Optimal. Leaf size=157 \[ -\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d (a+b \text {ArcSin}(c+d x))^2}-\frac {e}{2 b^2 d (a+b \text {ArcSin}(c+d x))}+\frac {e (c+d x)^2}{b^2 d (a+b \text {ArcSin}(c+d x))}+\frac {e \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 d}-\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^3 d} \]
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Rubi [A]
time = 0.23, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {4889, 12,
4729, 4807, 4731, 4491, 3384, 3380, 3383, 4737} \begin {gather*} \frac {e \sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^3 d}-\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c+d x))}{b}\right )}{b^3 d}+\frac {e (c+d x)^2}{b^2 d (a+b \text {ArcSin}(c+d x))}-\frac {e}{2 b^2 d (a+b \text {ArcSin}(c+d x))}-\frac {e \sqrt {1-(c+d x)^2} (c+d x)}{2 b d (a+b \text {ArcSin}(c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4729
Rule 4731
Rule 4737
Rule 4807
Rule 4889
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}-\frac {e \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {(2 e) \text {Subst}\left (\int \frac {x}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {(2 e) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {(2 e) \text {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 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 {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^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 )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {e \text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 d}-\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{b^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 107, normalized size = 0.68 \begin {gather*} -\frac {e \left (-4 \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+\frac {b (2 (a+b \text {ArcSin}(c+d x)) \cos (2 \text {ArcSin}(c+d x))+b \sin (2 \text {ArcSin}(c+d x)))}{(a+b \text {ArcSin}(c+d x))^2}+4 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )\right )\right )}{4 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 263, normalized size = 1.68
method | result | size |
derivativedivides | \(-\frac {e \left (4 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b^{2}-4 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a b -8 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a b +2 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{2}-4 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{2}+\sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+2 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b \right )}{4 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(263\) |
default | \(-\frac {e \left (4 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b^{2}-4 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a b -8 \arcsin \left (d x +c \right ) \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a b +2 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \sinIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{2}-4 \cosineIntegral \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{2}+\sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+2 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b \right )}{4 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}}\) | \(263\) |
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^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\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. 888 vs.
\(2 (151) = 302\).
time = 0.59, size = 888, normalized size = 5.66 \begin {gather*} \frac {2 \, b^{2} e \arcsin \left (d x + c\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} - \frac {2 \, b^{2} e \arcsin \left (d x + c\right )^{2} \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {4 \, a b e \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} - \frac {4 \, a b e \arcsin \left (d x + c\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {2 \, a^{2} e \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {b^{2} e \arcsin \left (d x + c\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} - \frac {2 \, a^{2} e \cos \left (\frac {a}{b}\right )^{2} \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} b^{2} e \arcsin \left (d x + c\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {2 \, a b e \arcsin \left (d x + c\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} - \frac {\sqrt {-{\left (d x + c\right )}^{2} + 1} {\left (d x + c\right )} b^{2} e}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} + \frac {{\left ({\left (d x + c\right )}^{2} - 1\right )} a b e}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {b^{2} e \arcsin \left (d x + c\right )}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\right )}} + \frac {a^{2} e \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (d x + c\right )\right )}{b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d} + \frac {a b e}{2 \, {\left (b^{5} d \arcsin \left (d x + c\right )^{2} + 2 \, a b^{4} d \arcsin \left (d x + c\right ) + a^{2} b^{3} d\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}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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