Optimal. Leaf size=86 \[ \frac {i a^5 2^{\frac {m}{2}+5} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \, _2F_1\left (-\frac {m}{2}-4,\frac {m}{2};\frac {m+2}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]
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Rubi [A] time = 0.15, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3505, 3523, 70, 69} \[ \frac {i a^5 2^{\frac {m}{2}+5} (1+i \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \text {Hypergeometric2F1}\left (-\frac {m}{2}-4,\frac {m}{2},\frac {m+2}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^5 \, dx &=\left ((e \sec (c+d x))^m (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-m/2}\right ) \int (a-i a \tan (c+d x))^{m/2} (a+i a \tan (c+d x))^{5+\frac {m}{2}} \, dx\\ &=\frac {\left (a^2 (e \sec (c+d x))^m (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-m/2}\right ) \operatorname {Subst}\left (\int (a-i a x)^{-1+\frac {m}{2}} (a+i a x)^{4+\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{4+\frac {m}{2}} a^6 (e \sec (c+d x))^m (a-i a \tan (c+d x))^{-m/2} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{-m/2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{4+\frac {m}{2}} (a-i a x)^{-1+\frac {m}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i 2^{5+\frac {m}{2}} a^5 \, _2F_1\left (-4-\frac {m}{2},\frac {m}{2};\frac {2+m}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^m (1+i \tan (c+d x))^{-m/2}}{d m}\\ \end {align*}
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Mathematica [B] time = 12.92, size = 1165, normalized size = 13.55 \[ -\frac {i 2^{m+5} e^{-i (c-4 d x)} \left (2+3 e^{2 i c}\right ) \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \, _2F_1\left (1,-\frac {m}{2}-1;\frac {m+6}{2};-e^{2 i (c+d x)}\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (1+e^{2 i c}\right ) \left (1+e^{2 i (c+d x)}\right )^3 (m+4) (\cos (d x)+i \sin (d x))^5}+\frac {i 2^{m+5} e^{i (c-d m x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \left (e^{i d (m+4) x} (m+6) \, _2F_1\left (1,-\frac {m}{2}-2;\frac {m+6}{2};-e^{2 i (c+d x)}\right )-e^{i d (m+6) x} (m+4) \, _2F_1\left (1,-\frac {m}{2}-1;\frac {m+8}{2};-e^{2 i (c+d x)}\right )\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (1+e^{2 i c}\right ) \left (1+e^{2 i (c+d x)}\right )^4 (m+4) (m+6) (\cos (d x)+i \sin (d x))^5}-\frac {i 2^{m+5} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \left (1+e^{2 i (c+d x)}\right ) \, _2F_1\left (1,1-\frac {m}{2};\frac {m+2}{2};-e^{2 i (c+d x)}\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (e^{3 i c}+e^{5 i c}\right ) m (\cos (d x)+i \sin (d x))^5}+\frac {i 2^{m+5} e^{-i (3 c+d m x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \left (e^{i d m x} (m+2) \, _2F_1\left (1,-\frac {m}{2};\frac {m+2}{2};-e^{2 i (c+d x)}\right )-e^{i d (m+2) x} m \, _2F_1\left (1,1-\frac {m}{2};\frac {m+4}{2};-e^{2 i (c+d x)}\right )\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (1+e^{2 i c}\right ) m (m+2) (\cos (d x)+i \sin (d x))^5}+\frac {i 2^{m+5} e^{i (d x-4 c)} \left (1+4 e^{2 i c}\right ) \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m+1} \, _2F_1\left (1,-\frac {m}{2};\frac {m+4}{2};-e^{2 i (c+d x)}\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (1+e^{2 i c}\right ) (m+2) (\cos (d x)+i \sin (d x))^5}-\frac {3 i 2^{m+5} e^{-i (c+d m x)} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^m \left (e^{i d (m+2) x} (m+4) \, _2F_1\left (1,-\frac {m}{2}-1;\frac {m+4}{2};-e^{2 i (c+d x)}\right )-e^{i d (m+4) x} (m+2) \, _2F_1\left (1,-\frac {m}{2};\frac {m+6}{2};-e^{2 i (c+d x)}\right )\right ) (e \sec (c+d x))^m (i \tan (c+d x) a+a)^5 \sec ^{-m-5}(c+d x)}{d \left (1+e^{2 i c}\right ) \left (1+e^{2 i (c+d x)}\right )^2 (m+2) (m+4) (\cos (d x)+i \sin (d x))^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {32 \, a^{5} \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} e^{\left (10 i \, d x + 10 i \, c\right )}}{e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} \left (e \sec \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.21, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{5}\, dx \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} \left (e \sec \left (d x + c\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^5 \,d x \]
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 \[ i a^{5} \left (\int \left (- i \left (e \sec {\left (c + d x \right )}\right )^{m}\right )\, dx + \int 5 \left (e \sec {\left (c + d x \right )}\right )^{m} \tan {\left (c + d x \right )}\, dx + \int \left (- 10 \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{3}{\left (c + d x \right )}\right )\, dx + \int \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{5}{\left (c + d x \right )}\, dx + \int 10 i \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{2}{\left (c + d x \right )}\, dx + \int \left (- 5 i \left (e \sec {\left (c + d x \right )}\right )^{m} \tan ^{4}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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