Integrand size = 34, antiderivative size = 113 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=-\frac {e^{-d-e x} f F^{c (a+b x)} \sqrt {g \cosh (d+e x)}}{2 \sqrt {f \cosh (d+e x)} (e-b c \log (F))}+\frac {e^{d+e x} f F^{c (a+b x)} \sqrt {g \cosh (d+e x)}}{2 \sqrt {f \cosh (d+e x)} (e+b c \log (F))} \] Output:
-1/2*exp(-e*x-d)*f*F^(c*(b*x+a))*(g*cosh(e*x+d))^(1/2)/(f*cosh(e*x+d))^(1/ 2)/(e-b*c*ln(F))+1/2*exp(e*x+d)*f*F^(c*(b*x+a))*(g*cosh(e*x+d))^(1/2)/(f*c osh(e*x+d))^(1/2)/(e+b*c*ln(F))
Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=\frac {f F^{c (a+b x)} \sqrt {g \cosh (d+e x)} (-b c \cosh (d+e x) \log (F)+e \sinh (d+e x))}{\sqrt {f \cosh (d+e x)} (e-b c \log (F)) (e+b c \log (F))} \] Input:
Integrate[F^(c*(a + b*x))*Sqrt[f*Cosh[d + e*x]]*Sqrt[g*Cosh[d + e*x]],x]
Output:
(f*F^(c*(a + b*x))*Sqrt[g*Cosh[d + e*x]]*(-(b*c*Cosh[d + e*x]*Log[F]) + e* Sinh[d + e*x]))/(Sqrt[f*Cosh[d + e*x]]*(e - b*c*Log[F])*(e + b*c*Log[F]))
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2031, 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.
\(\displaystyle \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx\) |
\(\Big \downarrow \) 2031 |
\(\displaystyle \frac {f \sqrt {g \cosh (d+e x)} \int F^{c (a+b x)} \cosh (d+e x)dx}{\sqrt {f \cosh (d+e x)}}\) |
\(\Big \downarrow \) 5998 |
\(\displaystyle \frac {f \sqrt {g \cosh (d+e x)} \left (\frac {e \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}\right )}{\sqrt {f \cosh (d+e x)}}\) |
Input:
Int[F^(c*(a + b*x))*Sqrt[f*Cosh[d + e*x]]*Sqrt[g*Cosh[d + e*x]],x]
Output:
(f*Sqrt[g*Cosh[d + e*x]]*(-((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)))/Sqrt[f*Cosh[d + e*x]]
Int[(Fx_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Simp[a^(m + 1/ 2)*b^(n - 1/2)*(Sqrt[b*v]/Sqrt[a*v]) Int[v^(m + n)*Fx, x], x] /; FreeQ[{a , b, m}, x] && !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]
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 = 1.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12
method | result | size |
risch | \(\frac {\sqrt {f \left (1+{\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-e x -d}}\, \sqrt {g \left (1+{\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-e x -d}}\, \left (\ln \left (F \right ) b c \,{\mathrm e}^{2 e x +2 d}+b c \ln \left (F \right )-{\mathrm e}^{2 e x +2 d} e +e \right ) F^{c \left (b x +a \right )}}{2 \left (1+{\mathrm e}^{2 e x +2 d}\right ) \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\) | \(126\) |
Input:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*cosh(e*x+d))^(1/2),x,method=_RE TURNVERBOSE)
Output:
1/2*(f*(1+exp(2*e*x+2*d))*exp(-e*x-d))^(1/2)/(1+exp(2*e*x+2*d))*(g*(1+exp( 2*e*x+2*d))*exp(-e*x-d))^(1/2)*(ln(F)*b*c*exp(2*e*x+2*d)+b*c*ln(F)-exp(2*e *x+2*d)*e+e)/(b*c*ln(F)-e)/(e+b*c*ln(F))*F^(c*(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (100) = 200\).
Time = 0.10 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.84 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=\frac {\sqrt {f \cosh \left (e x + d\right )} \sqrt {g \cosh \left (e x + d\right )} {\left ({\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} + b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) - e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )\right )}}{e^{2} \cosh \left (e x + d\right )^{2} - {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \left (F\right )^{2} - {\left (b^{2} c^{2} \log \left (F\right )^{2} - e^{2}\right )} \sinh \left (e x + d\right )^{2} + e^{2} - 2 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - e^{2} \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right )} \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*cosh(e*x+d))^(1/2),x, alg orithm="fricas")
Output:
sqrt(f*cosh(e*x + d))*sqrt(g*cosh(e*x + d))*((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*cos h(e*x + d)*log(F) - e*cosh(e*x + d))*sinh(e*x + d) - e)*cosh((b*c*x + a*c) *log(F)) + (e*cosh(e*x + d)^2 - (b*c*log(F) - e)*sinh(e*x + d)^2 - (b*c*co sh(e*x + d)^2 + b*c)*log(F) - 2*(b*c*cosh(e*x + d)*log(F) - e*cosh(e*x + d ))*sinh(e*x + d) - e)*sinh((b*c*x + a*c)*log(F)))/(e^2*cosh(e*x + d)^2 - ( b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 - (b^2*c^2*log(F)^2 - e^2)*sin h(e*x + d)^2 + e^2 - 2*(b^2*c^2*cosh(e*x + d)*log(F)^2 - e^2*cosh(e*x + d) )*sinh(e*x + d))
Result contains complex when optimal does not.
