3.4.23 \(\int F^{c (a+b x)} \sinh ^2(d+e x) \, dx\) [323]

Optimal. Leaf size=132 \[ -\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^2(d+e x)}{4 e^2-b^2 c^2 \log ^2(F)} \]

[Out]

-2*e^2*F^(c*(b*x+a))/b/c/ln(F)/(4*e^2-b^2*c^2*ln(F)^2)+2*e*F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)/(4*e^2-b^2*c^
2*ln(F)^2)-b*c*F^(c*(b*x+a))*ln(F)*sinh(e*x+d)^2/(4*e^2-b^2*c^2*ln(F)^2)

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Rubi [A]
time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5584, 2225} \begin {gather*} -\frac {b c \log (F) \sinh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d +
e*x])/(4*e^2 - b^2*c^2*Log[F]^2) - (b*c*F^(c*(a + b*x))*Log[F]*Sinh[d + e*x]^2)/(4*e^2 - b^2*c^2*Log[F]^2)

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 5584

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(n_), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Sinh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] + (-Dist[n*(n - 1)*(e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)
), Int[F^(c*(a + b*x))*Sinh[d + e*x]^(n - 2), x], x] + Simp[e*n*F^(c*(a + b*x))*Cosh[d + e*x]*(Sinh[d + e*x]^(
n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0
] && GtQ[n, 1]

Rubi steps

\begin {align*} \int F^{c (a+b x)} \sinh ^2(d+e x) \, dx &=\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^2(d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{4 e^2-b^2 c^2 \log ^2(F)}\\ &=-\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh ^2(d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 86, normalized size = 0.65 \begin {gather*} \frac {F^{c (a+b x)} \left (4 e^2-b^2 c^2 \log ^2(F)+b^2 c^2 \cosh (2 (d+e x)) \log ^2(F)-2 b c e \log (F) \sinh (2 (d+e x))\right )}{-8 b c e^2 \log (F)+2 b^3 c^3 \log ^3(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(F^(c*(a + b*x))*(4*e^2 - b^2*c^2*Log[F]^2 + b^2*c^2*Cosh[2*(d + e*x)]*Log[F]^2 - 2*b*c*e*Log[F]*Sinh[2*(d + e
*x)]))/(-8*b*c*e^2*Log[F] + 2*b^3*c^3*Log[F]^3)

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Maple [A]
time = 1.08, size = 143, normalized size = 1.08

method	result	size
risch	\(\frac {\left (\ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{4 e x +4 d}-2 \ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 \ln \left (F \right ) b c e \,{\mathrm e}^{4 e x +4 d}+b^{2} c^{2} \ln \left (F \right )^{2}+2 \ln \left (F \right ) b c e +8 e^{2} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \left (F \right ) \left (b c \ln \left (F \right )-2 e \right ) \left (b c \ln \left (F \right )+2 e \right )}\)	\(143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(ln(F)^2*b^2*c^2*exp(4*e*x+4*d)-2*ln(F)^2*b^2*c^2*exp(2*e*x+2*d)-2*ln(F)*b*c*e*exp(4*e*x+4*d)+b^2*c^2*ln(F
)^2+2*ln(F)*b*c*e+8*e^2*exp(2*e*x+2*d))/b/c/ln(F)/(b*c*ln(F)-2*e)*exp(-2*e*x-2*d)/(b*c*ln(F)+2*e)*F^(c*(b*x+a)
)

