Optimal. Leaf size=102 \[ -\frac {(-6 B+i A) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}} \]
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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} \[ -\frac {(-6 B+i A) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}} \]
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))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{7/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2} (A+B x)}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}+\frac {(a (A+6 i B)) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{7 f (c-i c \tan (e+f x))^{7/2}}-\frac {(i A-6 B) (a+i a \tan (e+f x))^{5/2}}{35 c f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 12.02, size = 121, normalized size = 1.19 \[ \frac {a^2 \cos (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} (\cos (6 e+8 f x)+i \sin (6 e+8 f x)) ((B-6 i A) \cos (e+f x)-(A+6 i B) \sin (e+f x))}{35 c^4 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.89, size = 103, normalized size = 1.01 \[ \frac {{\left ({\left (-5 i \, A - 5 \, B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-12 i \, A + 2 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (-7 i \, A + 7 \, B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{70 \, c^{4} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 115, normalized size = 1.13 \[ -\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (i A \left (\tan ^{2}\left (f x +e \right )\right )+5 i B \tan \left (f x +e \right )-6 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A -5 A \tan \left (f x +e \right )-B \right )}{35 f \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 167, normalized size = 1.64 \[ -\frac {{\left (350 \, {\left (A - i \, B\right )} a^{2} \cos \left (9 \, f x + 9 \, e\right ) + 140 \, {\left (6 \, A + i \, B\right )} a^{2} \cos \left (7 \, f x + 7 \, e\right ) + 490 \, {\left (A + i \, B\right )} a^{2} \cos \left (5 \, f x + 5 \, e\right ) + {\left (350 i \, A + 350 \, B\right )} a^{2} \sin \left (9 \, f x + 9 \, e\right ) + {\left (840 i \, A - 140 \, B\right )} a^{2} \sin \left (7 \, f x + 7 \, e\right ) + {\left (490 i \, A - 490 \, B\right )} a^{2} \sin \left (5 \, f x + 5 \, e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (-4900 i \, c^{4} \cos \left (2 \, f x + 2 \, e\right ) + 4900 \, c^{4} \sin \left (2 \, f x + 2 \, e\right ) - 4900 i \, c^{4}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.28, size = 192, normalized size = 1.88 \[ -\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}-7\,B\,\cos \left (4\,e+4\,f\,x\right )+5\,B\,\cos \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (4\,e+4\,f\,x\right )-5\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,7{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,5{}\mathrm {i}\right )}{70\,c^3\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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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