3.4.53 \(\int f^{a+c x^2} \sinh ^3(d+f x^2) \, dx\) [353]

Optimal. Leaf size=171 \[ \frac {3 e^{-d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}-\frac {e^{-3 d} f^a \sqrt {\pi } \text {Erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^d f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {f+c \log (f)}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d} f^a \sqrt {\pi } \text {Erfi}\left (x \sqrt {3 f+c \log (f)}\right )}{16 \sqrt {3 f+c \log (f)}} \]

[Out]

3/16*f^a*erf(x*(f-c*ln(f))^(1/2))*Pi^(1/2)/exp(d)/(f-c*ln(f))^(1/2)-1/16*f^a*erf(x*(3*f-c*ln(f))^(1/2))*Pi^(1/
2)/exp(3*d)/(3*f-c*ln(f))^(1/2)-3/16*exp(d)*f^a*erfi(x*(f+c*ln(f))^(1/2))*Pi^(1/2)/(f+c*ln(f))^(1/2)+1/16*exp(
3*d)*f^a*erfi(x*(3*f+c*ln(f))^(1/2))*Pi^(1/2)/(3*f+c*ln(f))^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5623, 2325, 2236, 2235} \begin {gather*} \frac {3 \sqrt {\pi } e^{-d} f^a \text {Erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}-\frac {\sqrt {\pi } e^{-3 d} f^a \text {Erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 \sqrt {\pi } e^d f^a \text {Erfi}\left (x \sqrt {c \log (f)+f}\right )}{16 \sqrt {c \log (f)+f}}+\frac {\sqrt {\pi } e^{3 d} f^a \text {Erfi}\left (x \sqrt {c \log (f)+3 f}\right )}{16 \sqrt {c \log (f)+3 f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + c*x^2)*Sinh[d + f*x^2]^3,x]

[Out]

(3*f^a*Sqrt[Pi]*Erf[x*Sqrt[f - c*Log[f]]])/(16*E^d*Sqrt[f - c*Log[f]]) - (f^a*Sqrt[Pi]*Erf[x*Sqrt[3*f - c*Log[
f]]])/(16*E^(3*d)*Sqrt[3*f - c*Log[f]]) - (3*E^d*f^a*Sqrt[Pi]*Erfi[x*Sqrt[f + c*Log[f]]])/(16*Sqrt[f + c*Log[f
]]) + (E^(3*d)*f^a*Sqrt[Pi]*Erfi[x*Sqrt[3*f + c*Log[f]]])/(16*Sqrt[3*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 ^3\left (d+f x^2\right ) \, dx &=\int \left (-\frac {1}{8} e^{-3 d-3 f x^2} f^{a+c x^2}+\frac {3}{8} e^{-d-f x^2} f^{a+c x^2}-\frac {3}{8} e^{d+f x^2} f^{a+c x^2}+\frac {1}{8} e^{3 d+3 f x^2} f^{a+c x^2}\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 d-3 f x^2} f^{a+c x^2} \, dx\right )+\frac {1}{8} \int e^{3 d+3 f x^2} f^{a+c x^2} \, dx+\frac {3}{8} \int e^{-d-f x^2} f^{a+c x^2} \, dx-\frac {3}{8} \int e^{d+f x^2} f^{a+c x^2} \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 d+a \log (f)-x^2 (3 f-c \log (f))} \, dx\right )+\frac {1}{8} \int e^{3 d+a \log (f)+x^2 (3 f+c \log (f))} \, dx+\frac {3}{8} \int e^{-d+a \log (f)-x^2 (f-c \log (f))} \, dx-\frac {3}{8} \int e^{d+a \log (f)+x^2 (f+c \log (f))} \, dx\\ &=\frac {3 e^{-d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {f-c \log (f)}\right )}{16 \sqrt {f-c \log (f)}}-\frac {e^{-3 d} f^a \sqrt {\pi } \text {erf}\left (x \sqrt {3 f-c \log (f)}\right )}{16 \sqrt {3 f-c \log (f)}}-\frac {3 e^d f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {f+c \log (f)}\right )}{16 \sqrt {f+c \log (f)}}+\frac {e^{3 d} f^a \sqrt {\pi } \text {erfi}\left (x \sqrt {3 f+c \log (f)}\right )}{16 \sqrt {3 f+c \log (f)}}\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 272, normalized size = 1.59 \begin {gather*} \frac {f^a \sqrt {\pi } \left (3 \text {Erf}\left (x \sqrt {f-c \log (f)}\right ) \sqrt {f-c \log (f)} \left (9 f^3+9 c f^2 \log (f)-c^2 f \log ^2(f)-c^3 \log ^3(f)\right ) (\cosh (d)-\sinh (d))-(f-c \log (f)) \left (\text {Erf}\left (x \sqrt {3 f-c \log (f)}\right ) \sqrt {3 f-c \log (f)} \left (3 f^2+4 c f \log (f)+c^2 \log ^2(f)\right ) (\cosh (3 d)-\sinh (3 d))+(3 f-c \log (f)) \left (3 \text {Erfi}\left (x \sqrt {f+c \log (f)}\right ) \sqrt {f+c \log (f)} (3 f+c \log (f)) (\cosh (d)+\sinh (d))-\text {Erfi}\left (x \sqrt {3 f+c \log (f)}\right ) (f+c \log (f)) \sqrt {3 f+c \log (f)} (\cosh (3 d)+\sinh (3 d))\right )\right )\right )}{16 \left (9 f^4-10 c^2 f^2 \log ^2(f)+c^4 \log ^4(f)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[f^(a + c*x^2)*Sinh[d + f*x^2]^3,x]

