Optimal. Leaf size=431 \[ -\frac{\sqrt [4]{a+b} \left (\sqrt{b}-\sqrt{a+b}\right ) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{(a+b)^{3/4} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{\sqrt{b} d \sqrt{a+b} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )} \]
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Rubi [A] time = 0.302493, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3215, 1197, 1103, 1195} \[ -\frac{\sqrt [4]{a+b} \left (\sqrt{b}-\sqrt{a+b}\right ) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{(a+b)^{3/4} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{b^{3/4} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{\sqrt{b} d \sqrt{a+b} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )} \]
Antiderivative was successfully verified.
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Rule 3215
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\sin ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{\sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\sqrt{a+b} \operatorname{Subst}\left (\int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a+b}}}{\sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{\sqrt{b} d}-\frac{\left (1-\frac{\sqrt{a+b}}{\sqrt{b}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{\cos (c+d x) \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{\sqrt{b} \sqrt{a+b} d \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right )}-\frac{(a+b)^{3/4} \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right ) \sqrt{\frac{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (1+\frac{\sqrt{b}}{\sqrt{a+b}}\right )\right )}{b^{3/4} d \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac{\sqrt [4]{a+b} \left (\sqrt{b}-\sqrt{a+b}\right ) \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right ) \sqrt{\frac{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}{(a+b) \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (1+\frac{\sqrt{b}}{\sqrt{a+b}}\right )\right )}{2 b^{3/4} d \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\\ \end{align*}
Mathematica [C] time = 31.8742, size = 89374, normalized size = 207.36 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.525, size = 398, normalized size = 0.9 \begin{align*} -{\frac{1}{d}\sqrt{1-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{a+b} \left ( i\sqrt{a}\sqrt{b}+b \right ) }}\sqrt{1+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{a+b} \left ( i\sqrt{a}\sqrt{b}-b \right ) }}{\it EllipticF} \left ( \cos \left ( dx+c \right ) \sqrt{{\frac{1}{a+b} \left ( i\sqrt{a}\sqrt{b}+b \right ) }},\sqrt{-1-2\,{\frac{i\sqrt{a}\sqrt{b}-b}{a+b}}} \right ){\frac{1}{\sqrt{{\frac{1}{a+b} \left ( i\sqrt{a}\sqrt{b}+b \right ) }}}}{\frac{1}{\sqrt{a+b-2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}+b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}}}-2\,{\frac{a+b}{d\sqrt{a+b-2\,b \left ( \cos \left ( dx+c \right ) \right ) ^{2}+b \left ( \cos \left ( dx+c \right ) \right ) ^{4}} \left ( -2\,b+2\,i\sqrt{a}\sqrt{b} \right ) }\sqrt{1-{\frac{ \left ( i\sqrt{a}\sqrt{b}+b \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{a+b}}}\sqrt{1+{\frac{ \left ( i\sqrt{a}\sqrt{b}-b \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{a+b}}} \left ({\it EllipticF} \left ( \cos \left ( dx+c \right ) \sqrt{{\frac{i\sqrt{a}\sqrt{b}+b}{a+b}}},\sqrt{-1-2\,{\frac{i\sqrt{a}\sqrt{b}-b}{a+b}}} \right ) -{\it EllipticE} \left ( \cos \left ( dx+c \right ) \sqrt{{\frac{i\sqrt{a}\sqrt{b}+b}{a+b}}},\sqrt{-1-2\,{\frac{i\sqrt{a}\sqrt{b}-b}{a+b}}} \right ) \right ){\frac{1}{\sqrt{{\frac{i\sqrt{a}\sqrt{b}+b}{a+b}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (d x + c\right )^{3}}{\sqrt{b \sin \left (d x + c\right )^{4} + a}}\,{d x} \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{{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{\sqrt{b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}, x\right ) \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(-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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