Integrand size = 22, antiderivative size = 70 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\frac {e^{-2 d-2 e x} F^{c (a+b x)}}{4 (2 e-b c \log (F))}+\frac {e^{2 d+2 e x} F^{c (a+b x)}}{4 (2 e+b c \log (F))} \] Output:
exp(-2*e*x-2*d)*F^(c*(b*x+a))/(8*e-4*b*c*ln(F))+exp(2*e*x+2*d)*F^(c*(b*x+a ))/(8*e+4*b*c*ln(F))
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\frac {F^{c (a+b x)} (2 e \cosh (2 (d+e x))-b c \log (F) \sinh (2 (d+e x)))}{2 \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x],x]
Output:
(F^(c*(a + b*x))*(2*e*Cosh[2*(d + e*x)] - b*c*Log[F]*Sinh[2*(d + e*x)]))/( 2*(4*e^2 - b^2*c^2*Log[F]^2))
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.29, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6035, 27, 5997}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 6035 |
\(\displaystyle \int \frac {1}{2} \sinh (2 d+2 e x) F^{c (a+b x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int F^{c (a+b x)} \sinh (2 d+2 e x)dx\) |
\(\Big \downarrow \) 5997 |
\(\displaystyle \frac {1}{2} \left (\frac {2 e \cosh (2 d+2 e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh (2 d+2 e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}\right )\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x],x]
Output:
((2*e*F^(c*(a + b*x))*Cosh[2*d + 2*e*x])/(4*e^2 - b^2*c^2*Log[F]^2) - (b*c *F^(c*(a + b*x))*Log[F]*Sinh[2*d + 2*e*x])/(4*e^2 - b^2*c^2*Log[F]^2))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[( d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)) , Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g} , x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 6.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13
method | result | size |
risch | \(\frac {\left (\ln \left (F \right ) b c \,{\mathrm e}^{4 e x +4 d}-2 \,{\mathrm e}^{4 e x +4 d} e -b c \ln \left (F \right )-2 e \right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 \left (b c \ln \left (F \right )-2 e \right ) \left (2 e +b c \ln \left (F \right )\right )}\) | \(79\) |
orering | \(\frac {2 b c \ln \left (F \right ) F^{c \left (b x +a \right )} \cosh \left (e x +d \right ) \sinh \left (e x +d \right )}{b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}}-\frac {F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right ) \sinh \left (e x +d \right )+F^{c \left (b x +a \right )} e \sinh \left (e x +d \right )^{2}+F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{2} e}{b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}}\) | \(134\) |
Input:
int(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/4*(ln(F)*b*c*exp(4*e*x+4*d)-2*exp(4*e*x+4*d)*e-b*c*ln(F)-2*e)/(b*c*ln(F) -2*e)*exp(-2*e*x-2*d)/(2*e+b*c*ln(F))*F^(c*(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (63) = 126\).
