3.4.24 \(\int F^{c (a+b x)} \sinh (d+e x) \, dx\) [324]

Optimal. Leaf size=75 \[ \frac {e F^{c (a+b x)} \cosh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5582} \begin {gather*} \frac {e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Sinh[d + e*x],x]

[Out]

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

Rule 5582

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

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.67 \begin {gather*} \frac {F^{c (a+b x)} (e \cosh (d+e x)-b c \log (F) \sinh (d+e x))}{(e-b c \log (F)) (e+b c \log (F))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.18, size = 77, normalized size = 1.03

method	result	size
risch	\(\frac {\left (\ln \left (F \right ) b c \,{\mathrm e}^{2 e x +2 d}-b c \ln \left (F \right )-e \,{\mathrm e}^{2 e x +2 d}-e \right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 67, normalized size = 0.89 \begin {gather*} \frac {F^{a c} e^{\left (b c x \log \left (F\right ) + x e + d\right )}}{2 \, {\left (b c \log \left (F\right ) + e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - x e\right )}}{2 \, {\left (b c e^{d} \log \left (F\right ) - e^{\left (d + 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (78) = 156\).
time = 0.39, size = 376, normalized size = 5.01 \begin {gather*} -\frac {{\left ({\left (\cosh \left (1\right ) + \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (b c \log \left (F\right ) - \cosh \left (1\right ) - \sinh \left (1\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (b c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right ) - {\left (\cosh \left (1\right ) + \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left ({\left (\cosh \left (1\right ) + \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (b c \log \left (F\right ) - \cosh \left (1\right ) - \sinh \left (1\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (b c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right ) - {\left (\cosh \left (1\right ) + \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + \cosh \left (1\right ) + \sinh \left (1\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \, {\left (b^{2} c^{2} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) \log \left (F\right )^{2} - {\left (\cosh \left (1\right )^{2} + 2 \, \cosh \left (1\right ) \sinh \left (1\right ) + \sinh \left (1\right )^{2}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + {\left (b^{2} c^{2} \log \left (F\right )^{2} - \cosh \left (1\right )^{2} - 2 \, \cosh \left (1\right ) \sinh \left (1\right ) - \sinh \left (1\right )^{2}\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(((cosh(1) + sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2 - (b*c*log(F) - cosh(1) - sinh(1))*sinh(x*cosh(1)
+ x*sinh(1) + d)^2 - (b*c*cosh(x*cosh(1) + x*sinh(1) + d)^2 - b*c)*log(F) - 2*(b*c*cosh(x*cosh(1) + x*sinh(1)
+ d)*log(F) - (cosh(1) + sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d) + cosh(1)
+ sinh(1))*cosh((b*c*x + a*c)*log(F)) + ((cosh(1) + sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2 - (b*c*log(F) -
cosh(1) - sinh(1))*sinh(x*cosh(1) + x*sinh(1) + d)^2 - (b*c*cosh(x*cosh(1) + x*sinh(1) + d)^2 - b*c)*log(F) -
2*(b*c*cosh(x*cosh(1) + x*sinh(1) + d)*log(F) - (cosh(1) + sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*c
osh(1) + x*sinh(1) + d) + cosh(1) + sinh(1))*sinh((b*c*x + a*c)*log(F)))/(b^2*c^2*cosh(x*cosh(1) + x*sinh(1) +
d)*log(F)^2 - (cosh(1)^2 + 2*cosh(1)*sinh(1) + sinh(1)^2)*cosh(x*cosh(1) + x*sinh(1) + d) + (b^2*c^2*log(F)^2
- cosh(1)^2 - 2*cosh(1)*sinh(1) - sinh(1)^2)*sinh(x*cosh(1) + x*sinh(1) + d))

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Sympy [C] Result contains complex when optimal does not.
time = 2.47, size = 416, normalized size = 5.55 \begin {gather*} \begin {cases} \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} x \sinh {\left (d + e x \right )}}{2} - \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} x \cosh {\left (d + e x \right )}}{2} + \frac {\left (-1\right )^{a c} \left (-1\right )^{- \frac {i e x}{\pi }} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: F = -1 \wedge b = - \frac {i e}{\pi c} \\x \sinh {\left (d \right )} & \text {for}\: F = 1 \wedge e = 0 \\\frac {b c \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \log {\left (e^{- \frac {e}{b c}} \right )} \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (e^{- \frac {e}{b c}} \right )}^{2} - e^{2}} - \frac {e \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (e^{- \frac {e}{b c}} \right )}^{2} - e^{2}} & \text {for}\: F = e^{- \frac {e}{b c}} \\\frac {b c \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \log {\left (e^{\frac {e}{b c}} \right )} \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (e^{\frac {e}{b c}} \right )}^{2} - e^{2}} - \frac {e \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (e^{\frac {e}{b c}} \right )}^{2} - e^{2}} & \text {for}\: F = e^{\frac {e}{b c}} \\\frac {F^{a c} F^{b c x} b c \log {\left (F \right )} \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} - \frac {F^{a c} F^{b c x} e \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} & \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),x)

