3.32 \(\int (d+e x) \sinh ^2(a+b x+c x^2) \, dx\)

Optimal. Leaf size=160 \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(d+e x)^2}{4 e} \]

[Out]

-1/4*(e*x+d)^2/e+1/8*e*sinh(2*c*x^2+2*b*x+2*a)/c+1/32*(-b*e+2*c*d)*exp(-2*a+1/2*b^2/c)*erf(1/2*(2*c*x+b)*2^(1/
2)/c^(1/2))*2^(1/2)*Pi^(1/2)/c^(3/2)+1/32*(-b*e+2*c*d)*exp(2*a-1/2*b^2/c)*erfi(1/2*(2*c*x+b)*2^(1/2)/c^(1/2))*
2^(1/2)*Pi^(1/2)/c^(3/2)

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Rubi [A]  time = 0.15, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5394, 5383, 5375, 2234, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e) \text {Erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e) \text {Erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {(d+e x)^2}{4 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*Sinh[a + b*x + c*x^2]^2,x]

[Out]

-(d + e*x)^2/(4*e) + ((2*c*d - b*e)*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^
(3/2)) + ((2*c*d - b*e)*E^(2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(16*c^(3/2)) + (e*
Sinh[2*a + 2*b*x + 2*c*x^2])/(8*c)

Rule 2204

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 2205

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 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 5375

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 5383

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*Sinh[a + b*x + c*x^2])/
(2*c), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Cosh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
e - 2*c*d, 0]

Rule 5394

Int[((d_.) + (e_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_), x_Symbol] :> Int[ExpandTrigReduce
[(d + e*x)^m, Sinh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]

Rubi steps

\begin {align*} \int (d+e x) \sinh ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {1}{2} (-d-e x)+\frac {1}{2} (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=-\frac {(d+e x)^2}{4 e}+\frac {1}{2} \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=-\frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(2 c d-b e) \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=-\frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(2 c d-b e) \int e^{-2 a-2 b x-2 c x^2} \, dx}{8 c}+\frac {(2 c d-b e) \int e^{2 a+2 b x+2 c x^2} \, dx}{8 c}\\ &=-\frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {\left ((2 c d-b e) e^{2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(2 b+4 c x)^2}{8 c}} \, dx}{8 c}+\frac {\left ((2 c d-b e) e^{-2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx}{8 c}\\ &=-\frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {(2 c d-b e) e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 177, normalized size = 1.11 \[ \frac {\sqrt {2 \pi } (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+\sqrt {2 \pi } (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\sinh \left (2 a-\frac {b^2}{2 c}\right )+\cosh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} (e \sinh (2 (a+x (b+c x)))-2 c x (2 d+e x))}{32 c^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*Sinh[a + b*x + c*x^2]^2,x]

[Out]

((2*c*d - b*e)*Sqrt[2*Pi]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] - Sinh[2*a - b^2/(2*c)]) +
(2*c*d - b*e)*Sqrt[2*Pi]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a - b^2/(2*c)] + Sinh[2*a - b^2/(2*c)])
+ 4*Sqrt[c]*(-2*c*x*(2*d + e*x) + e*Sinh[2*(a + x*(b + c*x))]))/(32*c^(3/2))

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fricas [B]  time = 0.58, size = 777, normalized size = 4.86 \[ \frac {2 \, c e \cosh \left (c x^{2} + b x + a\right )^{4} + 8 \, c e \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right )^{3} + 2 \, c e \sinh \left (c x^{2} + b x + a\right )^{4} - \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x + b\right )}}{2 \, \sqrt {c}}\right ) - 8 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )^{2} - 4 \, {\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - 3 \, c e \cosh \left (c x^{2} + b x + a\right )^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, c e + 8 \, {\left (c e \cosh \left (c x^{2} + b x + a\right )^{3} - 2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )\right )} \sinh \left (c x^{2} + b x + a\right )}{32 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, c^{2} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*c*e*cosh(c*x^2 + b*x + a)^4 + 8*c*e*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a)^3 + 2*c*e*sinh(c*x^2 +
b*x + a)^4 - sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b*
e)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((2*c*d - b*e)*cosh(-1/2*(b^2 - 4*a*c)/c) + (2*c*d - b
*e)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^
2 - 4*a*c)/c) + (2*c*d - b*e)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(-c
)*erf(1/2*sqrt(2)*(2*c*x + b)*sqrt(-c)/c) + sqrt(2)*sqrt(pi)*((2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*
(b^2 - 4*a*c)/c) - (2*c*d - b*e)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((2*c*d - b*e)*cosh(-1/2
*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((2*c*d - b*e)*cosh(
c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) - (2*c*d - b*e)*cosh(c*x^2 + b*x + a)*sinh(-1/2*(b^2 - 4*a*c)/c))*
sinh(c*x^2 + b*x + a))*sqrt(c)*erf(1/2*sqrt(2)*(2*c*x + b)/sqrt(c)) - 8*(c^2*e*x^2 + 2*c^2*d*x)*cosh(c*x^2 + b
*x + a)^2 - 4*(2*c^2*e*x^2 + 4*c^2*d*x - 3*c*e*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 - 2*c*e + 8*(c
*e*cosh(c*x^2 + b*x + a)^3 - 2*(c^2*e*x^2 + 2*c^2*d*x)*cosh(c*x^2 + b*x + a))*sinh(c*x^2 + b*x + a))/(c^2*cosh
(c*x^2 + b*x + a)^2 + 2*c^2*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x + a) + c^2*sinh(c*x^2 + b*x + a)^2)

