3.4.43 \(\int f^{a+b x} \sinh ^2(d+f x^2) \, dx\) [343]

Optimal. Leaf size=148 \[ \frac {1}{8} e^{-2 d+\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} f^{-\frac {1}{2}+a} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \]

[Out]

-1/2*f^(b*x+a)/b/ln(f)+1/16*exp(-2*d+1/8*b^2*ln(f)^2/f)*f^(-1/2+a)*erf(1/4*(4*f*x-b*ln(f))*2^(1/2)/f^(1/2))*2^
(1/2)*Pi^(1/2)+1/16*exp(2*d-1/8*b^2*ln(f)^2/f)*f^(-1/2+a)*erfi(1/4*(4*f*x+b*ln(f))*2^(1/2)/f^(1/2))*2^(1/2)*Pi
^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5623, 2225, 2325, 2266, 2236, 2235} \begin {gather*} \frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{\frac {b^2 \log ^2(f)}{8 f}-2 d} \text {Erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{2}} f^{a-\frac {1}{2}} e^{2 d-\frac {b^2 \log ^2(f)}{8 f}} \text {Erfi}\left (\frac {b \log (f)+4 f x}{2 \sqrt {2} \sqrt {f}}\right )-\frac {f^{a+b x}}{2 b \log (f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x)*Sinh[d + f*x^2]^2,x]

[Out]

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 2266

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

________________________________________________________________________________________

Mathematica [A]
time = 0.55, size = 149, normalized size = 1.01 \begin {gather*} \frac {1}{16} f^a \left (-\frac {8 f^{b x}}{b \log (f)}+\frac {e^{\frac {b^2 \log ^2(f)}{8 f}} \sqrt {2 \pi } \text {Erf}\left (\frac {4 f x-b \log (f)}{2 \sqrt {2} \sqrt {f}}\right ) (\cosh (2 d)-\sinh (2 d))}{\sqrt {f}}+\frac {e^{-\frac {b^2 \log ^2(f)}{8 f}} \sqrt {2 \pi } \text {Erfi}\left (\frac {4 f x+b \log (f)}{2 \sqrt {2} \sqrt {f}}\right ) (\cosh (2 d)+\sinh (2 d))}{\sqrt {f}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^a*((-8*f^(b*x))/(b*Log[f]) + (E^((b^2*Log[f]^2)/(8*f))*Sqrt[2*Pi]*Erf[(4*f*x - b*Log[f])/(2*Sqrt[2]*Sqrt[f]
)]*(Cosh[2*d] - Sinh[2*d]))/Sqrt[f] + (Sqrt[2*Pi]*Erfi[(4*f*x + b*Log[f])/(2*Sqrt[2]*Sqrt[f])]*(Cosh[2*d] + Si
nh[2*d]))/(E^((b^2*Log[f]^2)/(8*f))*Sqrt[f])))/16

________________________________________________________________________________________

Maple [A]
time = 2.94, size = 126, normalized size = 0.85

method	result	size
risch	\(-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{\frac {b^{2} \ln \left (f \right )^{2}-16 d f}{8 f}} \sqrt {2}\, \erf \left (-\sqrt {2}\, \sqrt {f}\, x +\frac {b \ln \left (f \right ) \sqrt {2}}{4 \sqrt {f}}\right )}{16 \sqrt {f}}-\frac {\sqrt {\pi }\, f^{a} {\mathrm e}^{-\frac {b^{2} \ln \left (f \right )^{2}-16 d f}{8 f}} \erf \left (-\sqrt {-2 f}\, x +\frac {b \ln \left (f \right )}{2 \sqrt {-2 f}}\right )}{8 \sqrt {-2 f}}-\frac {f^{a} f^{b x}}{2 b \ln \left (f \right )}\)	\(126\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*Pi^(1/2)*f^a*exp(1/8*(b^2*ln(f)^2-16*d*f)/f)*2^(1/2)/f^(1/2)*erf(-2^(1/2)*f^(1/2)*x+1/4*b*ln(f)*2^(1/2)/
f^(1/2))-1/8*Pi^(1/2)*f^a*exp(-1/8*(b^2*ln(f)^2-16*d*f)/f)/(-2*f)^(1/2)*erf(-(-2*f)^(1/2)*x+1/2*b*ln(f)/(-2*f)
^(1/2))-1/2*f^a*f^(b*x)/b/ln(f)

