Optimal. Leaf size=110 \[ -\frac{\tan ^3(c+d x)}{4 d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac{3 \tan (c+d x)}{8 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} d (a+b)^{5/2}} \]
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Rubi [A] time = 0.0956382, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3187, 288, 205} \[ -\frac{\tan ^3(c+d x)}{4 d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2}-\frac{3 \tan (c+d x)}{8 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} d (a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (a+(a+b) x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\tan ^3(c+d x)}{4 (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{\left (a+(a+b) x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 (a+b) d}\\ &=-\frac{\tan ^3(c+d x)}{4 (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}-\frac{3 \tan (c+d x)}{8 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{8 (a+b)^2 d}\\ &=\frac{3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{8 \sqrt{a} (a+b)^{5/2} d}-\frac{\tan ^3(c+d x)}{4 (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}-\frac{3 \tan (c+d x)}{8 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.24913, size = 97, normalized size = 0.88 \[ \frac{\frac{3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{5/2}}+\frac{\sin (2 (c+d x)) ((2 a+5 b) \cos (2 (c+d x))-8 a-5 b)}{(a+b)^2 (2 a-b \cos (2 (c+d x))+b)^2}}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 136, normalized size = 1.2 \begin{align*} -{\frac{5\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( a \left ( \tan \left ( dx+c \right ) \right ) ^{2}+ \left ( \tan \left ( dx+c \right ) \right ) ^{2}b+a \right ) ^{2} \left ( a+b \right ) }}-{\frac{3\,a\tan \left ( dx+c \right ) }{8\,d \left ( a \left ( \tan \left ( dx+c \right ) \right ) ^{2}+ \left ( \tan \left ( dx+c \right ) \right ) ^{2}b+a \right ) ^{2} \left ({a}^{2}+2\,ab+{b}^{2} \right ) }}+{\frac{3}{8\,d \left ({a}^{2}+2\,ab+{b}^{2} \right ) }\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \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 = 1.97294, size = 1567, normalized size = 14.25 \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 = 1.17635, size = 205, normalized size = 1.86 \begin{align*} \frac{\frac{3 \,{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt{a^{2} + a b}}\right )\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a^{2} + a b}} - \frac{5 \, a \tan \left (d x + c\right )^{3} + 5 \, b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )}{{\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}{\left (a^{2} + 2 \, a b + b^{2}\right )}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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