Integrand size = 45, antiderivative size = 152 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {i A+B}{f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 i A+B) \sqrt {c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 i A+B) \sqrt {c-i c \tan (e+f x)}}{3 a c f \sqrt {a+i a \tan (e+f x)}} \]
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Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47, 37} \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {B+i A}{f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(B+2 i A) \sqrt {c-i c \tan (e+f x)}}{3 a c f \sqrt {a+i a \tan (e+f x)}}+\frac {(B+2 i A) \sqrt {c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}} \]
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Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{5/2} (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))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(a (2 A-i B)) \text {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} \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))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 i A+B) \sqrt {c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 A-i B) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 f} \\ & = -\frac {i A+B}{f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 i A+B) \sqrt {c-i c \tan (e+f x)}}{3 c f (a+i a \tan (e+f x))^{3/2}}+\frac {(2 i A+B) \sqrt {c-i c \tan (e+f x)}}{3 a c f \sqrt {a+i a \tan (e+f x)}} \\ \end{align*}
Time = 3.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {A+i B+(-2 i A-B) \tan (e+f x)+(2 A-i B) \tan ^2(e+f x)}{3 a f (-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.40 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (2 i A \tan \left (f x +e \right )^{4}-i B \tan \left (f x +e \right )^{3}+B \tan \left (f x +e \right )^{4}+3 i A \tan \left (f x +e \right )^{2}+2 A \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right ) B +i A +2 A \tan \left (f x +e \right )-B \right )}{3 f \,a^{2} c \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}\) | \(152\) |
default | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (2 i A \tan \left (f x +e \right )^{4}-i B \tan \left (f x +e \right )^{3}+B \tan \left (f x +e \right )^{4}+3 i A \tan \left (f x +e \right )^{2}+2 A \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right ) B +i A +2 A \tan \left (f x +e \right )-B \right )}{3 f \,a^{2} c \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}\) | \(152\) |
parts | \(\frac {A \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (2 i \tan \left (f x +e \right )^{3}-2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )-3 \tan \left (f x +e \right )^{2}-1\right )}{3 f \,a^{2} c \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {i B \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (i \tan \left (f x +e \right )^{3}-\tan \left (f x +e \right )^{4}+i \tan \left (f x +e \right )+1\right )}{3 f \,a^{2} c \left (i-\tan \left (f x +e \right )\right )^{3} \left (i+\tan \left (f x +e \right )\right )^{2}}\) | \(211\) |
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {{\left (3 \, {\left (i \, A + B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 4 \, {\left (i \, A - B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} + 3 \, {\left (-i \, A + B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, {\left (i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - {\left (7 i \, A - B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, A + B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{12 \, a^{2} c f} \]
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\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
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Exception generated. \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {B \tan \left (f x + e\right ) + A}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
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Time = 9.36 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.12 \[ \int \frac {A+B \tan (e+f x)}{(a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (6\,A\,\sin \left (2\,e+2\,f\,x\right )-3\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,6{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}-B\,\cos \left (4\,e+4\,f\,x\right )-A\,3{}\mathrm {i}+A\,\sin \left (4\,e+4\,f\,x\right )+B\,\sin \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}\right )}{12\,a^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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