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Maxima [A]
time = 0.28, size = 98, normalized size = 0.74 \begin {gather*} \frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, x e + 2 \, d\right )}}{4 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, x e\right )}}{4 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e^{\left (2 \, d + 1\right )}\right )}} - \frac {F^{b c x + a c}}{2 \, b c \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*F^(a*c)*e^(b*c*x*log(F) + 2*x*e + 2*d)/(b*c*log(F) + 2*e) + 1/4*F^(a*c)*e^(b*c*x*log(F) - 2*x*e)/(b*c*e^(2
*d)*log(F) - 2*e^(2*d + 1)) - 1/2*F^(b*c*x + a*c)/(b*c*log(F))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (128) = 256\).
time = 0.39, size = 1115, normalized size = 8.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(((b^2*c^2*log(F)^2 - 2*(b*c*cosh(1) + b*c*sinh(1))*log(F))*sinh(x*cosh(1) + x*sinh(1) + d)^4 + 4*(b^2*c^2
*cosh(x*cosh(1) + x*sinh(1) + d)*log(F)^2 - 2*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)*log(
F))*sinh(x*cosh(1) + x*sinh(1) + d)^3 + 8*(cosh(1)^2 + 2*cosh(1)*sinh(1) + sinh(1)^2)*cosh(x*cosh(1) + x*sinh(
1) + d)^2 + (b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d)^4 - 2*b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d)^2 + b^2*c^2
)*log(F)^2 - 2*(6*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2*log(F) - (3*b^2*c^2*cosh(x*cos
h(1) + x*sinh(1) + d)^2 - b^2*c^2)*log(F)^2 - 4*cosh(1)^2 - 8*cosh(1)*sinh(1) - 4*sinh(1)^2)*sinh(x*cosh(1) +
x*sinh(1) + d)^2 - 2*((b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^4 - b*c*cosh(1) - b*c*sinh(1
))*log(F) - 4*(2*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^3*log(F) - (b^2*c^2*cosh(x*cosh(1
) + x*sinh(1) + d)^3 - b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d))*log(F)^2 - 4*(cosh(1)^2 + 2*cosh(1)*sinh(1) +
sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d))*cosh((b*c*x + a*c)*log(F)) + ((b^
2*c^2*log(F)^2 - 2*(b*c*cosh(1) + b*c*sinh(1))*log(F))*sinh(x*cosh(1) + x*sinh(1) + d)^4 + 4*(b^2*c^2*cosh(x*c
osh(1) + x*sinh(1) + d)*log(F)^2 - 2*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)*log(F))*sinh(
x*cosh(1) + x*sinh(1) + d)^3 + 8*(cosh(1)^2 + 2*cosh(1)*sinh(1) + sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d)^2
+ (b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d)^4 - 2*b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d)^2 + b^2*c^2)*log(F)^
2 - 2*(6*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2*log(F) - (3*b^2*c^2*cosh(x*cosh(1) + x*
sinh(1) + d)^2 - b^2*c^2)*log(F)^2 - 4*cosh(1)^2 - 8*cosh(1)*sinh(1) - 4*sinh(1)^2)*sinh(x*cosh(1) + x*sinh(1)
+ d)^2 - 2*((b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^4 - b*c*cosh(1) - b*c*sinh(1))*log(F)
- 4*(2*(b*c*cosh(1) + b*c*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^3*log(F) - (b^2*c^2*cosh(x*cosh(1) + x*sin
h(1) + d)^3 - b^2*c^2*cosh(x*cosh(1) + x*sinh(1) + d))*log(F)^2 - 4*(cosh(1)^2 + 2*cosh(1)*sinh(1) + sinh(1)^2
)*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d))*sinh((b*c*x + a*c)*log(F)))/(b^3*c^3*cosh(
x*cosh(1) + x*sinh(1) + d)^2*log(F)^3 - 4*(b*c*cosh(1)^2 + 2*b*c*cosh(1)*sinh(1) + b*c*sinh(1)^2)*cosh(x*cosh(
1) + x*sinh(1) + d)^2*log(F) + (b^3*c^3*log(F)^3 - 4*(b*c*cosh(1)^2 + 2*b*c*cosh(1)*sinh(1) + b*c*sinh(1)^2)*l
og(F))*sinh(x*cosh(1) + x*sinh(1) + d)^2 + 2*(b^3*c^3*cosh(x*cosh(1) + x*sinh(1) + d)*log(F)^3 - 4*(b*c*cosh(1
)^2 + 2*b*c*cosh(1)*sinh(1) + b*c*sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d)*log(F))*sinh(x*cosh(1) + x*sinh(1
) + d))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (119) = 238\).
time = 14.04, size = 1052, normalized size = 7.97 \begin {gather*} \begin {cases} \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} - \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: F = 1 \\\frac {b^{2} c^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {2 e}{b c}} \right )}^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{- \frac {2 e}{b c}} \right )}} - \frac {2 b c e \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {2 e}{b c}} \right )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{- \frac {2 e}{b c}} \right )}} - \frac {2 e^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{- \frac {2 e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{- \frac {2 e}{b c}}\right )^{a c} \left (e^{- \frac {2 e}{b c}}\right )^{b c x} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{- \frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{- \frac {2 e}{b