Integrand size = 31, antiderivative size = 918 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 2.36 (sec) , antiderivative size = 918, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {5972, 5976, 37, 5969, 12, 186, 54, 98, 95, 214, 5993, 5992, 5893, 5980, 3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\frac {b f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {c x-1} \sqrt {c x+1}}+\frac {a f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {c x+1}}{\sqrt {c f-g} \sqrt {c x-1}}\right ) c^2}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {c x-1} \sqrt {c x+1}}-\frac {b f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {c x-1} \sqrt {c x+1}}-\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \sqrt {d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \sqrt {-\frac {1-c x}{c x+1}} \sqrt {c x+1} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {c x-1} (f+g x)}-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \sqrt {c x-1} \sqrt {c x+1} (f+g x)^2 c}-\frac {\left (f x c^2+g\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b \left (c^2 f^2-g^2\right ) \sqrt {c x-1} \sqrt {c x+1} (f+g x)^2 c} \]
[In]
[Out]
Rule 12
Rule 31
Rule 37
Rule 54
Rule 95
Rule 98
Rule 186
Rule 214
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3401
Rule 3405
Rule 5893
Rule 5969
Rule 5972
Rule 5976
Rule 5980
Rule 5992
Rule 5993
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (2 g+2 c^2 f x\right ) (a+b \text {arccosh}(c x))^2}{(f+g x)^3} \, dx}{2 b c \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\sqrt {d-c^2 d x^2} \int \frac {\left (g+c^2 f x\right )^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\sqrt {d-c^2 d x^2} \int \left (\frac {a \left (g+c^2 f x\right )^2}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {b \left (g+c^2 f x\right )^2 \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (g+c^2 f x\right )^2 \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\left (a \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^4 f^2}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (-c^2 f^2+g^2\right )^2}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {2 c^2 f \left (-c^2 f^2+g^2\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {c^4 f^2 \text {arccosh}(c x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (-c^2 f^2+g^2\right )^2 \text {arccosh}(c x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {2 c^2 f \left (-c^2 f^2+g^2\right ) \text {arccosh}(c x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x} (f+g x)}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\left (a c^4 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^4 f^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (a \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2} \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 a c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \int \frac {\text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}+\frac {\left (a c^2 f \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c \left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{(c f+g \cosh (x))^2} \, dx,x,\text {arccosh}(c x)\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 a c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c f-g-(c f+g) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {4 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 a c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c f-g-(c f+g) x^2} \, dx,x,\frac {\sqrt {1+c x}}{\sqrt {-1+c x}}\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{c f+g \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\sinh (x)}{c f+g \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{g \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\text {arccosh}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\text {arccosh}(c x)\right )}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^x x}{2 c f+2 e^x g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\text {arccosh}(c x)\right )}{g \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^2 f \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \sqrt {-\frac {1-c x}{1+c x}} \sqrt {1+c x} \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{g \sqrt {-1+c x} (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \left (c^2 f^2-g^2\right ) \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {\left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {-1+c x} \sqrt {1+c x} (f+g x)^2}-\frac {2 a c^2 f \sqrt {d-c^2 d x^2} \text {arctanh}\left (\frac {\sqrt {c f+g} \sqrt {1+c x}}{\sqrt {c f-g} \sqrt {-1+c x}}\right )}{\sqrt {c f-g} g^2 \sqrt {c f+g} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \text {arccosh}(c x) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 5.30 (sec) , antiderivative size = 1139, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\frac {-\frac {2 a g \sqrt {d-c^2 d x^2}}{f+g x}+2 a c \sqrt {d} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {2 a c^2 \sqrt {d} f \log (f+g x)}{\sqrt {-c^2 f^2+g^2}}-\frac {2 a c^2 \sqrt {d} 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 {-c^2 f^2+g^2}}+b c \sqrt {d-c^2 d x^2} \left (-\frac {2 g \text {arccosh}(c x)}{c f+c g x}+\frac {\text {arccosh}(c x)^2}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {2 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 )}{\sqrt {-c^2 f^2+g^2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )}{2 g^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1822\) vs. \(2(864)=1728\).
Time = 2.11 (sec) , antiderivative size = 1823, normalized size of antiderivative = 1.99
method | result | size |
default | \(\text {Expression too large to display}\) | \(1823\) |
parts | \(\text {Expression too large to display}\) | \(1823\) |
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{{\left (f+g\,x\right )}^2} \,d x \]
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