Integrand size = 45, antiderivative size = 157 \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.28 (sec) , antiderivative size = 157, 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} \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {-B+i A}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(-B+2 i A) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}-\frac {(-B+2 i A) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}} \]
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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)^{3/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}+\frac {((2 A+i B) c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {(2 A+i B) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f} \\ & = \frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {(2 i A-B) \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}} \\ \end{align*}
Time = 4.82 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.62 \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {A-i B+(2 i A-B) \tan (e+f x)+(2 A+i B) \tan ^2(e+f x)}{3 c f (i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.41 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {i A \,{\mathrm e}^{4 i \left (f x +e \right )}+B \,{\mathrm e}^{4 i \left (f x +e \right )}+6 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-3 i A +3 B}{12 c \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(110\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i A \tan \left (f x +e \right )^{4}-i B \tan \left (f x +e \right )^{3}-B \tan \left (f x +e \right )^{4}+3 i A \tan \left (f x +e \right )^{2}-2 A \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right ) B +i A -2 A \tan \left (f x +e \right )+B \right )}{3 f a \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(151\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i A \tan \left (f x +e \right )^{4}-i B \tan \left (f x +e \right )^{3}-B \tan \left (f x +e \right )^{4}+3 i A \tan \left (f x +e \right )^{2}-2 A \tan \left (f x +e \right )^{3}-i \tan \left (f x +e \right ) B +i A -2 A \tan \left (f x +e \right )+B \right )}{3 f a \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(151\) |
parts | \(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i \tan \left (f x +e \right )^{3}+2 \tan \left (f x +e \right )^{4}+2 i \tan \left (f x +e \right )+3 \tan \left (f x +e \right )^{2}+1\right )}{3 f a \,c^{2} \left (i-\tan \left (f x +e \right )\right )^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (i \tan \left (f x +e \right )^{3}+\tan \left (f x +e \right )^{4}+i \tan \left (f x +e \right )-1\right )}{3 f a \,c^{2} \left (i-\tan \left (f x +e \right )\right )^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(209\) |
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Time = 0.25 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {{\left ({\left (-i \, A - B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-7 i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 4 \, {\left (-i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 \, {\left (i \, A + B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, {\left (-i \, A - B\right )} e^{\left (i \, f x + i \, e\right )} + 3 i \, A - 3 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{12 \, a c^{2} f} \]
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\[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\int { \frac {B \tan \left (f x + e\right ) + A}{\sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.73 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.93 \[ \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (2\,A\,\sin \left (2\,e+2\,f\,x\right )+A\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-2\,B\,\cos \left (2\,e+2\,f\,x\right )-A\,3{}\mathrm {i}+B\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{6\,a\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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