Optimal. Leaf size=355 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac {9683 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{4096 \sqrt {2} a^{5/2} d}+\frac {5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d}-\frac {1491 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{4096 a^3 d} \]
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Rubi [A] time = 0.33, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3887, 472, 579, 583, 522, 203} \[ \frac {5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac {1491 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{4096 a^3 d}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac {9683 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 472
Rule 522
Rule 579
Rule 583
Rule 3887
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^5} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^4 d}\\ &=-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\operatorname {Subst}\left (\int \frac {5 a-11 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^5 d}\\ &=-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\operatorname {Subst}\left (\int \frac {-51 a^2-243 a^3 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{96 a^6 d}\\ &=-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\operatorname {Subst}\left (\int \frac {-1509 a^3-3045 a^4 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{768 a^7 d}\\ &=-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\operatorname {Subst}\left (\int \frac {-16761 a^4-22905 a^5 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{3072 a^8 d}\\ &=\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {-13419 a^5-50283 a^6 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{18432 a^8 d}\\ &=-\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {\operatorname {Subst}\left (\int \frac {60309 a^6-13419 a^7 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{36864 a^8 d}\\ &=-\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^2 d}+\frac {9683 \operatorname {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{4096 a^2 d}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac {9683 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{4096 \sqrt {2} a^{5/2} d}-\frac {1491 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{4096 a^3 d}+\frac {5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac {1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{2048 a^4 d}-\frac {145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{1024 a^4 d}-\frac {9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{256 a^4 d}-\frac {\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}\\ \end {align*}
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Mathematica [C] time = 24.16, size = 5646, normalized size = 15.90 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.71, size = 868, normalized size = 2.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.69, size = 331, normalized size = 0.93 \[ -\frac {3 \, {\left (2 \, {\left (4 \, {\left (\frac {2 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {19 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {369 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {2989 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {512 \, \sqrt {2} {\left (12 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 21 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + 11 \, a^{2}\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}^{3} \sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{24576 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.56, size = 1066, normalized size = 3.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^4}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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