Optimal. Leaf size=252 \[ \frac {e^5 (e \tan (c+d x))^{m-5} \, _2F_1\left (1,\frac {m-5}{2};\frac {m-3}{2};-\tan ^2(c+d x)\right )}{a^3 d (5-m)}-\frac {e^5 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac {m-2}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left (\frac {m-5}{2},\frac {m-2}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m)}-\frac {3 e^5 \sec (c+d x) \cos ^2(c+d x)^{\frac {m-4}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left (\frac {m-5}{2},\frac {m-4}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m)}+\frac {3 e^5 (e \tan (c+d x))^{m-5}}{a^3 d (5-m)} \]
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Rubi [A] time = 0.34, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3888, 3886, 3476, 364, 2617, 2607, 32} \[ \frac {e^5 (e \tan (c+d x))^{m-5} \, _2F_1\left (1,\frac {m-5}{2};\frac {m-3}{2};-\tan ^2(c+d x)\right )}{a^3 d (5-m)}-\frac {e^5 \sec ^3(c+d x) \cos ^2(c+d x)^{\frac {m-2}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left (\frac {m-5}{2},\frac {m-2}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m)}-\frac {3 e^5 \sec (c+d x) \cos ^2(c+d x)^{\frac {m-4}{2}} (e \tan (c+d x))^{m-5} \, _2F_1\left (\frac {m-5}{2},\frac {m-4}{2};\frac {m-3}{2};\sin ^2(c+d x)\right )}{a^3 d (5-m)}+\frac {3 e^5 (e \tan (c+d x))^{m-5}}{a^3 d (5-m)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 364
Rule 2607
Rule 2617
Rule 3476
Rule 3886
Rule 3888
Rubi steps
\begin {align*} \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx &=\frac {e^6 \int (-a+a \sec (c+d x))^3 (e \tan (c+d x))^{-6+m} \, dx}{a^6}\\ &=\frac {e^6 \int \left (-a^3 (e \tan (c+d x))^{-6+m}+3 a^3 \sec (c+d x) (e \tan (c+d x))^{-6+m}-3 a^3 \sec ^2(c+d x) (e \tan (c+d x))^{-6+m}+a^3 \sec ^3(c+d x) (e \tan (c+d x))^{-6+m}\right ) \, dx}{a^6}\\ &=-\frac {e^6 \int (e \tan (c+d x))^{-6+m} \, dx}{a^3}+\frac {e^6 \int \sec ^3(c+d x) (e \tan (c+d x))^{-6+m} \, dx}{a^3}+\frac {\left (3 e^6\right ) \int \sec (c+d x) (e \tan (c+d x))^{-6+m} \, dx}{a^3}-\frac {\left (3 e^6\right ) \int \sec ^2(c+d x) (e \tan (c+d x))^{-6+m} \, dx}{a^3}\\ &=-\frac {3 e^5 \cos ^2(c+d x)^{\frac {1}{2} (-4+m)} \, _2F_1\left (\frac {1}{2} (-5+m),\frac {1}{2} (-4+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}-\frac {e^5 \cos ^2(c+d x)^{\frac {1}{2} (-2+m)} \, _2F_1\left (\frac {1}{2} (-5+m),\frac {1}{2} (-2+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) \sec ^3(c+d x) (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int (e x)^{-6+m} \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {e^7 \operatorname {Subst}\left (\int \frac {x^{-6+m}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{a^3 d}\\ &=\frac {3 e^5 (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}+\frac {e^5 \, _2F_1\left (1,\frac {1}{2} (-5+m);\frac {1}{2} (-3+m);-\tan ^2(c+d x)\right ) (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}-\frac {3 e^5 \cos ^2(c+d x)^{\frac {1}{2} (-4+m)} \, _2F_1\left (\frac {1}{2} (-5+m),\frac {1}{2} (-4+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) \sec (c+d x) (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}-\frac {e^5 \cos ^2(c+d x)^{\frac {1}{2} (-2+m)} \, _2F_1\left (\frac {1}{2} (-5+m),\frac {1}{2} (-2+m);\frac {1}{2} (-3+m);\sin ^2(c+d x)\right ) \sec ^3(c+d x) (e \tan (c+d x))^{-5+m}}{a^3 d (5-m)}\\ \end {align*}
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Mathematica [F] time = 11.60, size = 0, normalized size = 0.00 \[ \int \frac {(e \tan (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (e \tan \left (d x + c\right )\right )^{m}}{a^{3} \sec \left (d x + c\right )^{3} + 3 \, a^{3} \sec \left (d x + c\right )^{2} + 3 \, a^{3} \sec \left (d x + c\right ) + a^{3}}, x\right ) \]
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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \tan \left (d x +c \right )\right )^{m}}{\left (a +a \sec \left (d x +c \right )\right )^{3}}\, dx \]
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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{m}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m}{a^3\,{\left (\cos \left (c+d\,x\right )+1\right )}^3} \,d x \]
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 \[ \frac {\int \frac {\left (e \tan {\left (c + d x \right )}\right )^{m}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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