Optimal. Leaf size=362 \[ \frac{2 d e \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac{3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac{2 d e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.545986, antiderivative size = 354, normalized size of antiderivative = 0.98, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4745, 4621, 4723, 3303, 3299, 3302, 4631} \[ \frac{2 d e \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}+\frac{e^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}-\frac{3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{2 d e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}-\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{d^2 \sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 4745
Rule 4621
Rule 4723
Rule 3303
Rule 3299
Rule 3302
Rule 4631
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac{d^2}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac{2 d e x}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac{e^2 x^2}{\left (a+b \sin ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac{1}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac{x}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac{x^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ &=-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{\left (c d^2\right ) \int \frac{x}{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}+\frac{(2 d e) \operatorname{Subst}\left (\int \frac{\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}+\frac{e^2 \operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 (a+b x)}+\frac{3 \sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac{\left (3 e^2\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac{\left (2 d e \cos \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 x)\right )}{b c^2}+\frac{\left (2 d e \sin \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 x)\right )}{b c^2}\\ &=-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 d e \cos \left (\frac{2 a}{b}\right ) \text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}+\frac{2 d e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}-\frac{\left (d^2 \cos \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 x)\right )}{b c}-\frac{\left (e^2 \cos \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 x)\right )}{4 b c^3}+\frac{\left (3 e^2 \cos \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 x)\right )}{4 b c^3}+\frac{\left (d^2 \sin \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 x)\right )}{b c}+\frac{\left (e^2 \sin \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 x)\right )}{4 b c^3}-\frac{\left (3 e^2 \sin \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 x)\right )}{4 b c^3}\\ &=-\frac{d^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 d e x \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac{e^2 x^2 \sqrt{1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac{2 d e \cos \left (\frac{2 a}{b}\right ) \text{Ci}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}+\frac{d^2 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{b^2 c}+\frac{e^2 \text{Ci}\left (\frac{a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac{a}{b}\right )}{4 b^2 c^3}-\frac{3 e^2 \text{Ci}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac{3 a}{b}\right )}{4 b^2 c^3}-\frac{d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac{e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac{2 d e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \sin ^{-1}(c x)\right )}{b^2 c^2}+\frac{3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (\frac{3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end{align*}
Mathematica [A] time = 1.84988, size = 290, normalized size = 0.8 \[ -\frac{-\sin \left (\frac{a}{b}\right ) \left (4 c^2 d^2+e^2\right ) \text{CosIntegral}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+4 c^2 d^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )+\frac{4 b c^2 d^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{8 b c^2 d e x \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac{4 b c^2 e^2 x^2 \sqrt{1-c^2 x^2}}{a+b \sin ^{-1}(c x)}-8 c d e \cos \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+3 e^2 \sin \left (\frac{3 a}{b}\right ) \text{CosIntegral}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )-8 c d e \sin \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )+e^2 \cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\sin ^{-1}(c x)\right )-3 e^2 \cos \left (\frac{3 a}{b}\right ) \text{Si}\left (3 \left (\frac{a}{b}+\sin ^{-1}(c x)\right )\right )}{4 b^2 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 526, normalized size = 1.5 \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{e^{2} x^{2} + 2 \, d e x + d^{2}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, 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*} \int \frac{\left (d + e x\right )^{2}}{\left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.53935, size = 1713, normalized size = 4.73 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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