Integrand size = 18, antiderivative size = 93 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {F^{c (a+b x)}}{2 b c \log (F)}-\frac {e^{-2 d-2 e x} F^{c (a+b x)}}{4 (2 e-b c \log (F))}+\frac {e^{2 d+2 e x} F^{c (a+b x)}}{4 (2 e+b c \log (F))} \] Output:
1/2*F^(c*(b*x+a))/b/c/ln(F)-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 = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {F^{c (a+b x)} \left (-4 e^2+b^2 c^2 \log ^2(F)+b^2 c^2 \cosh (2 (d+e x)) \log ^2(F)-2 b c e \log (F) \sinh (2 (d+e x))\right )}{-8 b c e^2 \log (F)+2 b^3 c^3 \log ^3(F)} \] Input:
Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]
Output:
(F^(c*(a + b*x))*(-4*e^2 + b^2*c^2*Log[F]^2 + b^2*c^2*Cosh[2*(d + e*x)]*Lo g[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)
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6000, 2624}
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.
\(\displaystyle \int \cosh ^2(d+e x) F^{c (a+b x)} \, dx\) |
\(\Big \downarrow \) 6000 |
\(\displaystyle \frac {2 e^2 \int F^{c (a+b x)}dx}{4 e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle -\frac {b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}\) |
Input:
Int[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]
Output:
(2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^( c*(a + b*x))*Cosh[d + e*x]^2*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)
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Int[Cosh[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symb ol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Cosh[d + e*x]^n/(e^2*n^2 - b^2*c ^2*Log[F]^2)), x] + (Simp[e*n*F^(c*(a + b*x))*Sinh[d + e*x]*(Cosh[d + e*x]^ (n - 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] + Simp[n*(n - 1)*(e^2/(e^2*n^2 - b^2*c^2*Log[F]^2)) Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x], 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]
Time = 0.99 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97
method | result | size |
parallelrisch | \(-\frac {2 F^{c \left (b x +a \right )} \left (-\frac {c^{2} b^{2} \ln \left (F \right )^{2} \cosh \left (2 e x +2 d \right )}{2}-\frac {b^{2} c^{2} \ln \left (F \right )^{2}}{2}+\ln \left (F \right ) b c e \sinh \left (2 e x +2 d \right )+2 e^{2}\right )}{2 c^{3} b^{3} \ln \left (F \right )^{3}-8 e^{2} b c \ln \left (F \right )}\) | \(90\) |
risch | \(\frac {\left (\ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{4 e x +4 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 \ln \left (F \right ) b c e \,{\mathrm e}^{4 e x +4 d}+b^{2} c^{2} \ln \left (F \right )^{2}+2 \ln \left (F \right ) b c e -8 e^{2} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 \ln \left (F \right ) b c \left (b c \ln \left (F \right )-2 e \right ) \left (2 e +b c \ln \left (F \right )\right )}\) | \(143\) |
orering | \(\frac {\left (3 b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right ) F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{2}}{\left (b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right ) \ln \left (F \right ) b c}-\frac {3 \left (F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right )^{2}+2 F^{c \left (b x +a \right )} \cosh \left (e x +d \right ) e \sinh \left (e x +d \right )\right )}{b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}}+\frac {F^{c \left (b x +a \right )} b^{2} c^{2} \ln \left (F \right )^{2} \cosh \left (e x +d \right )^{2}+4 F^{c \left (b x +a \right )} b c \ln \left (F \right ) \cosh \left (e x +d \right ) e \sinh \left (e x +d \right )+2 F^{c \left (b x +a \right )} e^{2} \sinh \left (e x +d \right )^{2}+2 F^{c \left (b x +a \right )} \cosh \left (e x +d \right )^{2} e^{2}}{\left (b^{2} c^{2} \ln \left (F \right )^{2}-4 e^{2}\right ) \ln \left (F \right ) b c}\) | \(266\) |
Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-2*F^(c*(b*x+a))*(-1/2*c^2*b^2*ln(F)^2*cosh(2*e*x+2*d)-1/2*b^2*c^2*ln(F)^2 +ln(F)*b*c*e*sinh(2*e*x+2*d)+2*e^2)/(2*c^3*b^3*ln(F)^3-8*e^2*b*c*ln(F))
Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (84) = 168\).
