Optimal. Leaf size=222 \[ -\frac{2 a^5 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^2 d \left (a^2-b^2\right )^{5/2}}+\frac{4 a^3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^2 d \left (a^2-b^2\right )^{5/2}}-\frac{a^4 \cos (c+d x)}{b d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}-\frac{x}{b^2} \]
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Rubi [A] time = 0.364495, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2897, 2648, 2664, 12, 2660, 618, 204} \[ -\frac{2 a^5 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^2 d \left (a^2-b^2\right )^{5/2}}+\frac{4 a^3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^2 d \left (a^2-b^2\right )^{5/2}}-\frac{a^4 \cos (c+d x)}{b d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac{\cos (c+d x)}{2 d (a+b)^2 (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 d (a-b)^2 (\sin (c+d x)+1)}-\frac{x}{b^2} \]
Antiderivative was successfully verified.
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Rule 2897
Rule 2648
Rule 2664
Rule 12
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x) \tan ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \left (-\frac{1}{b^2}-\frac{1}{2 (a+b)^2 (-1+\sin (c+d x))}+\frac{1}{2 (a-b)^2 (1+\sin (c+d x))}+\frac{a^4}{b^2 \left (-a^2+b^2\right ) (a+b \sin (c+d x))^2}+\frac{2 \left (a^5-2 a^3 b^2\right )}{b^2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}\right ) \, dx\\ &=-\frac{x}{b^2}+\frac{\int \frac{1}{1+\sin (c+d x)} \, dx}{2 (a-b)^2}-\frac{\int \frac{1}{-1+\sin (c+d x)} \, dx}{2 (a+b)^2}+\frac{\left (2 a^3 \left (a^2-2 b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )^2}-\frac{a^4 \int \frac{1}{(a+b \sin (c+d x))^2} \, dx}{b^2 \left (a^2-b^2\right )}\\ &=-\frac{x}{b^2}+\frac{\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}-\frac{a^4 \cos (c+d x)}{b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac{a^4 \int \frac{a}{a+b \sin (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )^2}+\frac{\left (4 a^3 \left (a^2-2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^2 \left (a^2-b^2\right )^2 d}\\ &=-\frac{x}{b^2}+\frac{\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}-\frac{a^4 \cos (c+d x)}{b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac{a^5 \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )^2}-\frac{\left (8 a^3 \left (a^2-2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^2 \left (a^2-b^2\right )^2 d}\\ &=-\frac{x}{b^2}+\frac{4 a^3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2} d}+\frac{\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}-\frac{a^4 \cos (c+d x)}{b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac{\left (2 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^2 \left (a^2-b^2\right )^2 d}\\ &=-\frac{x}{b^2}+\frac{4 a^3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2} d}+\frac{\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}-\frac{a^4 \cos (c+d x)}{b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac{\left (4 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^2 \left (a^2-b^2\right )^2 d}\\ &=-\frac{x}{b^2}-\frac{2 a^5 \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2} d}+\frac{4 a^3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2} d}+\frac{\cos (c+d x)}{2 (a+b)^2 d (1-\sin (c+d x))}-\frac{\cos (c+d x)}{2 (a-b)^2 d (1+\sin (c+d x))}-\frac{a^4 \cos (c+d x)}{b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.03369, size = 236, normalized size = 1.06 \[ \frac{-\frac{-2 a^2 b^2 (c+d x)+a^4 (c+d x)+2 a b^3+b^4 (c+d x)}{\left (b^3-a^2 b\right )^2}+\frac{2 a^3 \left (a^2-4 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^2 \left (a^2-b^2\right )^{5/2}}-\frac{a^4 \cos (c+d x)}{b (a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{(a+b)^2 \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{\sin \left (\frac{1}{2} (c+d x)\right )}{(a-b)^2 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.131, size = 303, normalized size = 1.4 \begin{align*} -{\frac{1}{d \left ( a+b \right ) ^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{b}^{2}}}-2\,{\frac{{a}^{3}\tan \left ( 1/2\,dx+c/2 \right ) }{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}-2\,{\frac{{a}^{4}}{bd \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+2\,{\frac{{a}^{5}}{d{b}^{2} \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-8\,{\frac{{a}^{3}}{d \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{d \left ( a-b \right ) ^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}} \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 [A] time = 2.01788, size = 1577, normalized size = 7.1 \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 [A] time = 1.218, size = 356, normalized size = 1.6 \begin{align*} \frac{\frac{2 \,{\left (a^{5} - 4 \, a^{3} b^{2}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sqrt{a^{2} - b^{2}}} - \frac{2 \,{\left (2 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{4} - 2 \, a^{2} b^{2}\right )}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a\right )}} - \frac{d x + c}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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