Optimal. Leaf size=212 \[ \frac {2 (2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{15 a^2 c f \sqrt {a+i a \tan (e+f x)}}-\frac {B+i A}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 (2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3588, 78, 45, 37} \[ \frac {2 (2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{15 a^2 c f \sqrt {a+i a \tan (e+f x)}}-\frac {B+i A}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 (2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 B+3 i A) \sqrt {c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 3588
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {A+B x}{(a+i a x)^{7/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(a (3 A-2 i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{7/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac {(2 (3 A-2 i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac {i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac {2 (3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 (3 A-2 i B)) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{15 a f}\\ &=-\frac {i A+B}{f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{5 c f (a+i a \tan (e+f x))^{5/2}}+\frac {2 (3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{15 a c f (a+i a \tan (e+f x))^{3/2}}+\frac {2 (3 i A+2 B) \sqrt {c-i c \tan (e+f x)}}{15 a^2 c f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.78, size = 132, normalized size = 0.62 \[ -\frac {\sec (e+f x) \sqrt {c-i c \tan (e+f x)} (-i (3 A-2 i B) (5 \sin (e+f x)-3 \sin (3 (e+f x)))+(-30 A+5 i B) \cos (e+f x)+(6 A-9 i B) \cos (3 (e+f x)))}{60 a^2 c f (\tan (e+f x)-i) \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.42, size = 158, normalized size = 0.75 \[ \frac {{\left ({\left (-15 i \, A - 15 \, B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-48 i \, A + 8 \, B\right )} e^{\left (7 i \, f x + 7 i \, e\right )} + 30 i \, A e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-48 i \, A + 8 \, B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (60 i \, A + 10 \, B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (18 i \, A - 8 \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 3 i \, A - 3 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-5 i \, f x - 5 i \, e\right )}}{120 \, a^{3} c f} \]
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 \[ \int \frac {B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 186, normalized size = 0.88 \[ -\frac {\sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (4 i B \left (\tan ^{5}\left (f x +e \right )\right )+12 i A \left (\tan ^{4}\left (f x +e \right )\right )-6 A \left (\tan ^{5}\left (f x +e \right )\right )+2 i B \left (\tan ^{3}\left (f x +e \right )\right )+8 B \left (\tan ^{4}\left (f x +e \right )\right )+18 i A \left (\tan ^{2}\left (f x +e \right )\right )-3 A \left (\tan ^{3}\left (f x +e \right )\right )-2 i B \tan \left (f x +e \right )+7 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A +3 A \tan \left (f x +e \right )-B \right )}{15 f c \,a^{3} \left (\tan \left (f x +e \right )+i\right )^{2} \left (-\tan \left (f x +e \right )+i\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.72, size = 246, normalized size = 1.16 \[ \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (15\,B\,\cos \left (2\,e+2\,f\,x\right )-15\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,45{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,15{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}-A\,15{}\mathrm {i}-5\,B\,\cos \left (4\,e+4\,f\,x\right )-3\,B\,\cos \left (6\,e+6\,f\,x\right )+45\,A\,\sin \left (2\,e+2\,f\,x\right )+15\,A\,\sin \left (4\,e+4\,f\,x\right )+3\,A\,\sin \left (6\,e+6\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,15{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,5{}\mathrm {i}+B\,\sin \left (6\,e+6\,f\,x\right )\,3{}\mathrm {i}\right )}{120\,a^3\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \tan {\left (e + f x \right )}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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