)^(1/2)+c*b)/b)*Ei(1,(d*(-a*b)^(1/2)+(d*x+c)*b-c*b)/b)*a-1/2/d/b*exp(-d*x
-c)*x+1/2/d/b*exp(d*x+c)*x-1/2/d^2/b*exp(-d*x-c)-1/2/d^2/b*exp(d*x+c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (169) = 338\).

Time = 0.26 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.40 \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=\frac {4 \, b d x \sinh \left (d x + c\right ) - 4 \, b \cosh \left (d x + c\right ) - {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \cosh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x - \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x + \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (c + \sqrt {-\frac {a d^{2}}{b}}\right ) + {\left ({\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (d x + \sqrt {-\frac {a d^{2}}{b}}\right ) - {\left (a d^{2} \cosh \left (d x + c\right )^{2} - a d^{2} \sinh \left (d x + c\right )^{2}\right )} {\rm Ei}\left (-d x - \sqrt {-\frac {a d^{2}}{b}}\right )\right )} \sinh \left (-c + \sqrt {-\frac {a d^{2}}{b}}\right )}{4 \, {\left (b^{2} d^{2} \cosh \left (d x + c\right )^{2} - b^{2} d^{2} \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x^2+a),x, algorithm="fricas")

[Out]

1/4*(4*b*d*x*sinh(d*x + c) - 4*b*cosh(d*x + c) - ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(d*x - sqr
t(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^2/b)))*cosh(c + sqrt(-a*d^2
/b)) - ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) + (a*d^2*cosh(d*x + c)^2 - a*
d^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*cosh(-c + sqrt(-a*d^2/b)) - ((a*d^2*cosh(d*x + c)^2 - a*d^2*si
nh(d*x + c)^2)*Ei(d*x - sqrt(-a*d^2/b)) - (a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(-d*x + sqrt(-a*d^
2/b)))*sinh(c + sqrt(-a*d^2/b)) + ((a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(d*x + sqrt(-a*d^2/b)) -
(a*d^2*cosh(d*x + c)^2 - a*d^2*sinh(d*x + c)^2)*Ei(-d*x - sqrt(-a*d^2/b)))*sinh(-c + sqrt(-a*d^2/b)))/(b^2*d^2
*cosh(d*x + c)^2 - b^2*d^2*sinh(d*x + c)^2)

Sympy [F]

\[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^{3} \cosh {\left (c + d x \right )}}{a + b x^{2}}\, dx \]

[In]

integrate(x**3*cosh(d*x+c)/(b*x**2+a),x)

[Out]

Integral(x**3*cosh(c + d*x)/(a + b*x**2), x)

Maxima [F]

\[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{3} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

1/2*((d*x^3*e^(2*c) - x^2*e^(2*c))*e^(d*x) - (d*x^3 + x^2)*e^(-d*x))/(b*d^2*x^2*e^c + a*d^2*e^c) - 1/2*integra
te(2*(a*d*x^2*e^c - a*x*e^c)*e^(d*x)/(b^2*d^2*x^4 + 2*a*b*d^2*x^2 + a^2*d^2), x) + 1/2*integrate(2*(a*d*x^2 +
a*x)*e^(-d*x)/(b^2*d^2*x^4*e^c + 2*a*b*d^2*x^2*e^c + a^2*d^2*e^c), x)

Giac [F]

\[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=\int { \frac {x^{3} \cosh \left (d x + c\right )}{b x^{2} + a} \,d x } \]

[In]

integrate(x^3*cosh(d*x+c)/(b*x^2+a),x, algorithm="giac")

[Out]

integrate(x^3*cosh(d*x + c)/(b*x^2 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \cosh (c+d x)}{a+b x^2} \, dx=\int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{b\,x^2+a} \,d x \]

[In]

int((x^3*cosh(c + d*x))/(a + b*x^2),x)

[Out]

int((x^3*cosh(c + d*x))/(a + b*x^2), x)