Optimal. Leaf size=157 \[ \frac {e \sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}-\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e \sqrt {1-(c+d x)^2} (c+d x)}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2} \]
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Rubi [A] time = 0.33, 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 = {4805, 12, 4633, 4719, 4635, 4406, 3303, 3299, 3302, 4641} \[ \frac {e \sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{b^3 d}-\frac {e \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c+d x)\right )}{b^3 d}+\frac {e (c+d x)^2}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac {e \sqrt {1-(c+d x)^2} (c+d x)}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 4633
Rule 4635
Rule 4641
Rule 4719
Rule 4805
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \sin ^{-1}(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e x}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.62, size = 107, normalized size = 0.68 \[ -\frac {e \left (-4 \sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+4 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\sin ^{-1}(c+d x)\right )\right )+\frac {b \left (2 \cos \left (2 \sin ^{-1}(c+d x)\right ) \left (a+b \sin ^{-1}(c+d x)\right )+b \sin \left (2 \sin ^{-1}(c+d x)\right )\right )}{\left (a+b \sin ^{-1}(c+d x)\right )^2}\right )}{4 b^3 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d e x + c e}{b^{3} \arcsin \left (d x + c\right )^{3} + 3 \, a b^{2} \arcsin \left (d x + c\right )^{2} + 3 \, a^{2} b \arcsin \left (d x + c\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.58, size = 902, normalized size = 5.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 263, normalized size = 1.68 \[ -\frac {e \left (4 \arcsin \left (d x +c \right )^{2} \Si \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b^{2}-4 \arcsin \left (d x +c \right )^{2} \Ci \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b^{2}+8 \arcsin \left (d x +c \right ) \Si \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a b -8 \arcsin \left (d x +c \right ) \Ci \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a b +2 \arcsin \left (d x +c \right ) \cos \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+4 \Si \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a^{2}-4 \Ci \left (2 \arcsin \left (d x +c \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a^{2}+\sin \left (2 \arcsin \left (d x +c \right )\right ) b^{2}+2 \cos \left (2 \arcsin \left (d x +c \right )\right ) a b \right )}{4 d \left (a +b \arcsin \left (d x +c \right )\right )^{2} b^{3}} \]
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 \[ \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^3} \,d x \]
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 \[ e \left (\int \frac {c}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} + 3 a^{2} b \operatorname {asin}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asin}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asin}^{3}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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