Optimal. Leaf size=164 \[ -\frac {\sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{6 b^4 d}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c+d x)}{b}\right )}{6 b^4 d}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 0.27, antiderivative size = 160, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4803, 4621, 4719, 4723, 3303, 3299, 3302} \[ -\frac {\sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{6 b^4 d}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{6 b^4 d}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4621
Rule 4719
Rule 4723
Rule 4803
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^{-1}(c+d x)\right )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}-\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \left (a+b \sin ^{-1}(x)\right )} \, dx,x,c+d x\right )}{6 b^3 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{6 b^3 d}-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac {\sqrt {1-(c+d x)^2}}{3 b d \left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {c+d x}{6 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac {\sqrt {1-(c+d x)^2}}{6 b^3 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {\text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right ) \sin \left (\frac {a}{b}\right )}{6 b^4 d}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )}{6 b^4 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 134, normalized size = 0.82 \[ \frac {-\frac {2 b^3 \sqrt {1-(c+d x)^2}}{\left (a+b \sin ^{-1}(c+d x)\right )^3}+\frac {b^2 (c+d x)}{\left (a+b \sin ^{-1}(c+d x)\right )^2}-\sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )+\frac {b \sqrt {1-(c+d x)^2}}{a+b \sin ^{-1}(c+d x)}}{6 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 1112, normalized size = 6.78 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 270, normalized size = 1.65 \[ \frac {-\frac {\sqrt {1-\left (d x +c \right )^{2}}}{3 \left (a +b \arcsin \left (d x +c \right )\right )^{3} b}+\frac {\arcsin \left (d x +c \right )^{2} \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b^{2}-\arcsin \left (d x +c \right )^{2} \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b^{2}+2 \arcsin \left (d x +c \right ) \Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a b -2 \arcsin \left (d x +c \right ) \Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a b +\arcsin \left (d x +c \right ) \sqrt {1-\left (d x +c \right )^{2}}\, b^{2}+\Si \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a^{2}-\Ci \left (\arcsin \left (d x +c \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a^{2}+\sqrt {1-\left (d x +c \right )^{2}}\, a b +\left (d x +c \right ) b^{2}}{6 \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^4} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {asin}{\left (c + d x \right )}\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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