Optimal. Leaf size=159 \[ -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcCos}(c x)}}{2 c}+x (a+b \text {ArcCos}(c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \]
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Rubi [A]
time = 0.18, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4716, 4768,
4720, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcCos}(c x)}}{2 c}+x (a+b \text {ArcCos}(c x))^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4716
Rule 4720
Rule 4768
Rubi steps
\begin {align*} \int \left (a+b \cos ^{-1}(c x)\right )^{3/2} \, dx &=x \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {1}{2} (3 b c) \int \frac {x \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac {1}{4} \left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \cos ^{-1}(c x)}} \, dx\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{4 c}\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{4 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cos ^{-1}(c x)\right )}{4 c}\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{2 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{2 c}\\ &=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.67, size = 295, normalized size = 1.86 \begin {gather*} \frac {-2 a e^{-\frac {i a}{b}} \sqrt {a+b \text {ArcCos}(c x)} \left (-\frac {\text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcCos}(c x))}{b}\right )}{\sqrt {-\frac {i (a+b \text {ArcCos}(c x))}{b}}}-\frac {e^{\frac {2 i a}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcCos}(c x))}{b}\right )}{\sqrt {\frac {i (a+b \text {ArcCos}(c x))}{b}}}\right )+b \left (2 \sqrt {a+b \text {ArcCos}(c x)} \left (-3 \sqrt {1-c^2 x^2}+2 c x \text {ArcCos}(c x)\right )+\sqrt {\frac {1}{b}} \sqrt {2 \pi } S\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )-\sqrt {\frac {1}{b}} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {1}{b}} \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcCos}(c x)}\right ) \left (-2 a \cos \left (\frac {a}{b}\right )+3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(123)=246\).
time = 0.26, size = 278, normalized size = 1.75
method | result | size |
default | \(\frac {-3 \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2}-3 \sqrt {-\frac {1}{b}}\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {\pi }\, b^{2}+4 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +6 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}+6 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b}{4 c \sqrt {a +b \arccos \left (c x \right )}}\) | \(278\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.16, size = 993, normalized size = 6.25 \begin {gather*} -\frac {i \, \sqrt {2} \sqrt {\pi } a^{2} b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} c} + \frac {\sqrt {2} \sqrt {\pi } a b^{3} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} c} + \frac {i \, \sqrt {2} \sqrt {\pi } a^{2} b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} c} + \frac {\sqrt {2} \sqrt {\pi } a b^{3} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{3}}{\sqrt {{\left | b \right |}}} + b^{2} \sqrt {{\left | b \right |}}\right )} c} - \frac {\sqrt {2} \sqrt {\pi } a b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{2 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} - \frac {3 i \, \sqrt {2} \sqrt {\pi } b^{3} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{8 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} - \frac {\sqrt {2} \sqrt {\pi } a b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{2 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {3 i \, \sqrt {2} \sqrt {\pi } b^{3} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{8 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} c} + \frac {i \, \sqrt {\pi } a^{2} b \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{{\left (\frac {i \, \sqrt {2} b^{2}}{\sqrt {{\left | b \right |}}} + \sqrt {2} b \sqrt {{\left | b \right |}}\right )} c} - \frac {i \, \sqrt {\pi } a^{2} b \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arccos \left (c x\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arccos \left (c x\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{{\left (-\frac {i \, \sqrt {2} b^{2}}{\sqrt {{\left | b \right |}}} + \sqrt {2} b \sqrt {{\left | b \right |}}\right )} c} + \frac {\sqrt {b \arccos \left (c x\right ) + a} b \arccos \left (c x\right ) e^{\left (i \, \arccos \left (c x\right )\right )}}{2 \, c} + \frac {\sqrt {b \arccos \left (c x\right ) + a} b \arccos \left (c x\right ) e^{\left (-i \, \arccos \left (c x\right )\right )}}{2 \, c} + \frac {\sqrt {b \arccos \left (c x\right ) + a} a e^{\left (i \, \arccos \left (c x\right )\right )}}{2 \, c} + \frac {3 i \, \sqrt {b \arccos \left (c x\right ) + a} b e^{\left (i \, \arccos \left (c x\right )\right )}}{4 \, c} + \frac {\sqrt {b \arccos \left (c x\right ) + a} a e^{\left (-i \, \arccos \left (c x\right )\right )}}{2 \, c} - \frac {3 i \, \sqrt {b \arccos \left (c x\right ) + a} b e^{\left (-i \, \arccos \left (c x\right )\right )}}{4 \, c} \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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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