Optimal. Leaf size=238 \[ -\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))}{3 c^2}-\frac {f \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{4 b c \sqrt {1-c^2 x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 238, 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 = {4862, 4848,
4742, 4738, 30, 4768} \begin {gather*} \frac {1}{2} f x \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))-\frac {f \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))^2}{4 b c \sqrt {1-c^2 x^2}}-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \text {ArcCos}(c x))}{3 c^2}+\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}-\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 4738
Rule 4742
Rule 4768
Rule 4848
Rule 4862
Rubi steps
\begin {align*} \int (f+g x) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx &=\frac {\sqrt {d-c^2 d x^2} \int (f+g x) \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {d-c^2 d x^2} \int \left (f \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )+g x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (g \sqrt {d-c^2 d x^2}\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}+\frac {\left (f \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}+\frac {\left (b c f \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1-c^2 x^2}}-\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^2 x^2}}\\ &=-\frac {b g x \sqrt {d-c^2 d x^2}}{3 c \sqrt {1-c^2 x^2}}+\frac {b c f x^2 \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}+\frac {b c g x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} f x \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )-\frac {g \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{3 c^2}-\frac {f \sqrt {d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{4 b c \sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 219, normalized size = 0.92 \begin {gather*} \frac {12 a \sqrt {d-c^2 d x^2} \left (3 c^2 f x+2 g \left (-1+c^2 x^2\right )\right )-36 a c \sqrt {d} f \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+\frac {2 b g \sqrt {d-c^2 d x^2} \left (-9 c x-12 \left (1-c^2 x^2\right )^{3/2} \text {ArcCos}(c x)+\cos (3 \text {ArcCos}(c x))\right )}{\sqrt {1-c^2 x^2}}+\frac {9 b c f \sqrt {d-c^2 d x^2} \left (-2 \text {ArcCos}(c x)^2+\cos (2 \text {ArcCos}(c x))+2 \text {ArcCos}(c x) \sin (2 \text {ArcCos}(c x))\right )}{\sqrt {1-c^2 x^2}}}{72 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.10, size = 628, normalized size = 2.64
method | result | size |
default | \(-\frac {a g \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+\frac {a f x \sqrt {-c^{2} d \,x^{2}+d}}{2}+\frac {a f d \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right )^{2} f}{4 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+4 c^{4} x^{4}-3 i \sqrt {-c^{2} x^{2}+1}\, x c -5 c^{2} x^{2}+1\right ) g \left (i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}-i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, x c +c^{2} x^{2}-1\right ) g \left (\arccos \left (c x \right )+i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, x c -1\right ) g \left (\arccos \left (c x \right )-i\right )}{8 c^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+i \sqrt {-c^{2} x^{2}+1}-2 c x \right ) f \left (-i+2 \arccos \left (c x \right )\right )}{16 c \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}-4 i \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}+3 i \sqrt {-c^{2} x^{2}+1}\, x c +1\right ) g \left (-i+3 \arccos \left (c x \right )\right )}{72 c^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(628\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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