Optimal. Leaf size=244 \[ -\frac{b^2 \sqrt{c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac{b^{3/2} \left (-5 a^2 d+4 a b c-b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2 \sqrt{c-i d}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2 \sqrt{c+i d}} \]
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Rubi [A] time = 0.8548, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3569, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac{b^2 \sqrt{c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x))}-\frac{b^{3/2} \left (-5 a^2 d+4 a b c-b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{3/2}}-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2 \sqrt{c-i d}}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2 \sqrt{c+i d}} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \frac{1}{(a+b \tan (e+f x))^2 \sqrt{c+d \tan (e+f x)}} \, dx &=-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac{\int \frac{\frac{1}{2} \left (-2 a b c+2 a^2 d+b^2 d\right )+b (b c-a d) \tan (e+f x)+\frac{1}{2} b^2 d \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac{\int \frac{-\left (a^2-b^2\right ) (b c-a d)+2 a b (b c-a d) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)}+\frac{\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\right )\right ) \int \frac{1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}+\frac{\int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac{\int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac{\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 (b c-a d) f}\\ &=-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}+\frac{i \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 f}-\frac{i \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 f}+\frac{\left (b^2 \left (4 a b c-5 a^2 d-b^2 d\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2 d (b c-a d) f}\\ &=-\frac{b^{3/2} \left (4 a b c-5 a^2 d-b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{3/2} f}-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-\frac{i c}{d}+\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{(a-i b)^2 d f}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1+\frac{i c}{d}-\frac{i x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{(a+i b)^2 d f}\\ &=-\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{(a-i b)^2 \sqrt{c-i d} f}+\frac{i \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{(a+i b)^2 \sqrt{c+i d} f}-\frac{b^{3/2} \left (4 a b c-5 a^2 d-b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{3/2} f}-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 2.25134, size = 258, normalized size = 1.06 \[ \frac{\frac{b^{3/2} \left (5 a^2 d-4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\left (a^2+b^2\right ) \sqrt{b c-a d}}-\frac{i \left (\frac{(a-i b)^2 (a d-b c) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{\sqrt{c+i d}}+\frac{(a+i b)^2 (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{\sqrt{c-i d}}\right )}{a^2+b^2}-\frac{b^2 \sqrt{c+d \tan (e+f x)}}{a+b \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.102, size = 6548, normalized size = 26.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \tan{\left (e + f x \right )}\right )^{2} \sqrt{c + d \tan{\left (e + f x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2} \sqrt{d \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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