Optimal. Leaf size=255 \[ \frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.55, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5836, 5822, 5683, 5676, 30, 5718} \[ \frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {c x-1} \sqrt {c x+1}}-\frac {g (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 5676
Rule 5683
Rule 5718
Rule 5822
Rule 5836
Rubi steps
\begin {align*} \int (f+g x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int \sqrt {-1+c x} \sqrt {1+c x} (f+g x) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )+g x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b g \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {g (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2}-\frac {f \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.05, size = 251, normalized size = 0.98 \[ \frac {12 a \sqrt {d-c^2 d x^2} \left (3 c^2 f x+2 g \left (c^2 x^2-1\right )\right )-36 a c \sqrt {d} f \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-\frac {9 b c f \sqrt {d-c^2 d x^2} \left (2 \cosh ^{-1}(c x)^2+\cosh \left (2 \cosh ^{-1}(c x)\right )-2 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {2 b g \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}}{72 c^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-c^{2} d x^{2} + d} {\left (a g x + a f + {\left (b g x + b f\right )} \operatorname {arcosh}\left (c x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.73, size = 461, normalized size = 1.81 \[ -\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right )}{3 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g \,\mathrm {arccosh}\left (c x \right ) x^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{2 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \,\mathrm {arccosh}\left (c x \right ) x}{2 \left (c x +1\right ) \left (c x -1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f}{8 \sqrt {c x +1}\, \sqrt {c x -1}\, c}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g c \,x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, g x}{3 \sqrt {c x +1}\, c \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f c \,x^{2}}{4 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, f \mathrm {arccosh}\left (c x \right )^{2}}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\sqrt {-c^{2} d x^{2} + d} x + \frac {\sqrt {d} \arcsin \left (c x\right )}{c}\right )} a f - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a g}{3 \, c^{2} d} + \int \sqrt {-c^{2} d x^{2} + d} b g x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {-c^{2} d x^{2} + d} b f \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________