Integrand size = 20, antiderivative size = 101 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=\frac {f F^{a c+b c x}}{b c \log (F)}-\frac {b c f F^{a c+b c x} \cosh (d+e x) \log (F)}{e^2-b^2 c^2 \log ^2(F)}+\frac {e f F^{a c+b c x} \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \]
f*F^(b*c*x+a*c)/b/c/ln(F)-b*c*f*F^(b*c*x+a*c)*cosh(e*x+d)*ln(F)/(e^2-b^2*c ^2*ln(F)^2)+e*f*F^(b*c*x+a*c)*sinh(e*x+d)/(e^2-b^2*c^2*ln(F)^2)
Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=\frac {f F^{c (a+b x)} \left (-e^2+b^2 c^2 \log ^2(F)+b^2 c^2 \cosh (d+e x) \log ^2(F)-b c e \log (F) \sinh (d+e x)\right )}{b c \log (F) (-e+b c \log (F)) (e+b c \log (F))} \]
(f*F^(c*(a + b*x))*(-e^2 + b^2*c^2*Log[F]^2 + b^2*c^2*Cosh[d + e*x]*Log[F] ^2 - b*c*e*Log[F]*Sinh[d + e*x]))/(b*c*Log[F]*(-e + b*c*Log[F])*(e + b*c*L og[F]))
Time = 0.40 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7292, 27, 7293, 2009}
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.
\(\displaystyle \int F^{c (a+b x)} (f \cosh (d+e x)+f) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int f (\cosh (d+e x)+1) F^{a c+b c x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle f \int F^{a c+b x c} (\cosh (d+e x)+1)dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle f \int \left (\cosh (d+e x) F^{a c+b x c}+F^{a c+b x c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle f \left (\frac {e \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac {F^{a c+b c x}}{b c \log (F)}\right )\) |
f*(F^(a*c + b*c*x)/(b*c*Log[F]) - (b*c*F^(a*c + b*c*x)*Cosh[d + e*x]*Log[F ])/(e^2 - b^2*c^2*Log[F]^2) + (e*F^(a*c + b*c*x)*Sinh[d + e*x])/(e^2 - b^2 *c^2*Log[F]^2))
3.9.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(\frac {f \,F^{c \left (b x +a \right )} \left (\cosh \left (e x +d \right ) b^{2} c^{2} \ln \left (F \right )^{2}+b^{2} c^{2} \ln \left (F \right )^{2}-\sinh \left (e x +d \right ) b c e \ln \left (F \right )-e^{2}\right )}{\left (b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}\right ) b c \ln \left (F \right )}\) | \(88\) |
risch | \(\frac {f \left (\ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{e x +d}+b^{2} c^{2} \ln \left (F \right )^{2}-\ln \left (F \right ) b c e \,{\mathrm e}^{2 e x +2 d}+\ln \left (F \right ) b c e -2 e^{2} {\mathrm e}^{e x +d}\right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 b c \ln \left (F \right ) \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\) | \(135\) |
f*F^(c*(b*x+a))/(b^2*c^2*ln(F)^2-e^2)/b/c/ln(F)*(cosh(e*x+d)*b^2*c^2*ln(F) ^2+b^2*c^2*ln(F)^2-sinh(e*x+d)*b*c*e*ln(F)-e^2)
Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (103) = 206\).