Time = 0.09 (sec) , antiderivative size = 433, normalized size of antiderivative = 6.19 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=-\frac {{\left (2 \, e \cosh \left (e x + d\right )^{4} - {\left (b c \log \left (F\right ) - 2 \, e\right )} \sinh \left (e x + d\right )^{4} - 4 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right )^{3} - 6 \, {\left (b c \cosh \left (e x + d\right )^{2} \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )^{2}\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{4} - b c\right )} \log \left (F\right ) - 4 \, {\left (b c \cosh \left (e x + d\right )^{3} \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )^{3}\right )} \sinh \left (e x + d\right ) + 2 \, e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left (2 \, e \cosh \left (e x + d\right )^{4} - {\left (b c \log \left (F\right ) - 2 \, e\right )} \sinh \left (e x + d\right )^{4} - 4 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right )^{3} - 6 \, {\left (b c \cosh \left (e x + d\right )^{2} \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )^{2}\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{4} - b c\right )} \log \left (F\right ) - 4 \, {\left (b c \cosh \left (e x + d\right )^{3} \log \left (F\right ) - 2 \, e \cosh \left (e x + d\right )^{3}\right )} \sinh \left (e x + d\right ) + 2 \, e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{4 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{2} \log \left (F\right )^{2} - 4 \, e^{2} \cosh \left (e x + d\right )^{2} + {\left (b^{2} c^{2} \log \left (F\right )^{2} - 4 \, e^{2}\right )} \sinh \left (e x + d\right )^{2} + 2 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - 4 \, e^{2} \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right )\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x, algorithm="fricas")
Output:
-1/4*((2*e*cosh(e*x + d)^4 - (b*c*log(F) - 2*e)*sinh(e*x + d)^4 - 4*(b*c*c osh(e*x + d)*log(F) - 2*e*cosh(e*x + d))*sinh(e*x + d)^3 - 6*(b*c*cosh(e*x + d)^2*log(F) - 2*e*cosh(e*x + d)^2)*sinh(e*x + d)^2 - (b*c*cosh(e*x + d) ^4 - b*c)*log(F) - 4*(b*c*cosh(e*x + d)^3*log(F) - 2*e*cosh(e*x + d)^3)*si nh(e*x + d) + 2*e)*cosh((b*c*x + a*c)*log(F)) + (2*e*cosh(e*x + d)^4 - (b* c*log(F) - 2*e)*sinh(e*x + d)^4 - 4*(b*c*cosh(e*x + d)*log(F) - 2*e*cosh(e *x + d))*sinh(e*x + d)^3 - 6*(b*c*cosh(e*x + d)^2*log(F) - 2*e*cosh(e*x + d)^2)*sinh(e*x + d)^2 - (b*c*cosh(e*x + d)^4 - b*c)*log(F) - 4*(b*c*cosh(e *x + d)^3*log(F) - 2*e*cosh(e*x + d)^3)*sinh(e*x + d) + 2*e)*sinh((b*c*x + a*c)*log(F)))/(b^2*c^2*cosh(e*x + d)^2*log(F)^2 - 4*e^2*cosh(e*x + d)^2 + (b^2*c^2*log(F)^2 - 4*e^2)*sinh(e*x + d)^2 + 2*(b^2*c^2*cosh(e*x + d)*log (F)^2 - 4*e^2*cosh(e*x + d))*sinh(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (61) = 122\).
Time = 1.15 (sec) , antiderivative size = 454, normalized size of antiderivative = 6.49 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\begin {cases} x \sinh {\left (d \right )} \cosh {\left (d \right )} & \text {for}\: F = 1 \wedge e = 0 \\F^{a c} x \sinh {\left (d \right )} \cosh {\left (d \right )} & \text {for}\: b = 0 \wedge e = 0 \\x \sinh {\left (d \right )} \cosh {\left (d \right )} & \text {for}\: c = 0 \wedge e = 0 \\\frac {F^{a c + b c x} x \sinh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{4} - \frac {F^{a c + b c x} x \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{2} + \frac {F^{a c + b c x} x \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{4} - \frac {F^{a c + b c x} \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{2 b c \log {\left (F \right )}} & \text {for}\: e = - \frac {b c \log {\left (F \right )}}{2} \\- \frac {F^{a c + b c x} x \sinh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{4} + \frac {F^{a c + b c x} x \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{2} - \frac {F^{a c + b c x} x \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{4} + \frac {F^{a c + b c x} \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{2 b c \log {\left (F \right )}} & \text {for}\: e = \frac {b c \log {\left (F \right )}}{2} \\\frac {F^{a c + b c x} b c \log {\left (F \right )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - 4 e^{2}} - \frac {F^{a c + b c x} e \sinh ^{2}{\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - 4 e^{2}} - \frac {F^{a c + b c x} e \cosh ^{2}{\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - 4 e^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x)
