3.4.52 \(\int f^{a+c x^2} \sinh ^2(d+f x^2) \, dx\) [352]

Optimal. Leaf size=128 \[ -\frac {f^a \sqrt {\pi } \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}}+\frac {e^{-2 d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {e^{2 d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {2 f+c \log (f)}\right )}{8 \sqrt {2 f+c \log (f)}} \]

[Out]

-1/4*f^a*erfi(x*c^(1/2)*ln(f)^(1/2))*Pi^(1/2)/c^(1/2)/ln(f)^(1/2)+1/8*f^a*erf(x*(2*f-c*ln(f))^(1/2))*Pi^(1/2)/
exp(2*d)/(2*f-c*ln(f))^(1/2)+1/8*exp(2*d)*f^a*erfi(x*(2*f+c*ln(f))^(1/2))*Pi^(1/2)/(2*f+c*ln(f))^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5623, 2235, 2325, 2236} \begin {gather*} \frac {\sqrt {\pi } e^{-2 d} f^a \text {Erf}\left (x \sqrt {2 f-c \log (f)}\right )}{8 \sqrt {2 f-c \log (f)}}+\frac {\sqrt {\pi } e^{2 d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+2 f}\right )}{8 \sqrt {c \log (f)+2 f}}-\frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right )}{4 \sqrt {c} \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(f^a*Sqrt[Pi]*Erfi[Sqrt[c]*x*Sqrt[Log[f]]])/(Sqrt[c]*Sqrt[Log[f]]) + (f^a*Sqrt[Pi]*Erf[x*Sqrt[2*f - c*Log
[f]]])/(8*E^(2*d)*Sqrt[2*f - c*Log[f]]) + (E^(2*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[2*f + c*Log[f]]])/(8*Sqrt[2*f + c*
Log[f]])

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 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rule 5623

Int[(F_)^(u_)*Sinh[v_]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^u, Sinh[v]^n, x], x] /; FreeQ[F, x] && (Linea
rQ[u, x] || PolyQ[u, x, 2]) && (LinearQ[v, x] || PolyQ[v, x, 2]) && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]
time = 0.38, size = 179, normalized size = 1.40 \begin {gather*} \frac {f^a \sqrt {\pi } \left (\text {Erfi}\left (\sqrt {c} x \sqrt {\log (f)}\right ) \left (8 f^2-2 c^2 \log ^2(f)\right )+\sqrt {c} \sqrt {\log (f)} \left (\text {Erf}\left (x \sqrt {2 f-c \log (f)}\right ) \sqrt {2 f-c \log (f)} (2 f+c \log (f)) (-\cosh (2 d)+\sinh (2 d))-\text {Erfi}\left (x \sqrt {2 f+c \log (f)}\right ) (2 f-c \log (f)) \sqrt {2 f+c \log (f)} (\cosh (2 d)+\sinh (2 d))\right )\right )}{8 \sqrt {c} \sqrt {\log (f)} \left (-4 f^2+c^2 \log ^2(f)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*Sqrt[Pi]*(Erfi[Sqrt[c]*x*Sqrt[Log[f]]]*(8*f^2 - 2*c^2*Log[f]^2) + Sqrt[c]*Sqrt[Log[f]]*(Erf[x*Sqrt[2*f -
c*Log[f]]]*Sqrt[2*f - c*Log[f]]*(2*f + c*Log[f])*(-Cosh[2*d] + Sinh[2*d]) - Erfi[x*Sqrt[2*f + c*Log[f]]]*(2*f
- c*Log[f])*Sqrt[2*f + c*Log[f]]*(Cosh[2*d] + Sinh[2*d]))))/(8*Sqrt[c]*Sqrt[Log[f]]*(-4*f^2 + c^2*Log[f]^2))

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Maple [A]
time = 1.43, size = 101, normalized size = 0.79