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 127, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {f} x - \frac {\sqrt {2} b \log \left (f\right )}{4 \, \sqrt {f}}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2}}{8 \, f} - 2 \, d\right )}}{16 \, \sqrt {f}} + \frac {\sqrt {2} \sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {2} \sqrt {-f} x - \frac {\sqrt {2} b \log \left (f\right )}{4 \, \sqrt {-f}}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2}}{8 \, f} + 2 \, d\right )}}{16 \, \sqrt {-f}} - \frac {f^{b x + a}}{2 \, b \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(2)*sqrt(pi)*f^a*erf(sqrt(2)*sqrt(f)*x - 1/4*sqrt(2)*b*log(f)/sqrt(f))*e^(1/8*b^2*log(f)^2/f - 2*d)/s
qrt(f) + 1/16*sqrt(2)*sqrt(pi)*f^a*erf(sqrt(2)*sqrt(-f)*x - 1/4*sqrt(2)*b*log(f)/sqrt(-f))*e^(-1/8*b^2*log(f)^
2/f + 2*d)/sqrt(-f) - 1/2*f^(b*x + a)/(b*log(f))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (114) = 228\).
time = 0.37, size = 278, normalized size = 1.88 \begin {gather*} -\frac {\sqrt {2} \sqrt {\pi } b \sqrt {-f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{4 \, f}\right ) \log \left (f\right ) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \cosh \left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \left (f\right )\right )}}{4 \, \sqrt {f}}\right ) \log \left (f\right ) + \sqrt {2} \sqrt {\pi } b \sqrt {f} \operatorname {erf}\left (-\frac {\sqrt {2} {\left (4 \, f x - b \log \left (f\right )\right )}}{4 \, \sqrt {f}}\right ) \log \left (f\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) - \sqrt {2} \sqrt {\pi } b \sqrt {-f} \operatorname {erf}\left (\frac {\sqrt {2} {\left (4 \, f x + b \log \left (f\right )\right )} \sqrt {-f}}{4 \, f}\right ) \log \left (f\right ) \sinh \left (\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right ) + 8 \, f \cosh \left ({\left (b x + a\right )} \log \left (f\right )\right ) + 8 \, f \sinh \left ({\left (b x + a\right )} \log \left (f\right )\right )}{16 \, b f \log \left (f\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(sqrt(2)*sqrt(pi)*b*sqrt(-f)*cosh(1/8*(b^2*log(f)^2 - 8*a*f*log(f) - 16*d*f)/f)*erf(1/4*sqrt(2)*(4*f*x +
b*log(f))*sqrt(-f)/f)*log(f) + sqrt(2)*sqrt(pi)*b*sqrt(f)*cosh(1/8*(b^2*log(f)^2 + 8*a*f*log(f) - 16*d*f)/f)*
erf(-1/4*sqrt(2)*(4*f*x - b*log(f))/sqrt(f))*log(f) + sqrt(2)*sqrt(pi)*b*sqrt(f)*erf(-1/4*sqrt(2)*(4*f*x - b*l
og(f))/sqrt(f))*log(f)*sinh(1/8*(b^2*log(f)^2 + 8*a*f*log(f) - 16*d*f)/f) - sqrt(2)*sqrt(pi)*b*sqrt(-f)*erf(1/
4*sqrt(2)*(4*f*x + b*log(f))*sqrt(-f)/f)*log(f)*sinh(1/8*(b^2*log(f)^2 - 8*a*f*log(f) - 16*d*f)/f) + 8*f*cosh(