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giac [A]  time = 0.15, size = 248, normalized size = 1.55 \[ -\frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {c}} - \frac {\sqrt {2} \sqrt {\pi } d \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{16 \, \sqrt {-c}} - \frac {1}{4} \, x^{2} e - \frac {1}{2} \, d x + \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c + 2 \, c}{2 \, c}\right )}}{\sqrt {c}} - 2 \, e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 1\right )}}{32 \, c} + \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c - 2 \, c}{2 \, c}\right )}}{\sqrt {-c}} + 2 \, e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 1\right )}}{32 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) - 1/16*sqrt(2)*
sqrt(pi)*d*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) - 1/4*x^2*e - 1/2*d*x + 1/
32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c + 2*c)/c)/sqrt(c) - 2*e^(-2*c
*x^2 - 2*b*x - 2*a + 1))/c + 1/32*(sqrt(2)*sqrt(pi)*b*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*
a*c - 2*c)/c)/sqrt(-c) + 2*e^(2*c*x^2 + 2*b*x + 2*a + 1))/c

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maple [A]  time = 0.09, size = 241, normalized size = 1.51 \[ -\frac {e \,x^{2}}{4}-\frac {d x}{2}+\frac {d \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{16 \sqrt {c}}-\frac {e \,{\mathrm e}^{-2 c \,x^{2}-2 b x -2 a}}{16 c}-\frac {e b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \erf \left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}-\frac {d \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \erf \left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{8 \sqrt {-2 c}}+\frac {e \,{\mathrm e}^{2 c \,x^{2}+2 b x +2 a}}{16 c}+\frac {e b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \erf \left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*sinh(c*x^2+b*x+a)^2,x)

[Out]

-1/4*e*x^2-1/2*d*x+1/16*d*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)/c^(1/2)*erf(2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)
/c^(1/2))-1/16*e/c*exp(-2*c*x^2-2*b*x-2*a)-1/32*e*b/c^(3/2)*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)*erf(2^(1/
2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))-1/8*d*Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*c)^(1/2)*x+b/(
-2*c)^(1/2))+1/16*e/c*exp(2*c*x^2+2*b*x+2*a)+1/16*e*b/c*Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*erf(-(-2*
c)^(1/2)*x+b/(-2*c)^(1/2))

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maxima [B]  time = 0.62, size = 299, normalized size = 1.87 \[ \frac {1}{16} \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} - 8 \, x\right )} d - \frac {1}{32} \, {\left (8 \, x^{2} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}}\right )} e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-c)*x - 1/2*sqrt(2)*b/sqrt(-c))*e^(2*a - 1/2*b^2/c)/sqrt(-c) + sqrt(2)
*sqrt(pi)*erf(sqrt(2)*sqrt(c)*x + 1/2*sqrt(2)*b/sqrt(c))*e^(-2*a + 1/2*b^2/c)/sqrt(c) - 8*x)*d - 1/32*(8*x^2 +
sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(sqrt(1/2)*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(3/2))
- sqrt(2)*e^(1/2*(2*c*x + b)^2/c)/sqrt(c))*e^(2*a - 1/2*b^2/c)/sqrt(c) + sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(
sqrt(1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + sqrt(2)*c*e^(-1/2*(2*c*x + b)^2/c)/
(-c)^(3/2))*e^(-2*a + 1/2*b^2/c)/sqrt(-c))*e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x + c*x^2)^2*(d + e*x),x)

[Out]

int(sinh(a + b*x + c*x^2)^2*(d + e*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x\right ) \sinh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*sinh(c*x**2+b*x+a)**2,x)

[Out]

Integral((d + e*x)*sinh(a + b*x + c*x**2)**2, x)

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