Integrand size = 45, antiderivative size = 102 \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B) \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {(i A-2 B) \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3669, 79, 37} \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {(-2 B+i A) \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}}-\frac {(B+i A) \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}} \]
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Rule 37
Rule 79
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {A+B x}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {(a (A+2 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) \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {(i A-2 B) \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}} \\ \end{align*}
Time = 4.94 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {\sqrt {a+i a \tan (e+f x)} (2 A+i B+(-i A+2 B) \tan (e+f x))}{3 c f (i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}} \]
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Time = 0.42 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (i A \,{\mathrm e}^{2 i \left (f x +e \right )}+B \,{\mathrm e}^{2 i \left (f x +e \right )}+3 i A -3 B \right )}{6 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(84\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i B \tan \left (f x +e \right )^{2}+3 i A \tan \left (f x +e \right )+A \tan \left (f x +e \right )^{2}-i B -3 B \tan \left (f x +e \right )-2 A \right )}{3 f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(100\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (2 i B \tan \left (f x +e \right )^{2}+3 i A \tan \left (f x +e \right )+A \tan \left (f x +e \right )^{2}-i B -3 B \tan \left (f x +e \right )-2 A \right )}{3 f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(100\) |
parts | \(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (3 i \tan \left (f x +e \right )+\tan \left (f x +e \right )^{2}-2\right )}{3 f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}+\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (3 i \tan \left (f x +e \right )+2 \tan \left (f x +e \right )^{2}-1\right )}{3 f \,c^{2} \left (i+\tan \left (f x +e \right )\right )^{3}}\) | \(145\) |
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Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=\frac {{\left ({\left (-i \, A - B\right )} e^{\left (5 i \, f x + 5 i \, e\right )} - 2 \, {\left (2 i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 \, {\left (i \, A - B\right )} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, c^{2} f} \]
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\[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int \frac {\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (A + B \tan {\left (e + f x \right )}\right )}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} \sqrt {i \, a \tan \left (f x + e\right ) + a}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 1.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx=-\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,3{}\mathrm {i}-3\,B+A\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+B\,\cos \left (2\,e+2\,f\,x\right )-A\,\sin \left (2\,e+2\,f\,x\right )+B\,\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{6\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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