(b*x + a)*log(f)) + 8*f*sinh((b*x + a)*log(f)))/(b*f*log(f))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 356, normalized size = 2.41 \begin {gather*} -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {f} {\left (4 \, x - \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (\frac {b^{2} \log \left (f\right )^{2} + 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {f}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{4} \, \sqrt {2} \sqrt {-f} {\left (4 \, x + \frac {b \log \left (f\right )}{f}\right )}\right ) e^{\left (-\frac {b^{2} \log \left (f\right )^{2} - 8 \, a f \log \left (f\right ) - 16 \, d f}{8 \, f}\right )}}{16 \, \sqrt {-f}} - {\left (\frac {2 \, b \cos \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi a\right ) \log \left ({\left | f \right |}\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )}^{2}} - \frac {{\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )} \sin \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} \, \pi a\right )}{4 \, b^{2} \log \left ({\left | f \right |}\right )^{2} + {\left (\pi b \mathrm {sgn}\left (f\right ) - \pi b\right )}^{2}}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} + i \, {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi b x \mathrm {sgn}\left (f\right ) - \frac {1}{2} i \, \pi b x + \frac {1}{2} i \, \pi a \mathrm {sgn}\left (f\right ) - \frac {1}{2} i \, \pi a\right )}}{2 i \, \pi b \mathrm {sgn}\left (f\right ) - 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b x \mathrm {sgn}\left (f\right ) + \frac {1}{2} i \, \pi b x - \frac {1}{2} i \, \pi a \mathrm {sgn}\left (f\right ) + \frac {1}{2} i \, \pi a\right )}}{-2 i \, \pi b \mathrm {sgn}\left (f\right ) + 2 i \, \pi b + 4 \, b \log \left ({\left | f \right |}\right )}\right )} e^{\left (b x \log \left ({\left | f \right |}\right ) + a \log \left ({\left | f \right |}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*sqrt(pi)*erf(-1/4*sqrt(2)*sqrt(f)*(4*x - b*log(f)/f))*e^(1/8*(b^2*log(f)^2 + 8*a*f*log(f) - 16*d
*f)/f)/sqrt(f) - 1/16*sqrt(2)*sqrt(pi)*erf(-1/4*sqrt(2)*sqrt(-f)*(4*x + b*log(f)/f))*e^(-1/8*(b^2*log(f)^2 - 8
*a*f*log(f) - 16*d*f)/f)/sqrt(-f) - (2*b*cos(-1/2*pi*b*x*sgn(f) + 1/2*pi*b*x - 1/2*pi*a*sgn(f) + 1/2*pi*a)*log
(abs(f))/(4*b^2*log(abs(f))^2 + (pi*b*sgn(f) - pi*b)^2) - (pi*b*sgn(f) - pi*b)*sin(-1/2*pi*b*x*sgn(f) + 1/2*pi
*b*x - 1/2*pi*a*sgn(f) + 1/2*pi*a)/(4*b^2*log(abs(f))^2 + (pi*b*sgn(f) - pi*b)^2))*e^(b*x*log(abs(f)) + a*log(
abs(f))) + I*(-I*e^(1/2*I*pi*b*x*sgn(f) - 1/2*I*pi*b*x + 1/2*I*pi*a*sgn(f) - 1/2*I*pi*a)/(2*I*pi*b*sgn(f) - 2*
I*pi*b + 4*b*log(abs(f))) + I*e^(-1/2*I*pi*b*x*sgn(f) + 1/2*I*pi*b*x - 1/2*I*pi*a*sgn(f) + 1/2*I*pi*a)/(-2*I*p
i*b*sgn(f) + 2*I*pi*b + 4*b*log(abs(f))))*e^(b*x*log(abs(f)) + a*log(abs(f)))

________________________________________________________________________________________

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

[Out]

int(f^(a + b*x)*sinh(d + f*x^2)^2, x)

________________________________________________________________________________________