Optimal. Leaf size=280 \[ -\frac{2 f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \sqrt{a^2-b^2}}+\frac{2 f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \sqrt{a^2-b^2}}-\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^2 \sqrt{a^2-b^2}}+\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.527671, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4422, 3323, 2264, 2190, 2279, 2391} \[ -\frac{2 f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^3 \sqrt{a^2-b^2}}+\frac{2 f^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^3 \sqrt{a^2-b^2}}-\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b d^2 \sqrt{a^2-b^2}}+\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{b d^2 \sqrt{a^2-b^2}}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 4422
Rule 3323
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{(2 f) \int \frac{e+f x}{a+b \sin (c+d x)} \, dx}{b d}\\ &=-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{(4 f) \int \frac{e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b d}\\ &=-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}-\frac{(4 i f) \int \frac{e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\sqrt{a^2-b^2} d}+\frac{(4 i f) \int \frac{e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{\sqrt{a^2-b^2} d}\\ &=-\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{\left (2 i f^2\right ) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^2}-\frac{\left (2 i f^2\right ) \int \log \left (1-\frac{2 i b e^{i (c+d x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{b \sqrt{a^2-b^2} d^2}\\ &=-\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a-2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{\left (2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 i b x}{2 a+2 \sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b \sqrt{a^2-b^2} d^3}\\ &=-\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}+\frac{2 i f (e+f x) \log \left (1-\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^2}-\frac{2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}+\frac{2 f^2 \text{Li}_2\left (\frac{i b e^{i (c+d x)}}{a+\sqrt{a^2-b^2}}\right )}{b \sqrt{a^2-b^2} d^3}-\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 3.08874, size = 311, normalized size = 1.11 \[ -\frac{(e+f x)^2}{b d (a+b \sin (c+d x))}+\frac{2 i f \left (-f \sqrt{a^2-b^2} \text{PolyLog}\left (2,\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )+f \sqrt{a^2-b^2} \text{PolyLog}\left (2,-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right )-i d \left (2 e \sqrt{b^2-a^2} \tan ^{-1}\left (\frac{i a+b e^{i (c+d x)}}{\sqrt{a^2-b^2}}\right )+f x \sqrt{a^2-b^2} \left (\log \left (1-\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}-i a}\right )-\log \left (1+\frac{b e^{i (c+d x)}}{\sqrt{b^2-a^2}+i a}\right )\right )\right )\right )}{b d^3 \sqrt{-\left (a^2-b^2\right )^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.892, size = 606, normalized size = 2.2 \begin{align*}{\frac{-2\,i \left ({f}^{2}{x}^{2}+2\,fex+{e}^{2} \right ){{\rm e}^{i \left ( dx+c \right ) }}}{bd \left ( b{{\rm e}^{2\,i \left ( dx+c \right ) }}-b+2\,ia{{\rm e}^{i \left ( dx+c \right ) }} \right ) }}+{\frac{4\,ife}{b{d}^{2}}\arctan \left ({\frac{2\,ib{{\rm e}^{i \left ( dx+c \right ) }}-2\,a}{2}{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}}+2\,{\frac{{f}^{2}x}{b{d}^{2}\sqrt{-{a}^{2}+{b}^{2}}}\ln \left ({\frac{ia+b{{\rm e}^{i \left ( dx+c \right ) }}-\sqrt{-{a}^{2}+{b}^{2}}}{ia-\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }+2\,{\frac{c{f}^{2}}{b{d}^{3}\sqrt{-{a}^{2}+{b}^{2}}}\ln \left ({\frac{ia+b{{\rm e}^{i \left ( dx+c \right ) }}-\sqrt{-{a}^{2}+{b}^{2}}}{ia-\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }-2\,{\frac{{f}^{2}x}{b{d}^{2}\sqrt{-{a}^{2}+{b}^{2}}}\ln \left ({\frac{ia+b{{\rm e}^{i \left ( dx+c \right ) }}+\sqrt{-{a}^{2}+{b}^{2}}}{ia+\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }-2\,{\frac{c{f}^{2}}{b{d}^{3}\sqrt{-{a}^{2}+{b}^{2}}}\ln \left ({\frac{ia+b{{\rm e}^{i \left ( dx+c \right ) }}+\sqrt{-{a}^{2}+{b}^{2}}}{ia+\sqrt{-{a}^{2}+{b}^{2}}}} \right ) }-{\frac{2\,i{f}^{2}}{b{d}^{3}}{\it dilog} \left ({ \left ( ia+b{{\rm e}^{i \left ( dx+c \right ) }}-\sqrt{-{a}^{2}+{b}^{2}} \right ) \left ( ia-\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}}+{\frac{2\,i{f}^{2}}{b{d}^{3}}{\it dilog} \left ({ \left ( ia+b{{\rm e}^{i \left ( dx+c \right ) }}+\sqrt{-{a}^{2}+{b}^{2}} \right ) \left ( ia+\sqrt{-{a}^{2}+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}}-{\frac{4\,i{f}^{2}c}{b{d}^{3}}\arctan \left ({\frac{2\,ib{{\rm e}^{i \left ( dx+c \right ) }}-2\,a}{2}{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.8509, size = 3263, normalized size = 11.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \cos \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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