Optimal. Leaf size=191 \[ -\frac {\sqrt {1-(c+d x)^2}}{4 b d (a+b \text {ArcSin}(c+d x))^4}+\frac {c+d x}{12 b^2 d (a+b \text {ArcSin}(c+d x))^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d (a+b \text {ArcSin}(c+d x))^2}-\frac {c+d x}{24 b^4 d (a+b \text {ArcSin}(c+d x))}+\frac {\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{24 b^5 d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{24 b^5 d} \]
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Rubi [A]
time = 0.20, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4887, 4717,
4807, 4719, 3384, 3380, 3383} \begin {gather*} \frac {\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{24 b^5 d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c+d x)}{b}\right )}{24 b^5 d}-\frac {c+d x}{24 b^4 d (a+b \text {ArcSin}(c+d x))}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d (a+b \text {ArcSin}(c+d x))^2}+\frac {c+d x}{12 b^2 d (a+b \text {ArcSin}(c+d x))^3}-\frac {\sqrt {1-(c+d x)^2}}{4 b d (a+b \text {ArcSin}(c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 4717
Rule 4719
Rule 4807
Rule 4887
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^{-1}(c+d x)\right )^5} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^5} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}-\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{4 b d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{12 b^2 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{24 b^3 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{24 b^4 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{24 b^5 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{4 b d \left (a+b \sin ^{-1}(c+d x)\right )^4}+\frac {c+d x}{12 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {\sqrt {1-(c+d x)^2}}{24 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {c+d x}{24 b^4 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{24 b^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 156, normalized size = 0.82 \begin {gather*} \frac {-\frac {6 b^4 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^4}+\frac {2 b^3 (c+d x)}{(a+b \text {ArcSin}(c+d x))^3}+\frac {b^2 \sqrt {1-(c+d x)^2}}{(a+b \text {ArcSin}(c+d x))^2}-\frac {b (c+d x)}{a+b \text {ArcSin}(c+d x)}+\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c+d x)\right )}{24 b^5 d} \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. \(386\) vs.
\(2(178)=356\).
time = 0.16, size = 387, normalized size = 2.03
method | result | size |
derivativedivides | \(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{4 \left (a +b \arcsin \left (d x +c \right )\right )^{4} b}+\frac {\arcsin \left (d x +c \right )^{3} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{3}+\arcsin \left (d x +c \right )^{3} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{3}+3 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,b^{2}+3 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,b^{2}-\arcsin \left (d x +c \right )^{2} b^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right ) \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2} b +3 \arcsin \left (d x +c \right ) \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2} b +\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{3}-2 \arcsin \left (d x +c \right ) a \,b^{2} \left (d x +c \right )+\sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{3}+\cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{3}+\sqrt {1-\left (d x +c \right )^{2}}\, a \,b^{2}-a^{2} b \left (d x +c \right )+2 \left (d x +c \right ) b^{3}}{24 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{5}}}{d}\) | \(387\) |
default | \(\frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{4 \left (a +b \arcsin \left (d x +c \right )\right )^{4} b}+\frac {\arcsin \left (d x +c \right )^{3} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{3}+\arcsin \left (d x +c \right )^{3} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{3}+3 \arcsin \left (d x +c \right )^{2} \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,b^{2}+3 \arcsin \left (d x +c \right )^{2} \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,b^{2}-\arcsin \left (d x +c \right )^{2} b^{3} \left (d x +c \right )+3 \arcsin \left (d x +c \right ) \sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2} b +3 \arcsin \left (d x +c \right ) \cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2} b +\sqrt {1-\left (d x +c \right )^{2}}\, \arcsin \left (d x +c \right ) b^{3}-2 \arcsin \left (d x +c \right ) a \,b^{2} \left (d x +c \right )+\sinIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{3}+\cosineIntegral \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{3}+\sqrt {1-\left (d x +c \right )^{2}}\, a \,b^{2}-a^{2} b \left (d x +c \right )+2 \left (d x +c \right ) b^{3}}{24 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b^{5}}}{d}\) | \(387\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{5}}\, dx \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. 1915 vs.
\(2 (175) = 350\).
time = 0.43, size = 1915, normalized size = 10.03 \begin {gather*} \text {Too large to display} \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 {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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