Optimal. Leaf size=133 \[ \frac {(c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{d}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d} \]
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Rubi [A]
time = 0.18, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4887, 4715,
4809, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {(c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4715
Rule 4809
Rule 4887
Rubi steps
\begin {align*} \int \sqrt {a+b \sin ^{-1}(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {b \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {b \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\left (b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}+\frac {\left (b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{d}\\ &=\frac {(c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{d}-\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{d}+\frac {\sqrt {b} \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.06, size = 129, normalized size = 0.97 \begin {gather*} \frac {b e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {3}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{2 d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 203, normalized size = 1.53
method | result | size |
default | \(-\frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +2 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) \arcsin \left (d x +c \right ) b +2 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a}{2 d \sqrt {a +b \arcsin \left (d x +c \right )}}\) | \(203\) |
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 \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, 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 = 0.69, size = 563, normalized size = 4.23 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\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 )} d} + \frac {i \, \sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{4 \, {\left (\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} + \frac {\sqrt {2} \sqrt {\pi } a b \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\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 )} d} - \frac {i \, \sqrt {2} \sqrt {\pi } b^{2} \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{4 \, {\left (-\frac {i \, b^{2}}{\sqrt {{\left | b \right |}}} + b \sqrt {{\left | b \right |}}\right )} d} - \frac {\sqrt {\pi } a \operatorname {erf}\left (-\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac {i \, a}{b}\right )}}{d {\left (\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {\sqrt {\pi } a \operatorname {erf}\left (\frac {i \, \sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac {i \, a}{b}\right )}}{d {\left (-\frac {i \, \sqrt {2} b}{\sqrt {{\left | b \right |}}} + \sqrt {2} \sqrt {{\left | b \right |}}\right )}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} e^{\left (-i \, \arcsin \left (d x + c\right )\right )}}{2 \, d} \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 \sqrt {a+b\,\mathrm {asin}\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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