Time = 0.09 (sec) , antiderivative size = 699, normalized size of antiderivative = 7.52 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx =\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="fricas")
Output:
1/4*(((b^2*c^2*log(F)^2 - 2*b*c*e*log(F))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d)*log(F)^2 - 2*b*c*e*cosh(e*x + d)*log(F) )*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e*x + d)^2*log(F) - (3*b^2*c^2*cosh(e *x + d)^2 + b^2*c^2)*log(F)^2 + 4*e^2)*sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*cosh(e*x + d)^3*log(F) + 4*e^2*cosh(e *x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^2*c^2*cosh(e*x + d))*log(F)^2)*sinh (e*x + d))*cosh((b*c*x + a*c)*log(F)) + ((b^2*c^2*log(F)^2 - 2*b*c*e*log(F ))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d)*log( F)^2 - 2*b*c*e*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e* x + d)^2*log(F) - (3*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 + 4*e^2)* sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*co sh(e*x + d)^3*log(F) + 4*e^2*cosh(e*x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^ 2*c^2*cosh(e*x + d))*log(F)^2)*sinh(e*x + d))*sinh((b*c*x + a*c)*log(F)))/ (b^3*c^3*cosh(e*x + d)^2*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)^2*log(F) + (b^ 3*c^3*log(F)^3 - 4*b*c*e^2*log(F))*sinh(e*x + d)^2 + 2*(b^3*c^3*cosh(e*x + d)*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)*log(F))*sinh(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (78) = 156\).
Time = 1.25 (sec) , antiderivative size = 707, normalized size of antiderivative = 7.60 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx =\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*cosh(e*x+d)**2,x)
Output:
Piecewise((x*cosh(d)**2, Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(e, 0)), (-x*s inh(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)), (F**(a*c)*(-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)**2/2 + s inh(d + e*x)*cosh(d + e*x)/(2*e)), Eq(b, 0)), (-x*sinh(d + e*x)**2/2 + x*c osh(d + e*x)**2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(c, 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*l og(F)/2 - d)**2/4 - F**(a*c + b*c*x)*sinh(b*c*x*log(F)/2 - d)**2/(b*c*log( F)) + 3*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*l og(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) ) + F**(a*c + b*c*x)*cosh(b*c*x*log(F)/2 + d)**2/(b*c*log(F)), Eq(e, b*c*l og(F)/2)), (F**(a*c + b*c*x)*b**2*c**2*log(F)**2*cosh(d + e*x)**2/(b**3*c* *3*log(F)**3 - 4*b*c*e**2*log(F)) - 2*F**(a*c + 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 + 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 + b*c*x)*e**2*cosh(d + e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e **2*log(F)), True))
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{4 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{4 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} + \frac {F^{b c x + a c}}{2 \, b c \log \left (F\right )} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,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)) + 1/2*F^(b*c *x + a*c)/(b*c*log(F))
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 889, normalized size of antiderivative = 9.56 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\text {Too large to display} \] Input:
integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="giac")
Output:
(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*p i*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)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*p i*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*pi*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*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))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(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(a bs(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*pi*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*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)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a...
Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=-\frac {2\,F^{a\,c+b\,c\,x}\,e^2-F^{a\,c+b\,c\,x}\,b^2\,c^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^2+2\,F^{a\,c+b\,c\,x}\,b\,c\,e\,\mathrm {cosh}\left (d+e\,x\right )\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )}{b^3\,c^3\,{\ln \left (F\right )}^3-4\,b\,c\,e^2\,\ln \left (F\right )} \] Input:
int(F^(c*(a + b*x))*cosh(d + e*x)^2,x)
Output:
-(2*F^(a*c + b*c*x)*e^2 - F^(a*c + b*c*x)*b^2*c^2*cosh(d + e*x)^2*log(F)^2 + 2*F^(a*c + b*c*x)*b*c*e*cosh(d + e*x)*sinh(d + e*x)*log(F))/(b^3*c^3*lo g(F)^3 - 4*b*c*e^2*log(F))
\[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=f^{a c} \left (\int f^{b c x} \cosh \left (e x +d \right )^{2}d x \right ) \] Input:
int(F^(c*(b*x+a))*cosh(e*x+d)^2,x)
Output:
f**(a*c)*int(f**(b*c*x)*cosh(d + e*x)**2,x)