Optimal. Leaf size=155 \[ -\frac {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
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Rubi [A] time = 0.26, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ -\frac {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
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))^{9/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2} (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))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (2 A+7 i B)) \operatorname {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(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))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}+\frac {(a (2 A+7 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 )}{63 c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 11.52, size = 135, normalized size = 0.87 \[ \frac {a^2 \cos (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} (\cos (7 e+9 f x)+i \sin (7 e+9 f x)) (-7 (2 A+7 i B) \sin (2 (e+f x))+7 (2 B-7 i A) \cos (2 (e+f x))-45 i A)}{630 c^5 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 123, normalized size = 0.79 \[ \frac {{\left ({\left (-35 i \, A - 35 \, B\right )} a^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + {\left (-125 i \, A - 35 \, B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-153 i \, A + 63 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (-63 i \, A + 63 \, B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{1260 \, c^{5} f} \]
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 \[ \int \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 138, normalized size = 0.89 \[ -\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (-47 A -42 i B \left (\tan ^{2}\left (f x +e \right )\right )-12 A \left (\tan ^{2}\left (f x +e \right )\right )-33 i A \tan \left (f x +e \right )+2 i A \left (\tan ^{3}\left (f x +e \right )\right )-42 B \tan \left (f x +e \right )-7 i B -7 B \left (\tan ^{3}\left (f x +e \right )\right )\right )}{315 f \,c^{5} \left (\tan \left (f x +e \right )+i\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 198, normalized size = 1.28 \[ -\frac {{\left ({\left (35 i \, A + 35 \, B\right )} a^{2} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 90 i \, A a^{2} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + {\left (63 i \, A - 63 \, B\right )} a^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 35 \, {\left (A - i \, B\right )} a^{2} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 90 \, A a^{2} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 63 \, {\left (A + i \, B\right )} a^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{1260 \, c^{\frac {9}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.00, size = 217, normalized size = 1.40 \[ -\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,63{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,90{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,35{}\mathrm {i}-63\,B\,\cos \left (4\,e+4\,f\,x\right )+35\,B\,\cos \left (8\,e+8\,f\,x\right )-63\,A\,\sin \left (4\,e+4\,f\,x\right )-90\,A\,\sin \left (6\,e+6\,f\,x\right )-35\,A\,\sin \left (8\,e+8\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,63{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,35{}\mathrm {i}\right )}{1260\,c^4\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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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