Time = 13.41 (sec) , antiderivative size = 644, normalized size of antiderivative = 5.70 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx =\text {Too large to display} \] Input:
integrate(F**(c*(b*x+a))*(f*cosh(e*x+d))**(1/2)*(g*cosh(e*x+d))**(1/2),x)
Output:
Piecewise((F**(a*c - e*x/log(F))*x*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e *x))*sinh(d + e*x)/(2*cosh(d + e*x)) + F**(a*c - e*x/log(F))*x*sqrt(f*cosh (d + e*x))*sqrt(g*cosh(d + e*x))/2 + F**(a*c - e*x/log(F))*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))*sinh(d + e*x)/(e*cosh(d + e*x)) + F**(a*c - e *x/log(F))*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))/(2*e), Eq(b, -e/(c* log(F)))), (-F**(a*c + e*x/log(F))*x*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))*sinh(d + e*x)/(2*cosh(d + e*x)) + F**(a*c + e*x/log(F))*x*sqrt(f*co sh(d + e*x))*sqrt(g*cosh(d + e*x))/2 + F**(a*c + e*x/log(F))*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))*sinh(d + e*x)/(e*cosh(d + e*x)) - F**(a*c + e*x/log(F))*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))/(2*e), Eq(b, e/(c *log(F)))), (F**(a*c + b*c*x)*sqrt(f*cosh(e*x + log(-I*exp(-e*x))))*sqrt(g *cosh(e*x + log(-I*exp(-e*x))))/(b*c*log(F)), Eq(d, log(-I*exp(-e*x)))), ( F**(a*c + b*c*x)*sqrt(f*cosh(e*x + log(I*exp(-e*x))))*sqrt(g*cosh(e*x + lo g(I*exp(-e*x))))/(b*c*log(F)), Eq(d, log(I*exp(-e*x)))), (F**(a*c + b*c*x) *b*c*sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))*log(F)*cosh(d + e*x)/(b** 2*c**2*log(F)**2*cosh(d + e*x) - e**2*cosh(d + e*x)) - F**(a*c + b*c*x)*e* sqrt(f*cosh(d + e*x))*sqrt(g*cosh(d + e*x))*sinh(d + e*x)/(b**2*c**2*log(F )**2*cosh(d + e*x) - e**2*cosh(d + e*x)), True))
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=\int { \sqrt {f \cosh \left (e x + d\right )} \sqrt {g \cosh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*cosh(e*x+d))^(1/2),x, alg orithm="maxima")
Output:
integrate(sqrt(f*cosh(e*x + d))*sqrt(g*cosh(e*x + d))*F^((b*x + a)*c), x)
\[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=\int { \sqrt {f \cosh \left (e x + d\right )} \sqrt {g \cosh \left (e x + d\right )} F^{{\left (b x + a\right )} c} \,d x } \] Input:
integrate(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*cosh(e*x+d))^(1/2),x, alg orithm="giac")
Output:
integrate(sqrt(f*cosh(e*x + d))*sqrt(g*cosh(e*x + d))*F^((b*x + a)*c), x)
Time = 2.80 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.51 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=-\frac {\sqrt {g\,\left (\frac {{\mathrm {e}}^{d+e\,x}}{2}+\frac {{\mathrm {e}}^{-d-e\,x}}{2}\right )}\,\left (\frac {F^{a\,c+b\,c\,x}\,\sqrt {f\,\left (\frac {{\mathrm {e}}^{d+e\,x}}{2}+\frac {{\mathrm {e}}^{-d-e\,x}}{2}\right )}\,\left (e+b\,c\,\ln \left (F\right )\right )}{e^2-b^2\,c^2\,{\ln \left (F\right )}^2}-\frac {F^{a\,c+b\,c\,x}\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\sqrt {f\,\left (\frac {{\mathrm {e}}^{d+e\,x}}{2}+\frac {{\mathrm {e}}^{-d-e\,x}}{2}\right )}\,\left (e-b\,c\,\ln \left (F\right )\right )}{e^2-b^2\,c^2\,{\ln \left (F\right )}^2}\right )}{{\mathrm {e}}^{2\,d+2\,e\,x}+1} \] Input:
int(F^(c*(a + b*x))*(f*cosh(d + e*x))^(1/2)*(g*cosh(d + e*x))^(1/2),x)
Output:
-((g*(exp(d + e*x)/2 + exp(- d - e*x)/2))^(1/2)*((F^(a*c + b*c*x)*(f*(exp( d + e*x)/2 + exp(- d - e*x)/2))^(1/2)*(e + b*c*log(F)))/(e^2 - b^2*c^2*log (F)^2) - (F^(a*c + b*c*x)*exp(2*d + 2*e*x)*(f*(exp(d + e*x)/2 + exp(- d - e*x)/2))^(1/2)*(e - b*c*log(F)))/(e^2 - b^2*c^2*log(F)^2)))/(exp(2*d + 2*e *x) + 1)
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.48 \[ \int F^{c (a+b x)} \sqrt {f \cosh (d+e x)} \sqrt {g \cosh (d+e x)} \, dx=\frac {\sqrt {g}\, f^{b c x +a c +\frac {1}{2}} \left (\cosh \left (e x +d \right ) \mathrm {log}\left (f \right ) b c -\sinh \left (e x +d \right ) e \right )}{\mathrm {log}\left (f \right )^{2} b^{2} c^{2}-e^{2}} \] Input:
int(F^(c*(b*x+a))*(f*cosh(e*x+d))^(1/2)*(g*cosh(e*x+d))^(1/2),x)
Output:
(sqrt(g)*f**((2*a*c + 2*b*c*x + 1)/2)*(cosh(d + e*x)*log(f)*b*c - sinh(d + e*x)*e))/(log(f)**2*b**2*c**2 - e**2)