Optimal. Leaf size=273 \[ \frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}} \]
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Rubi [A] time = 2.8362, antiderivative size = 273, normalized size of antiderivative = 1., 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 = {3565, 3655, 6725, 63, 217, 206, 93, 208} \[ \frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^{5/2}}{(c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \int \frac{\frac{1}{2} \left (b^3 c^2+a^3 c d-3 a b^2 c d+3 a^2 b d^2\right )+\frac{1}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)+\frac{1}{2} b^3 \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (b^3 c^2+a^3 c d-3 a b^2 c d+3 a^2 b d^2\right )+\frac{1}{2} d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x+\frac{1}{2} b^3 \left (c^2+d^2\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{b^3 \left (c^2+d^2\right )}{2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )+d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x}{2 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{d f}+\frac{\operatorname{Subst}\left (\int \frac{d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )+d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{\left (2 b^2\right ) \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 )}{d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )+i d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right )+i d \left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{(i a+b)^3 \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c-i d) f}-\frac{(i a-b)^3 \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c+i d) f}+\frac{\left (2 b^2\right ) \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 )}{d f}\\ &=\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{(i a+b)^3 \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 )}{(c-i d) f}-\frac{(i a-b)^3 \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 )}{(c+i d) f}\\ &=-\frac{i (a-i b)^{5/2} \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 )}{(c-i d)^{3/2} f}+\frac{i (a+i b)^{5/2} \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 )}{(c+i d)^{3/2} f}+\frac{2 b^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 6.14165, size = 1478, normalized size = 5.41 \[ \frac{i (a+i b) \left ((a+i b) \left (\frac{2 \sqrt{a+b \tan (e+f x)}}{(-c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{a+i b} \tan ^{-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 )}{(-c-i d)^{3/2}}\right )-\frac{2 (b c-a d) \left (\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right )^{3/2} \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \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 ) \left (-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \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{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\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}}-\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 ) \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 d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\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 )}{2 f}-\frac{i (i b-a) \left (\frac{2 (b c-a d) \left (\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}\right )^{3/2} \left (\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}\right )^2 \sqrt{\frac{b (c+d \tan (e+f x))}{b c-a d}} \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 ) \left (-\frac{\sqrt{b} \sqrt{d} \sqrt{a+b \tan (e+f x)} \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{b c-a d} \sqrt{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}} \sqrt{\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}}-\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 ) \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 d^2 \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)} \sqrt{\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}}-(i b-a) \left (\frac{2 \sqrt{a+b \tan (e+f x)}}{(c-i d) \sqrt{c+d \tan (e+f x)}}-\frac{2 \sqrt{i b-a} \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{i b-a} \sqrt{c+d \tan (e+f x)}}\right )}{(c-i d)^{3/2}}\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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(-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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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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