Integrand size = 16, antiderivative size = 75 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=-\frac {b c F^{c (a+b x)} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac {e F^{c (a+b x)} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \] Output:
-b*c*F^(c*(b*x+a))*cosh(e*x+d)*ln(F)/(e^2-b^2*c^2*ln(F)^2)+e*F^(c*(b*x+a)) *sinh(e*x+d)/(e^2-b^2*c^2*ln(F)^2)
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.67 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=\frac {F^{c (a+b x)} (-b c \cosh (d+e x) \log (F)+e \sinh (d+e x))}{(e-b c \log (F)) (e+b c \log (F))} \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x],x]
Output:
(F^(c*(a + b*x))*(-(b*c*Cosh[d + e*x]*Log[F]) + e*Sinh[d + e*x]))/((e - b* c*Log[F])*(e + b*c*Log[F]))
Time = 0.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5998}
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.
\(\displaystyle \int \cosh (d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 5998 |
\(\displaystyle \frac {e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x],x]
Output:
-((b*c*F^(c*(a + b*x))*Cosh[d + e*x]*Log[F])/(e^2 - b^2*c^2*Log[F]^2)) + ( e*F^(c*(a + b*x))*Sinh[d + e*x])/(e^2 - b^2*c^2*Log[F]^2)
Int[Cosh[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] : > Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2 )), x] + Simp[e*F^(c*(a + b*x))*(Sinh[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]
Time = 0.37 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(-\frac {\left (-\ln \left (F \right ) \cosh \left (e x +d \right ) b c +e \sinh \left (e x +d \right )\right ) F^{c \left (b x +a \right )}}{b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}}\) | \(52\) |
risch | \(\frac {\left (\ln \left (F \right ) b c \,{\mathrm e}^{2 e x +2 d}+b c \ln \left (F \right )-e \,{\mathrm e}^{2 e x +2 d}+e \right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\) | \(74\) |
orering | \(\frac {2 b c \ln \left (F \right ) F^{c \left (b x +a \right )} \cosh \left (e x +d \right )}{b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}}-\frac {F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right )+F^{c \left (b x +a \right )} e \sinh \left (e x +d \right )}{b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}}\) | \(101\) |
Input:
int(F^(c*(b*x+a))*cosh(e*x+d),x,method=_RETURNVERBOSE)
Output:
-(-ln(F)*cosh(e*x+d)*b*c+e*sinh(e*x+d))*F^(c*(b*x+a))/(b^2*c^2*ln(F)^2-e^2 )
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (77) = 154\).
Time = 0.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.28 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=-\frac {{\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - e^{2} \cosh \left (e x + d\right ) + {\left (b^{2} c^{2} \log \left (F\right )^{2} - e^{2}\right )} \sinh \left (e x + d\right )\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d),x, algorithm="fricas")
Output:
-1/2*((e*cosh(e*x + d)^2 - (b*c*log(F) - e)*sinh(e*x + d)^2 - (b*c*cosh(e* x + d)^2 + b*c)*log(F) - 2*(b*c*cosh(e*x + d)*log(F) - e*cosh(e*x + d))*si nh(e*x + d) - e)*cosh((b*c*x + a*c)*log(F)) + (e*cosh(e*x + d)^2 - (b*c*lo g(F) - e)*sinh(e*x + d)^2 - (b*c*cosh(e*x + d)^2 + b*c)*log(F) - 2*(b*c*co sh(e*x + d)*log(F) - e*cosh(e*x + d))*sinh(e*x + d) - e)*sinh((b*c*x + a*c )*log(F)))/(b^2*c^2*cosh(e*x + d)*log(F)^2 - e^2*cosh(e*x + d) + (b^2*c^2* log(F)^2 - e^2)*sinh(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (68) = 136\).
Time = 0.65 (sec) , antiderivative size = 265, normalized size of antiderivative = 3.53 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=\begin {cases} x \cosh {\left (d \right )} & \text {for}\: F = 1 \wedge e = 0 \\F^{a c} x \cosh {\left (d \right )} & \text {for}\: b = 0 \wedge e = 0 \\x \cosh {\left (d \right )} & \text {for}\: c = 0 \wedge e = 0 \\- \frac {F^{a c + b c x} x \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} x \cosh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2 b c \log {\left (F \right )}} & \text {for}\: e = - b c \log {\left (F \right )} \\- \frac {F^{a c + b c x} x \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{2} + \frac {F^{a c + b c x} x \cosh {\left (b c x \log {\left (F \right )} + d \right )}}{2} + \frac {F^{a c + b c x} \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{2 b c \log {\left (F \right )}} & \text {for}\: e = b c \log {\left (F \right )} \\\frac {F^{a c + b c x} b c \log {\left (F \right )} \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} - \frac {F^{a c + b c x} e \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d),x)
Output:
Piecewise((x*cosh(d), Eq(F, 1) & Eq(e, 0)), (F**(a*c)*x*cosh(d), Eq(b, 0) & Eq(e, 0)), (x*cosh(d), Eq(c, 0) & Eq(e, 0)), (-F**(a*c + b*c*x)*x*sinh(b *c*x*log(F) - d)/2 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F) - d)/2 + F**(a*c + b*c*x)*sinh(b*c*x*log(F) - d)/(2*b*c*log(F)), Eq(e, -b*c*log(F))), (-F* *(a*c + b*c*x)*x*sinh(b*c*x*log(F) + d)/2 + F**(a*c + b*c*x)*x*cosh(b*c*x* log(F) + d)/2 + F**(a*c + b*c*x)*sinh(b*c*x*log(F) + d)/(2*b*c*log(F)), Eq (e, b*c*log(F))), (F**(a*c + b*c*x)*b*c*log(F)*cosh(d + e*x)/(b**2*c**2*lo g(F)**2 - e**2) - F**(a*c + b*c*x)*e*sinh(d + e*x)/(b**2*c**2*log(F)**2 - e**2), True))
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{2 \, {\left (b c \log \left (F\right ) + e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{2 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d),x, algorithm="maxima")
Output:
1/2*F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + 1/2*F^(a*c)*e^(b *c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 597, normalized size of antiderivative = 7.96 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d),x, algorithm="giac")
Output:
(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*lo g(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(ab s(F)) + (b*c*log(abs(F)) + e)*x + d) + (2*(b*c*log(abs(F)) - e)*cos(-1/2*p i*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*s gn(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 )
Time = 2.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=-\frac {F^{a\,c+b\,c\,x}\,{\mathrm {e}}^{-d-e\,x}\,\left (e-e\,{\mathrm {e}}^{2\,d+2\,e\,x}+b\,c\,\ln \left (F\right )+b\,c\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \left (F\right )\right )}{2\,\left (e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x),x)
Output:
-(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))
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68 \[ \int F^{c (a+b x)} \cosh (d+e x) \, dx=\frac {f^{b c x +a c} \left (\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c -\sinh \left (e x +d \right ) e \right )}{\mathrm {log}\left (f \right )^{2} b^{2} c^{2}-e^{2}} \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d),x)
Output:
(f**(a*c + b*c*x)*(cosh(d + e*x)*log(f)*b*c - sinh(d + e*x)*e))/(log(f)**2 *b**2*c**2 - e**2)