Optimal. Leaf size=93 \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\csc ^2(a+b x) \sqrt {d \cos (a+b x)}}{2 b d}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}} \]
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Rubi [A] time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2565, 290, 329, 212, 206, 203} \[ -\frac {3 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\csc ^2(a+b x) \sqrt {d \cos (a+b x)}}{2 b d}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 290
Rule 329
Rule 2565
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d}\\ &=-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b d}\\ &=-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b}\\ &=-\frac {3 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b \sqrt {d}}-\frac {\sqrt {d \cos (a+b x)} \csc ^2(a+b x)}{2 b d}\\ \end {align*}
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Mathematica [C] time = 0.22, size = 69, normalized size = 0.74 \[ \frac {d \left (-\cot ^2(a+b x)\right )^{3/4} \left (\sqrt [4]{-\cot ^2(a+b x)}-\, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\csc ^2(a+b x)\right )\right )}{2 b (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 334, normalized size = 3.59 \[ \left [\frac {6 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {-d} \log \left (\frac {d \cos \left (b x + a\right )^{2} + 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}}, -\frac {6 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) - 3 \, {\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt {d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) - 8 \, \sqrt {d \cos \left (b x + a\right )}}{16 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 181, normalized size = 1.95 \[ \frac {\frac {6 \, \arctan \left (-\frac {\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \frac {3 \, \log \left ({\left | -\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} \right |}\right )}{\sqrt {-d}} + \frac {2 \, \sqrt {-d}}{{\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} - d} - \frac {\sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{d}}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 297, normalized size = 3.19 \[ -\frac {3 \ln \left (\frac {2 \sqrt {d}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}+4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1}\right )}{8 \sqrt {d}\, b}-\frac {\sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}}{16 b d \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}-\frac {3 \ln \left (\frac {2 \sqrt {d}\, \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}-4 d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 d}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\right )}{8 \sqrt {d}\, b}+\frac {3 \ln \left (\frac {-2 d +2 \sqrt {-d}\, \sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{\cos \left (\frac {b x}{2}+\frac {a}{2}\right )}\right )}{4 b \sqrt {-d}}-\frac {\sqrt {2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d -d}}{8 b d \cos \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}+\frac {\sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}}{16 b d \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 103, normalized size = 1.11 \[ \frac {\frac {4 \, \sqrt {d \cos \left (b x + a\right )} d^{2}}{d^{2} \cos \left (b x + a\right )^{2} - d^{2}} - 6 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right ) + 3 \, \sqrt {d} \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right )}{8 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^3\,\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\left (a + b x \right )}}{\sqrt {d \cos {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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