Optimal. Leaf size=776 \[ -\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt{a+b} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{4 \sqrt{-a} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d}-\frac{\sqrt [4]{b} \left (-\sqrt{b} \sqrt{a+b}+a+b\right ) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 a d \sqrt [4]{a+b} \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\sqrt [4]{b} (a+b)^{3/4} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt [4]{a+b} \left (\sqrt{b}-\sqrt{a+b}\right )^2 \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} \Pi \left (\frac{\left (\sqrt{b}+\sqrt{a+b}\right )^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{8 a \sqrt [4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}} \]
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Rubi [A] time = 1.04339, antiderivative size = 776, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3215, 1223, 1714, 1195, 1708, 1103, 1706} \[ -\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d \sqrt{a+b} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}\right )}{4 \sqrt{-a} d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}{2 a d}-\frac{\sqrt [4]{b} \left (-\sqrt{b} \sqrt{a+b}+a+b\right ) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 a d \sqrt [4]{a+b} \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}+\frac{\sqrt [4]{b} (a+b)^{3/4} \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{2 a d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}}-\frac{\sqrt [4]{a+b} \left (\sqrt{b}-\sqrt{a+b}\right )^2 \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right ) \sqrt{\frac{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}{(a+b) \left (\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}+1\right )^2}} \Pi \left (\frac{\left (\sqrt{b}+\sqrt{a+b}\right )^2}{4 \sqrt{b} \sqrt{a+b}};2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt [4]{a+b}}\right )|\frac{1}{2} \left (\frac{\sqrt{b}}{\sqrt{a+b}}+1\right )\right )}{8 a \sqrt [4]{b} d \sqrt{a+b \cos ^4(c+d x)-2 b \cos ^2(c+d x)+b}} \]
Antiderivative was successfully verified.
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Rule 3215
Rule 1223
Rule 1714
Rule 1195
Rule 1708
Rule 1103
Rule 1706
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x)}{\sqrt{a+b \sin ^4(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2 \sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\operatorname{Subst}\left (\int \frac{-a+b-2 b x^2+b x^4}{\left (1-x^2\right ) \sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d}\\ &=-\frac{\sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}-\frac{\operatorname{Subst}\left (\int \frac{-b (-a+b)+b^{3/2} \sqrt{a+b}+\left (2 b^2-b \left (b+\sqrt{b} \sqrt{a+b}\right )\right ) x^2}{\left (1-x^2\right ) \sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a b d}+\frac{\left (\sqrt{b} \sqrt{a+b}\right ) \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 )}{2 a d}\\ &=-\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt{a+b} d \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right )}-\frac{\sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\sqrt [4]{b} (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 )}{2 a d \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}+\frac{\left (\sqrt{a+b} \left (\sqrt{b}-\sqrt{a+b}\right )\right ) \operatorname{Subst}\left (\int \frac{1+\frac{\sqrt{b} x^2}{\sqrt{a+b}}}{\left (1-x^2\right ) \sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{2 a d}-\frac{\left (\sqrt{b} \left (a+b-\sqrt{b} \sqrt{a+b}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b-2 b x^2+b x^4}} \, dx,x,\cos (c+d x)\right )}{a \sqrt{a+b} d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{-a} \cos (c+d x)}{\sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\right )}{4 \sqrt{-a} d}-\frac{\sqrt{b} \cos (c+d x) \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}{2 a \sqrt{a+b} d \left (1+\frac{\sqrt{b} \cos ^2(c+d x)}{\sqrt{a+b}}\right )}-\frac{\sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)} \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\sqrt [4]{b} (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 )}{2 a d \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}-\frac{\sqrt [4]{b} \left (a+b-\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 a \sqrt [4]{a+b} 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 )^2 \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}} \Pi \left (\frac{\left (\sqrt{b}+\sqrt{a+b}\right )^2}{4 \sqrt{b} \sqrt{a+b}};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 )}{8 a \sqrt [4]{b} d \sqrt{a+b-2 b \cos ^2(c+d x)+b \cos ^4(c+d x)}}\\ \end{align*}
Mathematica [C] time = 32.7293, size = 119171, normalized size = 153.57 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.727, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( dx+c \right ) \right ) ^{3}{\frac{1}{\sqrt{a+b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}}}\, dx \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{\csc \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{\csc \left (d x + c\right )^{3}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (c + d x \right )}}{\sqrt{a + b \sin ^{4}{\left (c + d x \right )}}}\, dx \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{\csc \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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