[Out]

(f^a*Sqrt[Pi]*(3*Erf[x*Sqrt[f - c*Log[f]]]*Sqrt[f - c*Log[f]]*(9*f^3 + 9*c*f^2*Log[f] - c^2*f*Log[f]^2 - c^3*L
og[f]^3)*(Cosh[d] - Sinh[d]) - (f - c*Log[f])*(Erf[x*Sqrt[3*f - c*Log[f]]]*Sqrt[3*f - c*Log[f]]*(3*f^2 + 4*c*f
*Log[f] + c^2*Log[f]^2)*(Cosh[3*d] - Sinh[3*d]) + (3*f - c*Log[f])*(3*Erfi[x*Sqrt[f + c*Log[f]]]*Sqrt[f + c*Lo
g[f]]*(3*f + c*Log[f])*(Cosh[d] + Sinh[d]) - Erfi[x*Sqrt[3*f + c*Log[f]]]*(f + c*Log[f])*Sqrt[3*f + c*Log[f]]*
(Cosh[3*d] + Sinh[3*d])))))/(16*(9*f^4 - 10*c^2*f^2*Log[f]^2 + c^4*Log[f]^4))

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Maple [A]
time = 1.82, size = 144, normalized size = 0.84

method	result	size
risch	\(\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{3 d} \erf \left (\sqrt {-c \ln \left (f \right )-3 f}\, x \right )}{16 \sqrt {-c \ln \left (f \right )-3 f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-3 d} \erf \left (x \sqrt {3 f -c \ln \left (f \right )}\right )}{16 \sqrt {3 f -c \ln \left (f \right )}}+\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{-d} \erf \left (x \sqrt {f -c \ln \left (f \right )}\right )}{16 \sqrt {f -c \ln \left (f \right )}}-\frac {3 \sqrt {\pi }\, f^{a} {\mathrm e}^{d} \erf \left (\sqrt {-c \ln \left (f \right )-f}\, x \right )}{16 \sqrt {-c \ln \left (f \right )-f}}\)	\(144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*x^2+a)*sinh(f*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/16*Pi^(1/2)*f^a*exp(3*d)/(-c*ln(f)-3*f)^(1/2)*erf((-c*ln(f)-3*f)^(1/2)*x)-1/16*Pi^(1/2)*f^a*exp(-3*d)/(3*f-c
*ln(f))^(1/2)*erf(x*(3*f-c*ln(f))^(1/2))+3/16*Pi^(1/2)*f^a*exp(-d)/(f-c*ln(f))^(1/2)*erf(x*(f-c*ln(f))^(1/2))-
3/16*Pi^(1/2)*f^a*exp(d)/(-c*ln(f)-f)^(1/2)*erf((-c*ln(f)-f)^(1/2)*x)

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Maxima [A]
time = 0.27, size = 143, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x\right ) e^{\left (-d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} - \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) e^{\left (-3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, f}} - \frac {3 \, \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x\right ) e^{d}}{16 \, \sqrt {-c \log \left (f\right ) - 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)^3,x, algorithm="maxima")

[Out]

