\(\int x \sinh ((a+b x)^2) \, dx\) [89]

Optimal result

Integrand size = 10, antiderivative size = 54 \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\frac {\cosh \left ((a+b x)^2\right )}{2 b^2}+\frac {a \sqrt {\pi } \text {erf}(a+b x)}{4 b^2}-\frac {a \sqrt {\pi } \text {erfi}(a+b x)}{4 b^2} \]

[Out]

1/2*cosh((b*x+a)^2)/b^2+1/4*a*erf(b*x+a)*Pi^(1/2)/b^2-1/4*a*erfi(b*x+a)*Pi^(1/2)/b^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5472, 6874, 5406, 2235, 2236, 5428, 2718} \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\frac {\sqrt {\pi } a \text {erf}(a+b x)}{4 b^2}-\frac {\sqrt {\pi } a \text {erfi}(a+b x)}{4 b^2}+\frac {\cosh \left ((a+b x)^2\right )}{2 b^2} \]

[In]

Int[x*Sinh[(a + b*x)^2],x]

[Out]

Cosh[(a + b*x)^2]/(2*b^2) + (a*Sqrt[Pi]*Erf[a + b*x])/(4*b^2) - (a*Sqrt[Pi]*Erfi[a + b*x])/(4*b^2)

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2718

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

Rule 5406

Int[Sinh[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[1/2, Int[E^(c + d*x^n), x], x] - Dist[1/2, Int[E^(-c - d*
x^n), x], x] /; FreeQ[{c, d}, x] && IGtQ[n, 1]

Rule 5428

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 5472

Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(u_)^(n_)])^(p_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1]^(
m + 1), Subst[Int[(x - Coefficient[u, x, 0])^m*(a + b*Sinh[c + d*x^n])^p, x], x, u], x] /; FreeQ[{a, b, c, d,
n, p}, x] && LinearQ[u, x] && NeQ[u, x] && IntegerQ[m]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (-a+x) \sinh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \left (-a \sinh \left (x^2\right )+x \sinh \left (x^2\right )\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int x \sinh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2}-\frac {a \text {Subst}\left (\int \sinh \left (x^2\right ) \, dx,x,a+b x\right )}{b^2} \\ & = \frac {\text {Subst}\left (\int \sinh (x) \, dx,x,(a+b x)^2\right )}{2 b^2}+\frac {a \text {Subst}\left (\int e^{-x^2} \, dx,x,a+b x\right )}{2 b^2}-\frac {a \text {Subst}\left (\int e^{x^2} \, dx,x,a+b x\right )}{2 b^2} \\ & = \frac {\cosh \left ((a+b x)^2\right )}{2 b^2}+\frac {a \sqrt {\pi } \text {erf}(a+b x)}{4 b^2}-\frac {a \sqrt {\pi } \text {erfi}(a+b x)}{4 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\frac {\cosh \left ((a+b x)^2\right )}{2 b^2}-\frac {a \sqrt {\pi } (-\text {erf}(a+b x)+\text {erfi}(a+b x))}{4 b^2} \]

[In]

Integrate[x*Sinh[(a + b*x)^2],x]

[Out]

Cosh[(a + b*x)^2]/(2*b^2) - (a*Sqrt[Pi]*(-Erf[a + b*x] + Erfi[a + b*x]))/(4*b^2)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22

method	result	size
risch	\(\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{4 b^{2}}+\frac {a \,\operatorname {erf}\left (b x +a \right ) \sqrt {\pi }}{4 b^{2}}+\frac {{\mathrm e}^{\left (b x +a \right )^{2}}}{4 b^{2}}+\frac {i a \sqrt {\pi }\, \operatorname {erf}\left (i b x +i a \right )}{4 b^{2}}\)	\(66\)

[In]

int(x*sinh((b*x+a)^2),x,method=_RETURNVERBOSE)

[Out]

1/4/b^2*exp(-(b*x+a)^2)+1/4*a*erf(b*x+a)*Pi^(1/2)/b^2+1/4/b^2*exp((b*x+a)^2)+1/4*I*a/b^2*Pi^(1/2)*erf(I*a+I*b*
x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (44) = 88\).

Time = 0.25 (sec) , antiderivative size = 258, normalized size of antiderivative = 4.78 \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\frac {b \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \sqrt {\pi } \sqrt {-b^{2}} {\left (a \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + a \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) + \sqrt {\pi } \sqrt {b^{2}} {\left (a \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + a \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + 2 \, b \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + b \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + b}{4 \, {\left (b^{3} \cosh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right ) + b^{3} \sinh \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )\right )}} \]

[In]

integrate(x*sinh((b*x+a)^2),x, algorithm="fricas")

[Out]

1/4*(b*cosh(b^2*x^2 + 2*a*b*x + a^2)^2 + sqrt(pi)*sqrt(-b^2)*(a*cosh(b^2*x^2 + 2*a*b*x + a^2) + a*sinh(b^2*x^2
+ 2*a*b*x + a^2))*erf(sqrt(-b^2)*(b*x + a)/b) + sqrt(pi)*sqrt(b^2)*(a*cosh(b^2*x^2 + 2*a*b*x + a^2) + a*sinh(
b^2*x^2 + 2*a*b*x + a^2))*erf(sqrt(b^2)*(b*x + a)/b) + 2*b*cosh(b^2*x^2 + 2*a*b*x + a^2)*sinh(b^2*x^2 + 2*a*b*
x + a^2) + b*sinh(b^2*x^2 + 2*a*b*x + a^2)^2 + b)/(b^3*cosh(b^2*x^2 + 2*a*b*x + a^2) + b^3*sinh(b^2*x^2 + 2*a*
b*x + a^2))

Sympy [F]

\[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\int x \sinh {\left (a^{2} + 2 a b x + b^{2} x^{2} \right )}\, dx \]

[In]

integrate(x*sinh((b*x+a)**2),x)

[Out]

Integral(x*sinh(a**2 + 2*a*b*x + b**2*x**2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (44) = 88\).

