Optimal. Leaf size=104 \[ -\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{b d (a+b \text {ArcSin}(c+d x))}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^2 d}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^2 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4889, 12, 4727,
3384, 3380, 3383} \begin {gather*} \frac {e \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^2 d}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^2 d}-\frac {e \sqrt {1-(c+d x)^2} (c+d x)}{b d (a+b \text {ArcSin}(c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4727
Rule 4889
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e \text {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\left (e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}+\frac {\left (e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{b d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{b^2 d}+\frac {e \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{b^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 99, normalized size = 0.95 \begin {gather*} \frac {e \left (-\frac {b (c+d x) \sqrt {1-c^2-2 c d x-d^2 x^2}}{a+b \text {ArcSin}(c+d x)}+\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )+\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 151, normalized size = 1.45
method | result | size |
derivativedivides | \(\frac {e \left (2 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +2 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\sin \left (2 \arcsin \left (d x +c \right )\right ) b \right )}{2 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(151\) |
default | \(\frac {e \left (2 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +2 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\sin \left (2 \arcsin \left (d x +c \right )\right ) b \right )}{2 d \left (a +b \arcsin \left (d x +c \right )\right ) b^{2}}\) | \(151\) |
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*} e \left (\int \frac {c}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{2} + 2 a b \operatorname {asin}{\left (c + d x \right )} + b^{2} \operatorname {asin}^{2}{\left (c + d x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 341 vs.
\(2 (102) = 204\).
time = 0.48, size = 341, normalized size = 3.28 \begin {gather*} \frac {2 \, b e \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {2 \, b e \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {2 \, a e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} + \frac {2 \, a e \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} - \frac {b e \arcsin \left (d x + c\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} - \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b e}{b^{3} d \arcsin \left (d x + c\right ) + a b^{2} d} - \frac {a e \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{3} d \arcsin \left (d x + c\right ) + a b^{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 \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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