Time = 0.26 (sec) , antiderivative size = 430, normalized size of antiderivative = 4.26 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=-\frac {{\left (2 \, e^{2} f \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} f \cosh \left (e x + d\right )^{2} + 2 \, b^{2} c^{2} f \cosh \left (e x + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} - {\left (b^{2} c^{2} f \log \left (F\right )^{2} - b c e f \log \left (F\right )\right )} \sinh \left (e x + d\right )^{2} + {\left (b c e f \cosh \left (e x + d\right )^{2} - b c e f\right )} \log \left (F\right ) + 2 \, {\left (b c e f \cosh \left (e x + d\right ) \log \left (F\right ) + e^{2} f - {\left (b^{2} c^{2} f \cosh \left (e x + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2}\right )} \sinh \left (e x + d\right )\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left (2 \, e^{2} f \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} f \cosh \left (e x + d\right )^{2} + 2 \, b^{2} c^{2} f \cosh \left (e x + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2} - {\left (b^{2} c^{2} f \log \left (F\right )^{2} - b c e f \log \left (F\right )\right )} \sinh \left (e x + d\right )^{2} + {\left (b c e f \cosh \left (e x + d\right )^{2} - b c e f\right )} \log \left (F\right ) + 2 \, {\left (b c e f \cosh \left (e x + d\right ) \log \left (F\right ) + e^{2} f - {\left (b^{2} c^{2} f \cosh \left (e x + d\right ) + b^{2} c^{2} f\right )} \log \left (F\right )^{2}\right )} \sinh \left (e x + d\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \, {\left (b^{3} c^{3} \cosh \left (e x + d\right ) \log \left (F\right )^{3} - b c e^{2} \cosh \left (e x + d\right ) \log \left (F\right ) + {\left (b^{3} c^{3} \log \left (F\right )^{3} - b c e^{2} \log \left (F\right )\right )} \sinh \left (e x + d\right )\right )}} \]
-1/2*((2*e^2*f*cosh(e*x + d) - (b^2*c^2*f*cosh(e*x + d)^2 + 2*b^2*c^2*f*co sh(e*x + d) + b^2*c^2*f)*log(F)^2 - (b^2*c^2*f*log(F)^2 - b*c*e*f*log(F))* sinh(e*x + d)^2 + (b*c*e*f*cosh(e*x + d)^2 - b*c*e*f)*log(F) + 2*(b*c*e*f* cosh(e*x + d)*log(F) + e^2*f - (b^2*c^2*f*cosh(e*x + d) + b^2*c^2*f)*log(F )^2)*sinh(e*x + d))*cosh((b*c*x + a*c)*log(F)) + (2*e^2*f*cosh(e*x + d) - (b^2*c^2*f*cosh(e*x + d)^2 + 2*b^2*c^2*f*cosh(e*x + d) + b^2*c^2*f)*log(F) ^2 - (b^2*c^2*f*log(F)^2 - b*c*e*f*log(F))*sinh(e*x + d)^2 + (b*c*e*f*cosh (e*x + d)^2 - b*c*e*f)*log(F) + 2*(b*c*e*f*cosh(e*x + d)*log(F) + e^2*f - (b^2*c^2*f*cosh(e*x + d) + b^2*c^2*f)*log(F)^2)*sinh(e*x + d))*sinh((b*c*x + a*c)*log(F)))/(b^3*c^3*cosh(e*x + d)*log(F)^3 - b*c*e^2*cosh(e*x + d)*l og(F) + (b^3*c^3*log(F)^3 - b*c*e^2*log(F))*sinh(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (94) = 188\).
Time = 0.65 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.83 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=\begin {cases} x \left (f \cosh {\left (d \right )} + f\right ) & \text {for}\: F = 1 \wedge b = 0 \wedge c = 0 \wedge e = 0 \\f x + \frac {f \sinh {\left (d + e x \right )}}{e} & \text {for}\: F = 1 \\F^{a c} \left (f x + \frac {f \sinh {\left (d + e x \right )}}{e}\right ) & \text {for}\: b = 0 \\f x + \frac {f \sinh {\left (d + e x \right )}}{e} & \text {for}\: c = 0 \\- \frac {F^{a c + b c x} f x \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} f x \cosh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} f \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2 b c \log {\left (F \right )}} + \frac {F^{a c + b c x} f}{b c \log {\left (F \right )}} & \text {for}\: e = - b c \log {\left (F \right )} \\- \frac {F^{a c + b c x} f x \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{2} + \frac {F^{a c + b c x} f x \cosh {\left (b c x \log {\left (F \right )} + d \right )}}{2} + \frac {F^{a c + b c x} f \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{b c \log {\left (F \right )}} - \frac {F^{a c + b c x} f \cosh {\left (b c x \log {\left (F \right )} + d \right )}}{2 b c \log {\left (F \right )}} + \frac {F^{a c + b c x} f}{b c \log {\left (F \right )}} & \text {for}\: e = b c \log {\left (F \right )} \\\frac {F^{a c + b c x} b^{2} c^{2} f \log {\left (F \right )}^{2} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} + \frac {F^{a c + b c x} b^{2} c^{2} f \log {\left (F \right )}^{2}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} - \frac {F^{a c + b c x} b c e f \log {\left (F \right )} \sinh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} - \frac {F^{a c + b c x} e^{2} f}{b^{3} c^{3} \log {\left (F \right )}^{3} - b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \]
