Optimal. Leaf size=258 \[ \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {i \sqrt {a-i b} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} (a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f} \]
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Rubi [A] time = 2.19, antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3570, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {i \sqrt {a-i b} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} (a d+3 b c) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3570
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\int \frac {\frac {1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {1}{2} d (3 b c+a d) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x+\frac {1}{2} d (3 b c+a d) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\operatorname {Subst}\left (\int \left (\frac {d (3 b c+a d)}{2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\operatorname {Subst}\left (\int \frac {-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}+\frac {(d (3 b c+a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-2 a c d-b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 a c d+b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}+\frac {(d (3 b c+a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{b f}\\ &=\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\left ((i a+b) (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b) (c+i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {(d (3 b c+a d)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{b f}\\ &=\frac {\sqrt {d} (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\left ((i a+b) (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left ((i a-b) (c+i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac {i \sqrt {a-i b} (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} (3 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [B] time = 6.09, size = 1526, normalized size = 5.91 \[ -\frac {i (-a-i b) \left (-\frac {2 d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\frac {\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right )}{2 \sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)} \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )^{3/2}}+\frac {1}{2 \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )}\right ) \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )^{3/2}}{b \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-(-c-i d) \left (-\frac {2 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right )}{b^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 (-c-i d) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} \sqrt {c+i d}}\right )\right )}{2 f}-\frac {i (i b-a) \left (\frac {2 d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )^{3/2} \left (\frac {\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right )}{2 \sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)} \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )^{3/2}}+\frac {1}{2 \left (\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}+1\right )}\right )}{b \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}-(i d-c) \left (\frac {2 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt {c+d \tan (e+f x)}}-\frac {2 \sqrt {i d-c} \tanh ^{-1}\left (\frac {\sqrt {i d-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {i b-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {i b-a}}\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [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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giac [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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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \sqrt {a + b \tan {\left (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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