Optimal. Leaf size=439 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac {74461 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{8192 a^5 d}-\frac {2473 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{12288 a^5 d}-\frac {155 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^5 d}-\frac {7 \cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps
used = 12, 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^{5/2} d}+\frac {74461 \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {58077 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{40960 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{320 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {41693 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{49152 a^4 d}+\frac {8925 \cot (c+d x) \sqrt {a \sec (c+d x)+a}}{32768 a^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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))^{5/2}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^6} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{a^5 d}\\ &=-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {5 a-15 a^2 x^2}{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 )}{10 a^6 d}\\ &=-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-135 a^2-455 a^3 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 )}{160 a^7 d}\\ &=-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-4685 a^3-8525 a^4 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 )}{1920 a^8 d}\\ &=-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-80565 a^4-111285 a^5 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 )}{15360 a^9 d}\\ &=-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-871155 a^5-994035 a^6 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 )}{61440 a^{10} d}\\ &=\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}+\frac {\text {Subst}\left (\int \frac {-3126975 a^6-4355775 a^7 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 )}{614400 a^{10} d}\\ &=-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}-\frac {\text {Subst}\left (\int \frac {-2008125 a^7-9380925 a^8 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 )}{3686400 a^{10} d}\\ &=\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}+\frac {\text {Subst}\left (\int \frac {12737475 a^8-2008125 a^9 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 )}{7372800 a^{10} d}\\ &=\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 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^2 d}-\frac {74461 \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 )}{32768 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 {74461 \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{32768 \sqrt {2} a^{5/2} d}+\frac {8925 \cot (c+d x) \sqrt {a+a \sec (c+d x)}}{32768 a^3 d}-\frac {41693 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{49152 a^4 d}+\frac {58077 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{40960 a^5 d}-\frac {9467 \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}}{8192 a^5 d}-\frac {2473 \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}}{12288 a^5 d}-\frac {155 \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}}{3072 a^5 d}-\frac {7 \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}}{512 a^5 d}-\frac {\cos ^5(c+d x) \cot ^5(c+d x) \sec ^{10}\left (\frac {1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{320 a^5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 24.13, size = 5688, normalized size = 12.96 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1411\) vs.
\(2(385)=770\).
time = 0.27, size = 1412, normalized size = 3.22
method | result | size |
default | \(\text {Expression too large to display}\) | \(1412\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.29, size = 1023, normalized size = 2.33 \begin {gather*} \left [-\frac {1116915 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \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 ) + 983040 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \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 (639063 \, \cos \left (d x + c\right )^{8} + 681555 \, \cos \left (d x + c\right )^{7} - 986085 \, \cos \left (d x + c\right )^{6} - 1360025 \, \cos \left (d x + c\right )^{5} + 405445 \, \cos \left (d x + c\right )^{4} + 836921 \, \cos \left (d x + c\right )^{3} + 15305 \, \cos \left (d x + c\right )^{2} - 133875 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{1966080 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}, -\frac {1116915 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \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 ) + 983040 \, {\left (\cos \left (d x + c\right )^{7} + 3 \, \cos \left (d x + c\right )^{6} + \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + 3 \, \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 (639063 \, \cos \left (d x + c\right )^{8} + 681555 \, \cos \left (d x + c\right )^{7} - 986085 \, \cos \left (d x + c\right )^{6} - 1360025 \, \cos \left (d x + c\right )^{5} + 405445 \, \cos \left (d x + c\right )^{4} + 836921 \, \cos \left (d x + c\right )^{3} + 15305 \, \cos \left (d x + c\right )^{2} - 133875 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{983040 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.64, size = 412, normalized size = 0.94 \begin {gather*} \frac {{\left (2 \, {\left (4 \, {\left (6 \, {\left (\frac {8 \, \sqrt {2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {91 \, \sqrt {2}}{a^{3} \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 {3043 \, \sqrt {2}}{a^{3} \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 {47185 \, \sqrt {2}}{a^{3} \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 {349965 \, \sqrt {2}}{a^{3} \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 {1024 \, \sqrt {2} {\left (345 \, {\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} - 1230 \, {\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 + 1760 \, {\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} - 1150 \, {\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} + 299 \, 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} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{983040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________