Optimal. Leaf size=248 \[ -\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac{9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}-\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac{9 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2} \]
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Rubi [A] time = 0.577872, antiderivative size = 306, normalized size of antiderivative = 1.23, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {4805, 12, 4633, 4719, 4635, 4406, 3303, 3299, 3302, 4623} \[ -\frac{9 e^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{9 e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{e^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4633
Rule 4719
Rule 4635
Rule 4406
Rule 3303
Rule 3299
Rule 3302
Rule 4623
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^2 x^2}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+b \sin ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}+\frac{e^2 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}-\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{1-x^2} \left (a+b \sin ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 (a+b x)}-\frac{\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{2 b^2 d}+\frac{\left (e^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}+\frac{\left (e^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{\left (9 e^2\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{\left (9 e^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{\left (9 e^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}-\frac{\left (9 e^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}+\frac{\left (9 e^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac{e^2 (c+d x)^2 \sqrt{1-(c+d x)^2}}{2 b d \left (a+b \sin ^{-1}(c+d x)\right )^2}-\frac{e^2 (c+d x)}{b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}+\frac{3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \sin ^{-1}(c+d x)\right )}-\frac{9 e^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Ci}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}-\frac{9 e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{9 e^2 \sin \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c+d x)\right )}{8 b^3 d}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c+d x)}{b}\right )}{b^3 d}\\ \end{align*}
Mathematica [A] time = 0.799369, size = 219, normalized size = 0.88 \[ \frac{e^2 \left (-\frac{4 b^2 \sqrt{1-(c+d x)^2} (c+d x)^2}{\left (a+b \sin ^{-1}(c+d x)\right )^2}+8 \left (\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )+9 \left (-\cos \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+\cos \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )-\sin \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )+\sin \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c+d x)\right )\right )\right )+\frac{4 b \left (3 (c+d x)^3-2 (c+d x)\right )}{a+b \sin ^{-1}(c+d x)}\right )}{8 b^3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 474, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{2} \left (\int \frac{c^{2}}{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^{2} x^{2}}{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{2 c 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.99738, size = 2183, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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