Integrand size = 31, antiderivative size = 523 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {g \sqrt {-1+c x} \sqrt {-\frac {1-c x}{1+c x}} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {c^2 f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {c^2 f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b c^2 f \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c^2 f \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \]
-g*(c*x+1)^(3/2)*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*((c*x-1)/(c*x+1))^(1/2)/ (c^2*f^2-g^2)/(g*x+f)/(-c^2*d*x^2+d)^(1/2)+b*c*ln(g*x+f)*(c*x-1)^(1/2)*(c* x+1)^(1/2)/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)+c^2*f*(a+b*arccosh(c*x))*ln( 1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c*x-1)^( 1/2)*(c*x+1)^(1/2)/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-c^2*f*(a+b*arc cosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/ 2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+ b*c^2*f*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^ (1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/ 2)-b*c^2*f*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^ 2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^ (1/2)
Result contains complex when optimal does not.
Time = 4.25 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {a g \sqrt {d-c^2 d x^2}}{d \left (-c^2 f^2+g^2\right ) (f+g x)}-\frac {a c^2 f \log (f+g x)}{\sqrt {d} \left (-c^2 f^2+g^2\right )^{3/2}}-\frac {a c^2 f \log \left (d \left (g+c^2 f x\right )+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d} (c f-g) (c f+g) \sqrt {-c^2 f^2+g^2}}+\frac {b c \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (-\frac {g \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)}{(c f-g) (c f+g) (c f+c g x)}+\frac {\log \left (1+\frac {g x}{f}\right )}{c^2 f^2-g^2}+\frac {c f \left (2 \text {arccosh}(c x) \arctan \left (\frac {(c f+g) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 i \arccos \left (-\frac {c f}{g}\right ) \arctan \left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )+2 \left (\arctan \left (\frac {(c f+g) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\arctan \left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{-\frac {1}{2} \text {arccosh}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )-2 \left (\arctan \left (\frac {(c f+g) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\arctan \left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} \text {arccosh}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\arccos \left (-\frac {c f}{g}\right )+2 \arctan \left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (c f-g+i \sqrt {-c^2 f^2+g^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c f}{g}\right )-2 \arctan \left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-c f+g+i \sqrt {-c^2 f^2+g^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c f-i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )\right )\right )}{\left (-c^2 f^2+g^2\right )^{3/2}}\right )}{\sqrt {d-c^2 d x^2}} \]
-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*L og[f + g*x])/(Sqrt[d]*(-(c^2*f^2) + g^2)^(3/2)) - (a*c^2*f*Log[d*(g + c^2* f*x) + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/(Sqrt[d]*(c*f - g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) + (b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*( 1 + c*x)*(-((g*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*f - g)*(c*f + g)*(c*f + c*g*x))) + Log[1 + (g*x)/f]/(c^2*f^2 - g^2) + (c*f*(2* ArcCosh[c*x]*ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2 ]] - (2*I)*ArcCos[-((c*f)/g)]*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/S qrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[A rcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCos h[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E ^(ArcCosh[c*x]/2)*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] - 2*(A rcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[(( -(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^(ArcCos h[c*x]/2)*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - ( ArcCos[-((c*f)/g)] + 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c ^2*f^2) + g^2]])*Log[((c*f + g)*(c*f - g + I*Sqrt[-(c^2*f^2) + g^2])*(-1 + Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCos h[c*x]/2]))] - (ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c *x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(-(c*f) + g + I*Sqrt[-(...
Time = 1.59 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.69, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {6387, 6395, 3042, 3805, 26, 3042, 26, 3147, 16, 3801, 2694, 27, 2620, 2715, 2838}
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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2} (f+g x)^2} \, dx\) |
\(\Big \downarrow \) 6387 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1} (f+g x)^2}dx}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 6395 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(c f+c g x)^2}d\text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\left (c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )\right )^2}d\text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3805 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {i b g \int -\frac {i \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b g \int \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b g \int -\frac {i \cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c f-g \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {i b g \int \frac {\cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c f-g \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b \int \frac {1}{c f+c g x}d(c g x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3801 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 c e^{\text {arccosh}(c x)} f+e^{2 \text {arccosh}(c x)} g+g}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2694 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c f+e^{\text {arccosh}(c x)} g-\sqrt {c^2 f^2-g^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c f+e^{\text {arccosh}(c x)} g+\sqrt {c^2 f^2-g^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c f+e^{\text {arccosh}(c x)} g-\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c f+e^{\text {arccosh}(c x)} g+\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )de^{\text {arccosh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )de^{\text {arccosh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\) |
(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((g*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) *(a + b*ArcCosh[c*x]))/((c^2*f^2 - g^2)*(c*f + c*g*x))) + (b*Log[c*f + c*g *x])/(c^2*f^2 - g^2) + (2*c*f*((g*(((a + b*ArcCosh[c*x])*Log[1 + (E^ArcCos h[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (b*PolyLog[2, -((E^ArcCosh[c*x ]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g))/(2*Sqrt[c^2*f^2 - g^2]) - (g*(((a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) /g + (b*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g)) /(2*Sqrt[c^2*f^2 - g^2])))/(c^2*f^2 - g^2)))/Sqrt[d - c^2*d*x^2]
3.1.70.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f *x]))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[b*d*(m/(f*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])) Int[(f + g*x)^m* (-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[1/( c^(m + 1)*Sqrt[(-d1)*d2]) Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ [e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1977\) vs. \(2(519)=1038\).
Time = 1.91 (sec) , antiderivative size = 1978, normalized size of antiderivative = 3.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(1978\) |
parts | \(\text {Expression too large to display}\) | \(1978\) |
a/d/(c^2*f^2-g^2)/(x+f/g)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2 -g^2)/g^2)^(1/2)-a/g*c^2*f/(c^2*f^2-g^2)/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln(( -2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)* (-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g)) -b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f) *(c*x-1)*(c*x+1)*x*c^2*f+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2- 1)/(c^2*f^2-g^2)/(g*x+f)*x^3*c^4*f-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d /(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*c*g+b*(-d *(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x^2*c ^2*g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g* x+f)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*f-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x) /d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x*c^2*f-b*(-d*(c^2*x^2-1))^(1/2)*arcc osh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*g-b*c^2*(-d*(c^2*x^2-1))^(1/2 )*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g ^4*x^2+2*c^2*f^2*g^2-g^4)/d*(c^2*f^2-g^2)^(1/2)*arccosh(c*x)*ln((-(c*x+(c* x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^( 1/2)))*f+b*c^2*(-d*(c^2*x^2-1))^(1/2)*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^6*f^4 *x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)/d*(c^2*f^2-g ^2)^(1/2)*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^ 2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*f+2*b*c^3*(-d*(c^2*x^2-1))^(1/...
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2 *d*f*g*x^3 - 2*d*f*g*x - d*f^2 + (c^2*d*f^2 - d*g^2)*x^2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \]
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]