Optimal. Leaf size=137 \[ \frac {9 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2565, 290, 325, 329, 298, 203, 206} \[ \frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}+\frac {9 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 290
Rule 298
Rule 325
Rule 329
Rule 2565
Rubi steps
\begin {align*} \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^{7/2} \left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{x^{7/2} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d}\\ &=\frac {9}{10 b d (d \cos (a+b x))^{5/2}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{4 b d^3}\\ &=\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{4 b d^5}\\ &=\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{d^2}} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{2 b d^5}\\ &=\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{d-x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d^3}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{d+x^2} \, dx,x,\sqrt {d \cos (a+b x)}\right )}{4 b d^3}\\ &=\frac {9 \tan ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {d \cos (a+b x)}}{\sqrt {d}}\right )}{4 b d^{7/2}}+\frac {9}{10 b d (d \cos (a+b x))^{5/2}}+\frac {9}{2 b d^3 \sqrt {d \cos (a+b x)}}-\frac {\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 102, normalized size = 0.74 \[ \frac {45 \cot ^2(a+b x) \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};\csc ^2(a+b x)\right )+\left (-\cot ^2(a+b x)\right )^{3/4} \left (-5 \cot ^2(a+b x)+4 \sec ^2(a+b x)+40\right )}{10 b d^3 \left (-\cot ^2(a+b x)\right )^{3/4} \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 438, normalized size = 3.20 \[ \left [\frac {90 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) - 45 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {-d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {-d} {\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (45 \, \cos \left (b x + a\right )^{4} - 36 \, \cos \left (b x + a\right )^{2} - 4\right )} \sqrt {d \cos \left (b x + a\right )}}{80 \, {\left (b d^{4} \cos \left (b x + a\right )^{5} - b d^{4} \cos \left (b x + a\right )^{3}\right )}}, \frac {90 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt {d} \cos \left (b x + a\right )}\right ) + 45 \, {\left (\cos \left (b x + a\right )^{5} - \cos \left (b x + a\right )^{3}\right )} \sqrt {d} \log \left (\frac {d \cos \left (b x + a\right )^{2} - 4 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {d} {\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, {\left (45 \, \cos \left (b x + a\right )^{4} - 36 \, \cos \left (b x + a\right )^{2} - 4\right )} \sqrt {d \cos \left (b x + a\right )}}{80 \, {\left (b d^{4} \cos \left (b x + a\right )^{5} - b d^{4} \cos \left (b x + a\right )^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.41, size = 417, normalized size = 3.04 \[ \frac {\frac {90 \, \arctan \left (-\frac {\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} + \frac {45 \, \log \left ({\left | -\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} \right |}\right )}{\sqrt {-d}} + \frac {10 \, \sqrt {-d}}{{\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} - d} + \frac {5 \, \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{d} - \frac {32 \, {\left (15 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{4} - 40 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{3} \sqrt {-d} - 70 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} d + 40 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )} \sqrt {-d} d + 11 \, d^{2}\right )}}{{\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} - \sqrt {-d}\right )}^{5}}}{40 \, b d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 1165, normalized size = 8.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 134, normalized size = 0.98 \[ \frac {\frac {4 \, {\left (45 \, d^{4} \cos \left (b x + a\right )^{4} - 36 \, d^{4} \cos \left (b x + a\right )^{2} - 4 \, d^{4}\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {9}{2}} d^{2} - \left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}} d^{4}} + \frac {90 \, \arctan \left (\frac {\sqrt {d \cos \left (b x + a\right )}}{\sqrt {d}}\right )}{d^{\frac {5}{2}}} + \frac {45 \, \log \left (\frac {\sqrt {d \cos \left (b x + a\right )} - \sqrt {d}}{\sqrt {d \cos \left (b x + a\right )} + \sqrt {d}}\right )}{d^{\frac {5}{2}}}}{40 \, b d} \]
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 \[ \int \frac {1}{{\sin \left (a+b\,x\right )}^3\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]
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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