Optimal. Leaf size=387 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac {16363 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8192 a^2 d}-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3972, 483,
593, 597, 536, 209} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac {16363 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{8192 \sqrt {2} a^{3/2} d}+\frac {12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d}-\frac {8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac {21 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{8192 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 483
Rule 536
Rule 593
Rule 597
Rule 3972
Rubi steps
\begin {align*} \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \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 ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {3 a-13 a^2 x^2}{x^6 \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 {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-127 a^2-319 a^3 x^2}{x^6 \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 {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-3063 a^3-4599 a^4 x^2}{x^6 \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 {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {-36801 a^4-42945 a^5 x^2}{x^6 \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 {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}+\frac {\text {Subst}\left (\int \frac {-122565 a^5-184005 a^6 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 )}{30720 a^8 d}\\ &=-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {\text {Subst}\left (\int \frac {945 a^6-367695 a^7 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 )}{184320 a^8 d}\\ &=-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8192 a^2 d}-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}+\frac {\text {Subst}\left (\int \frac {738225 a^7+945 a^8 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 )}{368640 a^8 d}\\ &=-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8192 a^2 d}-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac {16363 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8192 a d}+\frac {2 \text {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 d}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac {16363 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{8192 \sqrt {2} a^{3/2} d}-\frac {21 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{8192 a^2 d}-\frac {8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac {12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac {2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac {511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac {29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac {\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 24.16, size = 5662, normalized size = 14.63 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1239\) vs.
\(2(338)=676\).
time = 0.22, size = 1240, normalized size = 3.20
method | result | size |
default | \(\text {Expression too large to display}\) | \(1240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.70, size = 955, normalized size = 2.47 \begin {gather*} \left [-\frac {245445 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 245760 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (d x + c\right )^{3} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right ) + 1}\right ) \sin \left (d x + c\right ) + 4 \, {\left (151041 \, \cos \left (d x + c\right )^{7} + 103524 \, \cos \left (d x + c\right )^{6} - 228999 \, \cos \left (d x + c\right )^{5} - 181256 \, \cos \left (d x + c\right )^{4} + 97611 \, \cos \left (d x + c\right )^{3} + 82340 \, \cos \left (d x + c\right )^{2} + 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{491520 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )}, -\frac {245445 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 245760 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right ) - a}\right ) \sin \left (d x + c\right ) + 2 \, {\left (151041 \, \cos \left (d x + c\right )^{7} + 103524 \, \cos \left (d x + c\right )^{6} - 228999 \, \cos \left (d x + c\right )^{5} - 181256 \, \cos \left (d x + c\right )^{4} + 97611 \, \cos \left (d x + c\right )^{3} + 82340 \, \cos \left (d x + c\right )^{2} + 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{245760 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\cot ^{6}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.42, size = 379, normalized size = 0.98 \begin {gather*} -\frac {5 \, {\left (2 \, {\left (4 \, {\left (\frac {6 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {65 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {1451 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \frac {13503 \, \sqrt {2}}{a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\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 {256 \, \sqrt {2} {\left (555 \, {\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 )}^{8} - 1950 \, {\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 )}^{6} a + 2780 \, {\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} a^{2} - 1810 \, {\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^{3} + 473 \, a^{4}\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 )}^{5} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{245760 \, d} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^6}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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