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-2 d} \erf \left (x \sqrt {2 f -c \ln \left (f \right )}\right )}{8 \sqrt {2 f -c \ln \left (f \right )}}+\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{2 d} \erf \left (\sqrt {-c \ln \left (f \right )-2 f}\, x \right )}{8 \sqrt {-c \ln \left (f \right )-2 f}}-\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-c \ln \left (f \right )}\, x \right )}{4 \sqrt {-c \ln \left (f \right )}}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+a)*sinh(f*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/8*Pi^(1/2)*f^a*exp(-2*d)/(2*f-c*ln(f))^(1/2)*erf(x*(2*f-c*ln(f))^(1/2))+1/8*Pi^(1/2)*f^a*exp(2*d)/(-c*ln(f)-
2*f)^(1/2)*erf((-c*ln(f)-2*f)^(1/2)*x)-1/4*f^a*Pi^(1/2)/(-c*ln(f))^(1/2)*erf((-c*ln(f))^(1/2)*x)

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Maxima [A]
time = 0.28, size = 100, normalized size = 0.78 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} + \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (-2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 2*f)*x)*e^(2*d)/sqrt(-c*log(f) - 2*f) + 1/8*sqrt(pi)*f^a*erf(sqrt(-c*log
(f) + 2*f)*x)*e^(-2*d)/sqrt(-c*log(f) + 2*f) - 1/4*sqrt(pi)*f^a*erf(sqrt(-c*log(f))*x)/sqrt(-c*log(f))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (98) = 196\).
time = 0.35, size = 254, normalized size = 1.98 \begin {gather*} -\frac {{\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) - 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} + 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) - 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) + {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \cosh \left (a \log \left (f\right ) + 2 \, d\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 2 \, c f \log \left (f\right )\right )} \sinh \left (a \log \left (f\right ) + 2 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 2 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) - 2 \, {\left (\sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \cosh \left (a \log \left (f\right )\right ) + \sqrt {\pi } {\left (c^{2} \log \left (f\right )^{2} - 4 \, f^{2}\right )} \sinh \left (a \log \left (f\right )\right )\right )} \sqrt {-c \log \left (f\right )} \operatorname {erf}\left (\sqrt {-c \log \left (f\right )} x\right )}{8 \, {\left (c^{3} \log \left (f\right )^{3} - 4 \, c f^{2} \log \left (f\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*cosh(a*log(f) - 2*d) + sqrt(pi)*(c^2*log(f)^2 + 2*c*f*log(f))*si
nh(a*log(f) - 2*d))*sqrt(-c*log(f) + 2*f)*erf(sqrt(-c*log(f) + 2*f)*x) + (sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f
))*cosh(a*log(f) + 2*d) + sqrt(pi)*(c^2*log(f)^2 - 2*c*f*log(f))*sinh(a*log(f) + 2*d))*sqrt(-c*log(f) - 2*f)*e
rf(sqrt(-c*log(f) - 2*f)*x) - 2*(sqrt(pi)*(c^2*log(f)^2 - 4*f^2)*cosh(a*log(f)) + sqrt(pi)*(c^2*log(f)^2 - 4*f
^2)*sinh(a*log(f)))*sqrt(-c*log(f))*erf(sqrt(-c*log(f))*x))/(c^3*log(f)^3 - 4*c*f^2*log(f))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + c x^{2}} \sinh ^{2}{\left (d + f x^{2} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 107, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-c \log \left (f\right )} x\right )}{4 \, \sqrt {-c \log \left (f\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 2 \, f} x\right ) e^{\left (a \log \left (f\right ) + 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) - 2 \, f}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 2 \, f} x\right ) e^{\left (a \log \left (f\right ) - 2 \, d\right )}}{8 \, \sqrt {-c \log \left (f\right ) + 2 \, f}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*f^a*erf(-sqrt(-c*log(f))*x)/sqrt(-c*log(f)) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) - 2*f)*x)*e^(a*log
(f) + 2*d)/sqrt(-c*log(f) - 2*f) - 1/8*sqrt(pi)*erf(-sqrt(-c*log(f) + 2*f)*x)*e^(a*log(f) - 2*d)/sqrt(-c*log(f
) + 2*f)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int f^{c\,x^2+a}\,{\mathrm {sinh}\left (f\,x^2+d\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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