Optimal. Leaf size=365 \[ \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {PolyLog}\left (2,-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.47, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5972, 5980,
3401, 2296, 2221, 2317, 2438} \begin {gather*} \frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2} \sqrt {c^2 f^2-g^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3401
Rule 5972
Rule 5980
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (f+g x)} \, dx}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}}\\ &=\frac {\left (2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (2 g \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\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,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\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,\cosh ^{-1}(c x)\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\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^{\cosh ^{-1}(c x)}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {-1+c x} \sqrt {1+c x}\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^{\cosh ^{-1}(c x)}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.31, size = 932, normalized size = 2.55 \begin {gather*} \frac {\frac {a \log (f+g x)}{\sqrt {d}}-\frac {a \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}}-\frac {b \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (2 \cosh ^{-1}(c x) \text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 i \text {ArcCos}\left (-\frac {c f}{g}\right ) \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 \left (\text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{-\frac {1}{2} \cosh ^{-1}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 \left (\text {ArcTan}\left (\frac {(c f+g) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} \cosh ^{-1}(c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )+2 \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(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} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {c f}{g}\right )-2 \text {ArcTan}\left (\frac {(-c f+g) \tanh \left (\frac {1}{2} \cosh ^{-1}(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} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text {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} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {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} \cosh ^{-1}(c x)\right )\right )}{g \left (c f+g+i \sqrt {-c^2 f^2+g^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt {d-c^2 d x^2}}}{\sqrt {-c^2 f^2+g^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 4.18, size = 754, normalized size = 2.07
method | result | size |
default | \(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (\frac {-\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} f^{2}-g^{2}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \dilog \left (\frac {\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )}{d \left (c^{4} f^{2} x^{2}-c^{2} g^{2} x^{2}-c^{2} f^{2}+g^{2}\right )}\) | \(754\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{\left (f+g\,x\right )\,\sqrt {d-c^2\,d\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________