Piecewise((x*(f*cosh(d) + f), Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(e, 0)), (f*x + f*sinh(d + e*x)/e, Eq(F, 1)), (F**(a*c)*(f*x + f*sinh(d + e*x)/e), Eq(b, 0)), (f*x + f*sinh(d + e*x)/e, Eq(c, 0)), (-F**(a*c + b*c*x)*f*x*sin h(b*c*x*log(F) - d)/2 + F**(a*c + b*c*x)*f*x*cosh(b*c*x*log(F) - d)/2 + F* *(a*c + b*c*x)*f*sinh(b*c*x*log(F) - d)/(2*b*c*log(F)) + F**(a*c + b*c*x)* f/(b*c*log(F)), Eq(e, -b*c*log(F))), (-F**(a*c + b*c*x)*f*x*sinh(b*c*x*log (F) + d)/2 + F**(a*c + b*c*x)*f*x*cosh(b*c*x*log(F) + d)/2 + F**(a*c + b*c *x)*f*sinh(b*c*x*log(F) + d)/(b*c*log(F)) - F**(a*c + b*c*x)*f*cosh(b*c*x* log(F) + d)/(2*b*c*log(F)) + F**(a*c + b*c*x)*f/(b*c*log(F)), Eq(e, b*c*lo g(F))), (F**(a*c + b*c*x)*b**2*c**2*f*log(F)**2*cosh(d + e*x)/(b**3*c**3*l og(F)**3 - b*c*e**2*log(F)) + F**(a*c + b*c*x)*b**2*c**2*f*log(F)**2/(b**3 *c**3*log(F)**3 - b*c*e**2*log(F)) - F**(a*c + b*c*x)*b*c*e*f*log(F)*sinh( d + e*x)/(b**3*c**3*log(F)**3 - b*c*e**2*log(F)) - F**(a*c + b*c*x)*e**2*f /(b**3*c**3*log(F)**3 - b*c*e**2*log(F)), True))
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.86 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=\frac {1}{2} \, f {\left (\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{b c \log \left (F\right ) + e} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{b c e^{d} \log \left (F\right ) - e e^{d}}\right )} + \frac {F^{b c x + a c} f}{b c \log \left (F\right )} \]
1/2*f*(F^(a*c)*e^(b*c*x*log(F) + e*x + d)/(b*c*log(F) + e) + F^(a*c)*e^(b* c*x*log(F) - e*x)/(b*c*e^d*log(F) - e*e^d)) + F^(b*c*x + a*c)*f/(b*c*log(F ))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 886, normalized size of antiderivative = 8.77 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=\text {Too large to display} \]
2*(2*b*c*f*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(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c) ^2) - (pi*b*c*sgn(F) - pi*b*c)*f*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*f*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)/(I*p i*b*c*sgn(F) - I*pi*b*c + 2*b*c*log(abs(F))) - I*f*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))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) + (2*(b*c*log(abs(F)) + e)*f*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)*f*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*p i*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 ) + I*(I*f*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*pi*b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F)) + 4*e ) - I*f*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*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F)) + 4*e) )*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + e)*x + d) + (2*(b*c*log(abs(F)) - e)*f*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/...
Time = 2.48 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33 \[ \int F^{c (a+b x)} (f+f \cosh (d+e x)) \, dx=-\frac {F^{b\,c\,x}\,F^{a\,c}\,f\,{\mathrm {e}}^{-d-e\,x}\,\left (b^2\,c^2\,{\ln \left (F\right )}^2-2\,e^2\,{\mathrm {e}}^{d+e\,x}+b\,c\,e\,\ln \left (F\right )+2\,b^2\,c^2\,{\mathrm {e}}^{d+e\,x}\,{\ln \left (F\right )}^2+b^2\,c^2\,{\mathrm {e}}^{2\,d+2\,e\,x}\,{\ln \left (F\right )}^2-b\,c\,e\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \left (F\right )\right )}{2\,b\,c\,\ln \left (F\right )\,\left (e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \]