Optimal. Leaf size=80 \[ -\frac {b^5 (b \tan (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2671, 276}
\begin {gather*} -\frac {b^5 (b \tan (e+f x))^{n-5}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{n-3}}{f (3-n)}-\frac {b (b \tan (e+f x))^{n-1}}{f (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2671
Rubi steps
\begin {align*} \int \csc ^6(e+f x) (b \tan (e+f x))^n \, dx &=\frac {b \text {Subst}\left (\int x^{-6+n} \left (b^2+x^2\right )^2 \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac {b \text {Subst}\left (\int \left (b^4 x^{-6+n}+2 b^2 x^{-4+n}+x^{-2+n}\right ) \, dx,x,b \tan (e+f x)\right )}{f}\\ &=-\frac {b^5 (b \tan (e+f x))^{-5+n}}{f (5-n)}-\frac {2 b^3 (b \tan (e+f x))^{-3+n}}{f (3-n)}-\frac {b (b \tan (e+f x))^{-1+n}}{f (1-n)}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 69, normalized size = 0.86 \begin {gather*} \frac {b \left (8-6 n+n^2+2 (-3+n) \cos (2 (e+f x))+\cos (4 (e+f x))\right ) \csc ^4(e+f x) (b \tan (e+f x))^{-1+n}}{f (-5+n) (-3+n) (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.52, size = 10922, normalized size = 136.52
method | result | size |
risch | \(\text {Expression too large to display}\) | \(10922\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 87, normalized size = 1.09 \begin {gather*} \frac {\frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 1\right )} \tan \left (f x + e\right )} + \frac {2 \, b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 3\right )} \tan \left (f x + e\right )^{3}} + \frac {b^{n} \tan \left (f x + e\right )^{n}}{{\left (n - 5\right )} \tan \left (f x + e\right )^{5}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 152 vs.
\(2 (75) = 150\).
time = 0.39, size = 152, normalized size = 1.90 \begin {gather*} \frac {{\left (8 \, \cos \left (f x + e\right )^{5} + 4 \, {\left (n - 5\right )} \cos \left (f x + e\right )^{3} + {\left (n^{2} - 8 \, n + 15\right )} \cos \left (f x + e\right )\right )} \left (\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right )^{n}}{{\left ({\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{4} + f n^{3} - 9 \, f n^{2} - 2 \, {\left (f n^{3} - 9 \, f n^{2} + 23 \, f n - 15 \, f\right )} \cos \left (f x + e\right )^{2} + 23 \, f n - 15 \, f\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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