Time = 0.42 (sec) , antiderivative size = 649, normalized size of antiderivative = 12.02 \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=\frac {1}{2} \, x^{2} \sinh \left ({\left (b x + a\right )}^{2}\right ) + \frac {1}{4} \, {\left (\frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} b^{3} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {5}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{3} b^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {5}{2}}} + \frac {2 \, a b^{3} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {5}{2}}}\right )} a}{\sqrt {-b^{2}}} + \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{3} b^{4} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {7}{2}}} - \frac {3 \, {\left (b^{2} x + a b\right )}^{3} a b^{4} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {7}{2}}} + \frac {3 \, a^{2} b^{4} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {7}{2}}} + \frac {b^{4} \Gamma \left (2, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{\left (-b^{2}\right )^{\frac {7}{2}}}\right )} b}{\sqrt {-b^{2}}} - \frac {a {\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{3} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} - \frac {2 \, a e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b^{2}} - \frac {{\left (b^{2} x + a b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{5} \left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac {3}{2}}}\right )}}{b} + \frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{3} {\left (\operatorname {erf}\left (\sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}\right ) - 1\right )}}{b^{4} \sqrt {-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}}} - \frac {3 \, a^{2} e^{\left (\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{b^{3}} + \frac {\Gamma \left (2, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{3}} - \frac {3 \, {\left (b^{2} x + a b\right )}^{3} a \Gamma \left (\frac {3}{2}, -\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{b^{6} \left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )^{\frac {3}{2}}}\right )} b \]

[In]

integrate(x*sinh((b*x+a)^2),x, algorithm="maxima")

[Out]

1/2*x^2*sinh((b*x + a)^2) + 1/4*((sqrt(pi)*(b^2*x + a*b)*a^2*b^3*(erf(sqrt((b^2*x + a*b)^2)/b) - 1)/(sqrt((b^2
*x + a*b)^2)*(-b^2)^(5/2)) - (b^2*x + a*b)^3*b^3*gamma(3/2, (b^2*x + a*b)^2/b^2)/(((b^2*x + a*b)^2)^(3/2)*(-b^
2)^(5/2)) + 2*a*b^3*e^(-(b^2*x + a*b)^2/b^2)/(-b^2)^(5/2))*a/sqrt(-b^2) + (sqrt(pi)*(b^2*x + a*b)*a^3*b^4*(erf
(sqrt((b^2*x + a*b)^2)/b) - 1)/(sqrt((b^2*x + a*b)^2)*(-b^2)^(7/2)) - 3*(b^2*x + a*b)^3*a*b^4*gamma(3/2, (b^2*
x + a*b)^2/b^2)/(((b^2*x + a*b)^2)^(3/2)*(-b^2)^(7/2)) + 3*a^2*b^4*e^(-(b^2*x + a*b)^2/b^2)/(-b^2)^(7/2) + b^4
*gamma(2, (b^2*x + a*b)^2/b^2)/(-b^2)^(7/2))*b/sqrt(-b^2) - a*(sqrt(pi)*(b^2*x + a*b)*a^2*(erf(sqrt(-(b^2*x +
a*b)^2/b^2)) - 1)/(b^3*sqrt(-(b^2*x + a*b)^2/b^2)) - 2*a*e^((b^2*x + a*b)^2/b^2)/b^2 - (b^2*x + a*b)^3*gamma(3
/2, -(b^2*x + a*b)^2/b^2)/(b^5*(-(b^2*x + a*b)^2/b^2)^(3/2)))/b + sqrt(pi)*(b^2*x + a*b)*a^3*(erf(sqrt(-(b^2*x
+ a*b)^2/b^2)) - 1)/(b^4*sqrt(-(b^2*x + a*b)^2/b^2)) - 3*a^2*e^((b^2*x + a*b)^2/b^2)/b^3 + gamma(2, -(b^2*x +
a*b)^2/b^2)/b^3 - 3*(b^2*x + a*b)^3*a*gamma(3/2, -(b^2*x + a*b)^2/b^2)/(b^6*(-(b^2*x + a*b)^2/b^2)^(3/2)))*b

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.83 \[ \int x \sinh \left ((a+b x)^2\right ) \, dx=-\frac {-\frac {i \, \sqrt {\pi } a \operatorname {erf}\left (i \, b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{b}}{4 \, b} - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{4 \, b} \]

[In]

integrate(x*sinh((b*x+a)^2),x, algorithm="giac")

[Out]

-1/4*(-I*sqrt(pi)*a*erf(I*b*(x + a/b))/b - e^(b^2*x^2 + 2*a*b*x + a^2)/b)/b - 1/4*(sqrt(pi)*a*erf(-b*(x + a/b)
)/b - e^(-b^2*x^2 - 2*a*b*x - a^2)/b)/b

Mupad [F(-1)]

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

[In]

int(x*sinh((a + b*x)^2),x)

[Out]

int(x*sinh((a + b*x)^2), x)