Output:
Piecewise((x*sinh(d)*cosh(d), Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*sinh(d)*co sh(d), Eq(b, 0) & Eq(e, 0)), (x*sinh(d)*cosh(d), Eq(c, 0) & Eq(e, 0)), (F* *(a*c + b*c*x)*x*sinh(b*c*x*log(F)/2 - d)**2/4 - F**(a*c + b*c*x)*x*sinh(b *c*x*log(F)/2 - d)*cosh(b*c*x*log(F)/2 - d)/2 + F**(a*c + b*c*x)*x*cosh(b* c*x*log(F)/2 - d)**2/4 - F**(a*c + b*c*x)*sinh(b*c*x*log(F)/2 - d)*cosh(b* c*x*log(F)/2 - d)/(2*b*c*log(F)), Eq(e, -b*c*log(F)/2)), (-F**(a*c + b*c*x )*x*sinh(b*c*x*log(F)/2 + d)**2/4 + F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/2 + d)*cosh(b*c*x*log(F)/2 + d)/2 - F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/2 + d)**2/4 + F**(a*c + b*c*x)*sinh(b*c*x*log(F)/2 + d)*cosh(b*c*x*log(F)/2 + d)/(2*b*c*log(F)), Eq(e, b*c*log(F)/2)), (F**(a*c + b*c*x)*b*c*log(F)*si nh(d + e*x)*cosh(d + e*x)/(b**2*c**2*log(F)**2 - 4*e**2) - F**(a*c + b*c*x )*e*sinh(d + e*x)**2/(b**2*c**2*log(F)**2 - 4*e**2) - F**(a*c + b*c*x)*e*c osh(d + e*x)**2/(b**2*c**2*log(F)**2 - 4*e**2), True))
Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{4 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{4 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x, algorithm="maxima")
Output:
1/4*F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) - 1/4*F^(a*c )*e^(b*c*x*log(F) - 2*e*x)/(b*c*e^(2*d)*log(F) - 2*e*e^(2*d))
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 613, normalized size of antiderivative = 8.76 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x, algorithm="giac")
Output:
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(ab s(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) + 1/4*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)) + 4*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)) + 4*e)) *e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) - 1/2*(2*(b*c*log(a bs(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*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) + 1/4*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)) - 4*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)) - 4*e))*e^(a*c*log(abs( F)) + (b*c*log(abs(F)) - 2*e)*x - 2*d)
Time = 2.91 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.37 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\frac {F^{a\,c+b\,c\,x}\,e\,\left (\frac {{\mathrm {e}}^{-2\,d-2\,e\,x}}{2}+\frac {{\mathrm {e}}^{2\,d+2\,e\,x}}{2}\right )+\frac {F^{a\,c+b\,c\,x}\,b\,c\,\ln \left (F\right )\,\left (\frac {{\mathrm {e}}^{-2\,d-2\,e\,x}}{2}-\frac {{\mathrm {e}}^{2\,d+2\,e\,x}}{2}\right )}{2}}{4\,e^2-b^2\,c^2\,{\ln \left (F\right )}^2} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x)*sinh(d + e*x),x)
Output:
(F^(a*c + b*c*x)*e*(exp(- 2*d - 2*e*x)/2 + exp(2*d + 2*e*x)/2) + (F^(a*c + b*c*x)*b*c*log(F)*(exp(- 2*d - 2*e*x)/2 - exp(2*d + 2*e*x)/2))/2)/(4*e^2 - b^2*c^2*log(F)^2)
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x) \, dx=\frac {f^{b c x +a c} \left (-\cosh \left (e x +d \right )^{2} e +\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) \sinh \left (e x +d \right ) b c -\sinh \left (e x +d \right )^{2} e \right )}{\mathrm {log}\left (f \right )^{2} b^{2} c^{2}-4 e^{2}} \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d),x)
Output:
(f**(a*c + b*c*x)*( - cosh(d + e*x)**2*e + cosh(d + e*x)*log(f)*sinh(d + e *x)*b*c - sinh(d + e*x)**2*e))/(log(f)**2*b**2*c**2 - 4*e**2)