Optimal. Leaf size=180 \[ -\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (15 a^2-b (10 a-b (3-2 p)) (1-2 p)\right ) \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 a^2 f} \]
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Rubi [A]
time = 0.13, antiderivative size = 176, normalized size of antiderivative = 0.98, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3744, 473, 464,
372, 371} \begin {gather*} -\frac {\left (15-\frac {b (1-2 p) (10 a-b (3-2 p))}{a^2}\right ) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )}{15 f}-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{p+1}}{5 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 464
Rule 473
Rule 3744
Rubi steps
\begin {align*} \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2 \left (a+b x^2\right )^p}{x^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}+\frac {\text {Subst}\left (\int \frac {\left (10 a-b (3-2 p)+5 a x^2\right ) \left (a+b x^2\right )^p}{x^4} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (-15 a^2-b (10 a-b (3-2 p)) (-3+2 (1+p))\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (\left (-15 a^2-b (10 a-b (3-2 p)) (-3+2 (1+p))\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac {(10 a-b (3-2 p)) \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{15 a^2 f}-\frac {\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{1+p}}{5 a f}-\frac {\left (15 a^2-b (10 a-b (3-2 p)) (1-2 p)\right ) \cot (e+f x) \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 a^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 141, normalized size = 0.78 \begin {gather*} -\frac {\cot (e+f x) \left (3 \cot ^4(e+f x) \, _2F_1\left (-\frac {5}{2},-p;-\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a}\right )+10 \cot ^2(e+f x) \, _2F_1\left (-\frac {3}{2},-p;-\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )+15 \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b \tan ^2(e+f x)}{a}\right )\right ) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{15 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (\csc ^{6}\left (f x +e \right )\right ) \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^2+a\right )}^p}{{\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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