[Out]

Piecewise(((-1)**(a*c)*x*sinh(d + e*x)/(2*(-1)**(I*e*x/pi)) - (-1)**(a*c)*x*cosh(d + e*x)/(2*(-1)**(I*e*x/pi))
+ (-1)**(a*c)*cosh(d + e*x)/(2*(-1)**(I*e*x/pi)*e), Eq(F, -1) & Eq(b, -I*e/(pi*c))), (x*sinh(d), Eq(F, 1) & E
q(e, 0)), (b*c*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*log(exp(-e/(b*c)))*sinh(d + e*x)/(b**2*c**2*log(exp
(-e/(b*c)))**2 - e**2) - e*exp(-e/(b*c))**(a*c)*exp(-e/(b*c))**(b*c*x)*cosh(d + e*x)/(b**2*c**2*log(exp(-e/(b*
c)))**2 - e**2), Eq(F, exp(-e/(b*c)))), (b*c*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*log(exp(e/(b*c)))*sinh(
d + e*x)/(b**2*c**2*log(exp(e/(b*c)))**2 - e**2) - e*exp(e/(b*c))**(a*c)*exp(e/(b*c))**(b*c*x)*cosh(d + e*x)/(
b**2*c**2*log(exp(e/(b*c)))**2 - e**2), Eq(F, exp(e/(b*c)))), (F**(a*c)*F**(b*c*x)*b*c*log(F)*sinh(d + e*x)/(b
**2*c**2*log(F)**2 - e**2) - F**(a*c)*F**(b*c*x)*e*cosh(d + e*x)/(b**2*c**2*log(F)**2 - e**2), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.43, size = 598, normalized size = 7.97 \begin {gather*} {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) + 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 ) + 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 ) + e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + e\right )} x + d\right )} + \frac {1}{2} 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 )}}{i \, \pi b c \mathrm {sgn}\left (F\right ) - i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right ) + 2 \, 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 )}}{-i \, \pi b c \mathrm {sgn}\left (F\right ) + i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right ) + 2 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) + e\right )} x + d\right )} - {\left (\frac {2 \, {\left (b c \log \left ({\left | F \right |}\right ) - 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 ) - 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 ) - e\right )}^{2}}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - e\right )} x - d\right )} + \frac {1}{2} 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 )}}{i \, \pi b c \mathrm {sgn}\left (F\right ) - i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right ) - 2 \, 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 )}}{-i \, \pi b c \mathrm {sgn}\left (F\right ) + i \, \pi b c + 2 \, b c \log \left ({\left | F \right |}\right ) - 2 \, e}\right )} e^{\left (a c \log \left ({\left | F \right |}\right ) + {\left (b c \log \left ({\left | F \right |}\right ) - e\right )} x - 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),x, algorithm="giac")

[Out]

(2*(b*c*log(abs(F)) + 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*sg
n(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 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)) + e)^2))*e^(a*c*log(abs(
F)) + (b*c*log(abs(F)) + e)*x + d) + 1/2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F)) + 2*e) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*p
i*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-I*pi*b*c*sgn(F) + I*pi*b*c + 2*b*c*log(abs(F)) + 2*e))*e^(a*c*
log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) - (2*(b*c*log(abs(F)) - 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)) - 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)) - e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d) + 1/2*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)/(I*pi*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))
- 2*e) + 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)/(-I*pi*b*c*sgn(F)
+ I*pi*b*c + 2*b*c*log(abs(F)) - 2*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) - e)*x - d)

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Mupad [B]
time = 0.66, size = 73, normalized size = 0.97 \begin {gather*} \frac {F^{a\,c+b\,c\,x}\,{\mathrm {e}}^{-d-e\,x}\,\left (e+e\,{\mathrm {e}}^{2\,d+2\,e\,x}+b\,c\,\ln \left (F\right )-b\,c\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \left (F\right )\right )}{2\,\left (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),x)

[Out]

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

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