Optimal. Leaf size=105 \[ \frac {e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d}-\frac {e \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} d} \]
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Rubi [A]
time = 0.13, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {4889, 12,
4731, 4491, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {\sqrt {\pi } e \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d}-\frac {\sqrt {\pi } e \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{2 \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4731
Rule 4889
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\sqrt {a+b \sin ^{-1}(c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {\left (e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}-\frac {\left (e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 d}\\ &=\frac {\left (e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d}-\frac {\left (e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{b d}\\ &=\frac {e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{2 \sqrt {b} d}-\frac {e \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{2 \sqrt {b} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.04, size = 134, normalized size = 1.28 \begin {gather*} -\frac {e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{4 \sqrt {2} d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 99, normalized size = 0.94
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, e \left (\cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )\right )}{4 d}\) | \(99\) |
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*} e \left (\int \frac {c}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx + \int \frac {d x}{\sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}}\, dx\right ) \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 = 0.78, size = 142, normalized size = 1.35 \begin {gather*} \frac {i \, \sqrt {\pi } e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, d {\left (\sqrt {b} - \frac {i \, b^{\frac {3}{2}}}{{\left | b \right |}}\right )}} - \frac {i \, \sqrt {\pi } e \operatorname {erf}\left (-\frac {\sqrt {b \arcsin \left (d x + c\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arcsin \left (d x + c\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, \sqrt {b} d {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \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 \frac {c\,e+d\,e\,x}{\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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