Optimal. Leaf size=514 \[ \frac {8 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a \csc (c+d x)+a}}+\frac {24 a \cot (c+d x)}{7 d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]
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Rubi [A] time = 0.30, antiderivative size = 514, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3806, 50, 63, 303, 218, 1877} \[ \frac {8 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {\csc ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\csc (c+d x)}+1}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\csc (c+d x)}-\sqrt {3}+1}{-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right )^2}} (a-a \csc (c+d x)) \sqrt {a \csc (c+d x)+a}}-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a \csc (c+d x)+a}}+\frac {24 a \cot (c+d x)}{7 d \left (-\sqrt [3]{\csc (c+d x)}+\sqrt {3}+1\right ) \sqrt {a \csc (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 218
Rule 303
Rule 1877
Rule 3806
Rubi steps
\begin {align*} \int \csc ^{\frac {5}{3}}(c+d x) \sqrt {a+a \csc (c+d x)} \, dx &=\frac {\left (a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {x^{2/3}}{\sqrt {a-a x}} \, dx,x,\csc (c+d x)\right )}{d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a+a \csc (c+d x)}}+\frac {\left (4 a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt {a-a x}} \, dx,x,\csc (c+d x)\right )}{7 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a+a \csc (c+d x)}}+\frac {\left (12 a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a+a \csc (c+d x)}}-\frac {\left (12 a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {3}-x}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}-\frac {\left (12 \sqrt {2 \left (2-\sqrt {3}\right )} a^2 \cot (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x^3}} \, dx,x,\sqrt [3]{\csc (c+d x)}\right )}{7 d \sqrt {a-a \csc (c+d x)} \sqrt {a+a \csc (c+d x)}}\\ &=\frac {24 a \cot (c+d x)}{7 d \left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {a+a \csc (c+d x)}}-\frac {6 a \cos (c+d x) \csc ^{\frac {5}{3}}(c+d x)}{7 d \sqrt {a+a \csc (c+d x)}}-\frac {12 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}+\frac {8 \sqrt {2} 3^{3/4} a^2 \cot (c+d x) \left (1-\sqrt [3]{\csc (c+d x)}\right ) \sqrt {\frac {1+\sqrt [3]{\csc (c+d x)}+\csc ^{\frac {2}{3}}(c+d x)}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}-\sqrt [3]{\csc (c+d x)}}{1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{7 d \sqrt {\frac {1-\sqrt [3]{\csc (c+d x)}}{\left (1+\sqrt {3}-\sqrt [3]{\csc (c+d x)}\right )^2}} (a-a \csc (c+d x)) \sqrt {a+a \csc (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 1.18, size = 120, normalized size = 0.23 \[ -\frac {2 \sqrt {a (\csc (c+d x)+1)} \left (3 (\csc (c+d x)+4)-8 \sqrt [3]{\csc (c+d x)} \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {3}{2};1-\csc (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{7 d \sqrt [3]{\csc (c+d x)} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {5}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {5}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.41, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{\frac {5}{3}}\left (d x +c \right )\right ) \sqrt {a +a \csc \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \csc \left (d x + c\right ) + a} \csc \left (d x + c\right )^{\frac {5}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+\frac {a}{\sin \left (c+d\,x\right )}}\,{\left (\frac {1}{\sin \left (c+d\,x\right )}\right )}^{5/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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