1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - 3*f)*x)*e^(3*d)/sqrt(-c*log(f) - 3*f) + 3/16*sqrt(pi)*f^a*erf(sqrt(-c*l
og(f) + f)*x)*e^(-d)/sqrt(-c*log(f) + f) - 1/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) + 3*f)*x)*e^(-3*d)/sqrt(-c*log
(f) + 3*f) - 3/16*sqrt(pi)*f^a*erf(sqrt(-c*log(f) - f)*x)*e^d/sqrt(-c*log(f) - f)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (135) = 270\).
time = 0.37, size = 492, normalized size = 2.88 \begin {gather*} \frac {{\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) - 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) - 3 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) - 3 \, d\right )\right )} \sqrt {-c \log \left (f\right ) + 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) - 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) - d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} + c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) - 9 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) - d\right )\right )} \sqrt {-c \log \left (f\right ) + f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) + f} x\right ) + 3 \, {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) + d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - c^{2} f \log \left (f\right )^{2} - 9 \, c f^{2} \log \left (f\right ) + 9 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) + d\right )\right )} \sqrt {-c \log \left (f\right ) - f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - f} x\right ) - {\left (\sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \cosh \left (a \log \left (f\right ) + 3 \, d\right ) + \sqrt {\pi } {\left (c^{3} \log \left (f\right )^{3} - 3 \, c^{2} f \log \left (f\right )^{2} - c f^{2} \log \left (f\right ) + 3 \, f^{3}\right )} \sinh \left (a \log \left (f\right ) + 3 \, d\right )\right )} \sqrt {-c \log \left (f\right ) - 3 \, f} \operatorname {erf}\left (\sqrt {-c \log \left (f\right ) - 3 \, f} x\right )}{16 \, {\left (c^{4} \log \left (f\right )^{4} - 10 \, c^{2} f^{2} \log \left (f\right )^{2} + 9 \, f^{4}\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)^3,x, algorithm="fricas")

[Out]

1/16*((sqrt(pi)*(c^3*log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*cosh(a*log(f) - 3*d) + sqrt(pi)*(c^3*
log(f)^3 + 3*c^2*f*log(f)^2 - c*f^2*log(f) - 3*f^3)*sinh(a*log(f) - 3*d))*sqrt(-c*log(f) + 3*f)*erf(sqrt(-c*lo
g(f) + 3*f)*x) - 3*(sqrt(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*cosh(a*log(f) - d) + sqr
t(pi)*(c^3*log(f)^3 + c^2*f*log(f)^2 - 9*c*f^2*log(f) - 9*f^3)*sinh(a*log(f) - d))*sqrt(-c*log(f) + f)*erf(sqr
t(-c*log(f) + f)*x) + 3*(sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*cosh(a*log(f) + d)
+ sqrt(pi)*(c^3*log(f)^3 - c^2*f*log(f)^2 - 9*c*f^2*log(f) + 9*f^3)*sinh(a*log(f) + d))*sqrt(-c*log(f) - f)*er
f(sqrt(-c*log(f) - f)*x) - (sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*cosh(a*log(f) +
3*d) + sqrt(pi)*(c^3*log(f)^3 - 3*c^2*f*log(f)^2 - c*f^2*log(f) + 3*f^3)*sinh(a*log(f) + 3*d))*sqrt(-c*log(f)
- 3*f)*erf(sqrt(-c*log(f) - 3*f)*x))/(c^4*log(f)^4 - 10*c^2*f^2*log(f)^2 + 9*f^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int f^{a + c x^{2}} \sinh ^{3}{\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)**3,x)

[Out]

Integral(f**(a + c*x**2)*sinh(d + f*x**2)**3, x)

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Giac [A]
time = 0.43, size = 155, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - 3 \, f} x\right ) e^{\left (a \log \left (f\right ) + 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - 3 \, f}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) - f} x\right ) e^{\left (a \log \left (f\right ) + d\right )}}{16 \, \sqrt {-c \log \left (f\right ) - f}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + f} x\right ) e^{\left (a \log \left (f\right ) - d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + f}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-c \log \left (f\right ) + 3 \, f} x\right ) e^{\left (a \log \left (f\right ) - 3 \, d\right )}}{16 \, \sqrt {-c \log \left (f\right ) + 3 \, 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)^3,x, algorithm="giac")

[Out]

-1/16*sqrt(pi)*erf(-sqrt(-c*log(f) - 3*f)*x)*e^(a*log(f) + 3*d)/sqrt(-c*log(f) - 3*f) + 3/16*sqrt(pi)*erf(-sqr
t(-c*log(f) - f)*x)*e^(a*log(f) + d)/sqrt(-c*log(f) - f) - 3/16*sqrt(pi)*erf(-sqrt(-c*log(f) + f)*x)*e^(a*log(
f) - d)/sqrt(-c*log(f) + f) + 1/16*sqrt(pi)*erf(-sqrt(-c*log(f) + 3*f)*x)*e^(a*log(f) - 3*d)/sqrt(-c*log(f) +
3*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 )}^3 \,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)^3,x)

[Out]

int(f^(a + c*x^2)*sinh(d + f*x^2)^3, x)

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