Integrand size = 29, antiderivative size = 127 \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {f \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4862, 4848, 4738, 4768, 8} \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {f \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}} \]
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Rule 8
Rule 4738
Rule 4768
Rule 4848
Rule 4862
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}} \\ & = \frac {\sqrt {1-c^2 x^2} \int \left (\frac {f (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}+\frac {g x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d-c^2 d x^2}} \\ & = \frac {\left (f \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arccos (c x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \int \frac {x (a+b \arccos (c x))}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d-c^2 d x^2}} \\ & = -\frac {g \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {f \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}}-\frac {\left (b g \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{c \sqrt {d-c^2 d x^2}} \\ & = -\frac {b g x \sqrt {1-c^2 x^2}}{c \sqrt {d-c^2 d x^2}}-\frac {g \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{c^2 \sqrt {d-c^2 d x^2}}-\frac {f \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.35 \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-2 \sqrt {d} g \left (a-a c^2 x^2+b c x \sqrt {1-c^2 x^2}\right )+2 b \sqrt {d} g \left (-1+c^2 x^2\right ) \arccos (c x)-b c \sqrt {d} f \sqrt {1-c^2 x^2} \arccos (c x)^2-2 a c f \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{2 c^2 \sqrt {d} \sqrt {d-c^2 d x^2}} \]
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Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arccos \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(247\) |
parts | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f}{2 c d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) g \left (\arccos \left (c x \right )-i\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(247\) |
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\[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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Exception generated. \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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none
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b f \arccos \left (c x\right ) \arcsin \left (c x\right )}{c \sqrt {d}} + \frac {b f \arcsin \left (c x\right )^{2}}{2 \, c \sqrt {d}} - \frac {b g x}{c \sqrt {d}} + \frac {a f \arcsin \left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {-c^{2} d x^{2} + d} b g \arccos \left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a g}{c^{2} d} \]
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\[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \arccos \left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x) (a+b \arccos (c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]
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