Optimal. Leaf size=102 \[ -\frac {(-8 B+i A) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3588, 78, 37} \[ -\frac {(-8 B+i A) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (A+8 i B)) \operatorname {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{9 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(i A-8 B) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 12.67, size = 121, normalized size = 1.19 \[ \frac {a^3 \cos (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} (\cos (8 e+11 f x)+i \sin (8 e+11 f x)) ((B-8 i A) \cos (e+f x)-(A+8 i B) \sin (e+f x))}{63 c^5 f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 103, normalized size = 1.01 \[ \frac {{\left ({\left (-7 i \, A - 7 \, B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + {\left (-16 i \, A + 2 \, B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-9 i \, A + 9 \, B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{126 \, c^{5} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 134, normalized size = 1.31 \[ -\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{3} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (8 i B \left (\tan ^{3}\left (f x +e \right )\right )+6 i A \left (\tan ^{2}\left (f x +e \right )\right )+A \left (\tan ^{3}\left (f x +e \right )\right )-6 i B \tan \left (f x +e \right )+15 B \left (\tan ^{2}\left (f x +e \right )\right )-8 i A +15 A \tan \left (f x +e \right )+B \right )}{63 f \,c^{5} \left (\tan \left (f x +e \right )+i\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.22, size = 167, normalized size = 1.64 \[ -\frac {{\left (882 \, {\left (A - i \, B\right )} a^{3} \cos \left (11 \, f x + 11 \, e\right ) + 252 \, {\left (8 \, A + i \, B\right )} a^{3} \cos \left (9 \, f x + 9 \, e\right ) + 1134 \, {\left (A + i \, B\right )} a^{3} \cos \left (7 \, f x + 7 \, e\right ) + {\left (882 i \, A + 882 \, B\right )} a^{3} \sin \left (11 \, f x + 11 \, e\right ) + {\left (2016 i \, A - 252 \, B\right )} a^{3} \sin \left (9 \, f x + 9 \, e\right ) + {\left (1134 i \, A - 1134 \, B\right )} a^{3} \sin \left (7 \, f x + 7 \, e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (-15876 i \, c^{5} \cos \left (2 \, f x + 2 \, e\right ) + 15876 \, c^{5} \sin \left (2 \, f x + 2 \, e\right ) - 15876 i \, c^{5}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.48, size = 192, normalized size = 1.88 \[ -\frac {a^3\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (6\,e+6\,f\,x\right )\,9{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,7{}\mathrm {i}-9\,B\,\cos \left (6\,e+6\,f\,x\right )+7\,B\,\cos \left (8\,e+8\,f\,x\right )-9\,A\,\sin \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (8\,e+8\,f\,x\right )-B\,\sin \left (6\,e+6\,f\,x\right )\,9{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,7{}\mathrm {i}\right )}{126\,c^4\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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