c}} \right )}} & \text {for}\: F = e^{- \frac {2 e}{b c}} \\\frac {b^{2} c^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \log {\left (e^{\frac {2 e}{b c}} \right )}^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{\frac {2 e}{b c}} \right )}} - \frac {2 b c e \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \log {\left (e^{\frac {2 e}{b c}} \right )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{\frac {2 e}{b c}} \right )}} - \frac {2 e^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{\frac {2 e}{b c}} \right )}} + \frac {2 e^{2} \left (e^{\frac {2 e}{b c}}\right )^{a c} \left (e^{\frac {2 e}{b c}}\right )^{b c x} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (e^{\frac {2 e}{b c}} \right )}^{3} - 4 b c e^{2} \log {\left (e^{\frac {2 e}{b c}} \right )}} & \text {for}\: F = e^{\frac {2 e}{b c}} \\F^{a c} \left (\frac {x \sinh ^{2}{\left (d + e x \right )}}{2} - \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e}\right ) & \text {for}\: b = 0 \\\frac {x \sinh ^{2}{\left (d + e x \right )}}{2} - \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: c = 0 \\\frac {F^{a c} F^{b c x} b^{2} c^{2} \log {\left (F \right )}^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} - \frac {2 F^{a c} F^{b c x} b c e \log {\left (F \right )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} - \frac {2 F^{a c} F^{b c x} e^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} + \frac {2 F^{a c} F^{b c x} e^{2} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(d + e*x)**2/2 - x*cosh(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(F, 1)), (b**2*c
**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*log(exp(-2*e/(b*c)))**2*sinh(d + e*x)**2/(b**3*c**3*log(ex
p(-2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(-2*e/(b*c)))) - 2*b*c*e*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)
*log(exp(-2*e/(b*c)))*sinh(d + e*x)*cosh(d + e*x)/(b**3*c**3*log(exp(-2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(-2*e
/(b*c)))) - 2*e**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*sinh(d + e*x)**2/(b**3*c**3*log(exp(-2*e/(b
*c)))**3 - 4*b*c*e**2*log(exp(-2*e/(b*c)))) + 2*e**2*exp(-2*e/(b*c))**(a*c)*exp(-2*e/(b*c))**(b*c*x)*cosh(d +
e*x)**2/(b**3*c**3*log(exp(-2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(-2*e/(b*c)))), Eq(F, exp(-2*e/(b*c)))), (b**2*
c**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*log(exp(2*e/(b*c)))**2*sinh(d + e*x)**2/(b**3*c**3*log(exp(
2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(2*e/(b*c)))) - 2*b*c*e*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*log(e
xp(2*e/(b*c)))*sinh(d + e*x)*cosh(d + e*x)/(b**3*c**3*log(exp(2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(2*e/(b*c))))
- 2*e**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*sinh(d + e*x)**2/(b**3*c**3*log(exp(2*e/(b*c)))**3 - 4
*b*c*e**2*log(exp(2*e/(b*c)))) + 2*e**2*exp(2*e/(b*c))**(a*c)*exp(2*e/(b*c))**(b*c*x)*cosh(d + e*x)**2/(b**3*c
**3*log(exp(2*e/(b*c)))**3 - 4*b*c*e**2*log(exp(2*e/(b*c)))), Eq(F, exp(2*e/(b*c)))), (F**(a*c)*(x*sinh(d + e*
x)**2/2 - x*cosh(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e)), Eq(b, 0)), (x*sinh(d + e*x)**2/2 - x*cosh
(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(c, 0)), (F**(a*c)*F**(b*c*x)*b**2*c**2*log(F)**2*sinh(d
+ e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)) - 2*F**(a*c)*F**(b*c*x)*b*c*e*log(F)*sinh(d + e*x)*cosh(d
+ e*x)/(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)) - 2*F**(a*c)*F**(b*c*x)*e**2*sinh(d + e*x)**2/(b**3*c**3*log
(F)**3 - 4*b*c*e**2*log(F)) + 2*F**(a*c)*F**(b*c*x)*e**2*cosh(d + e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e**2*lo
g(F)), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.46, size = 890, normalized size = 6.74 \begin {gather*} -{\left (\frac {2 \, b c \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right ) \log \left ({\left | F \right |}\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{4 \, b^{2} c^{2} \log \left ({\left | F \right |}\right )^{2} + {\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2}}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + i \, {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{2 i \, \pi b c \mathrm {sgn}\left (F\right ) - 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right )} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-2 i \, \pi b c \mathrm {sgn}\left (F\right ) + 2 i \, \pi b c + 4 \, b c \log \left ({\left | F \right |}\right )}\right )} e^{\left (b c x \log \left ({\left | F \right |}\right ) + a c \log \left ({\left | F \right |}\right )\right )} + \frac {1}{2} \, {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) + 2 \, e\right )} \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) + 2 \, e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) + 2 \, e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + 2 \, e\right )} x + 2 \, d\right )} + i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{4 i \, \pi b c \mathrm {sgn}\left (F\right ) - 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) + 16 \, e} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-4 i \, \pi b c \mathrm {sgn}\left (F\right ) + 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) + 16 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + 2 \, e\right )} x + 2 \, d\right )} + \frac {1}{2} \, {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) - 2 \, e\right )} \cos \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) - 2 \, e\right )}^{2}} - \frac {{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )} \sin \left (-\frac {1}{2} \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi b c x - \frac {1}{2} \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} \, \pi a c\right )}{{\left (\pi b c \mathrm {sgn}\left (F\right ) - \pi b c\right )}^{2} + 4 \, {\left (b c \log \left ({\left | F \right |}\right ) - 2 \, e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - 2 \, e\right )} x - 2 \, d\right )} + i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi b c x + \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) - \frac {1}{2} i \, \pi a c\right )}}{4 i \, \pi b c \mathrm {sgn}\left (F\right ) - 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) - 16 \, e} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi b c x \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi b c x - \frac {1}{2} i \, \pi a c \mathrm {sgn}\left (F\right ) + \frac {1}{2} i \, \pi a c\right )}}{-4 i \, \pi b c \mathrm {sgn}\left (F\right ) + 4 i \, \pi b c + 8 \, b c \log \left ({\left | F \right |}\right ) - 16 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - 2 \, e\right )} x - 2 \, d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*c*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)*log(abs(F))/(4*b^2*c^2*log(a
bs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/
2*pi*a*c*sgn(F) + 1/2*pi*a*c)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a
*c*log(abs(F))) + I*(-I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(2*I*p
i*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) + I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*s
gn(F) + 1/2*I*pi*a*c)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) + a*c*log(ab
s(F))) + 1/2*(2*(b*c*log(abs(F)) + 2*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a
*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sg
n(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^
2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1
/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*e) - I*e^(-1/2*I*p
i*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c
*log(abs(F)) + 16*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + 1/2*(2*(b*c*log(abs(F)) - 2*e)*c
os(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*
log(abs(F)) - 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) +
1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - 2*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F))
- 2*e)*x - 2*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(4*I*p
i*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*e) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*p
i*a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*e))*e^(a*c*log(abs(F))
+ (b*c*log(abs(F)) - 2*e)*x - 2*d)

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Mupad [B]
time = 0.92, size = 97, normalized size = 0.73 \begin {gather*} -\frac {F^{a\,c+b\,c\,x}\,\left (2\,e^2-\frac {b^2\,c^2\,{\ln \left (F\right )}^2}{2}+\frac {b^2\,c^2\,{\ln \left (F\right )}^2\,\mathrm {cosh}\left (2\,d+2\,e\,x\right )}{2}-b\,c\,e\,\ln \left (F\right )\,\mathrm {sinh}\left (2\,d+2\,e\,x\right )\right )}{b\,c\,\ln \left (F\right )\,\left (4\,e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(F^(a*c + b*c*x)*(2*e^2 - (b^2*c^2*log(F)^2)/2 + (b^2*c^2*log(F)^2*cosh(2*d + 2*e*x))/2 - b*c*e*log(F)*sinh(2
*d + 2*e*x)))/(b*c*log(F)*(4*e^2 - b^2*c^2*log(F)^2))

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