Optimal. Leaf size=273 \[ -\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}} \]
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Rubi [A] time = 3.07245, 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} \[ -\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{3/2} f}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}} \]
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{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{3/2}} \, dx &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \int \frac{\frac{1}{2} \left (3 b^2 c^2 d+a^2 d^3+a b c \left (c^2-3 d^2\right )\right )-\frac{1}{2} b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \tan (e+f x)+\frac{1}{2} \left (a^2+b^2\right ) d^3 \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} \left (3 b^2 c^2 d+a^2 d^3+a b c \left (c^2-3 d^2\right )\right )-\frac{1}{2} b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) x+\frac{1}{2} \left (a^2+b^2\right ) d^3 x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b \left (a^2+b^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{\left (a^2+b^2\right ) d^3}{2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{b \left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right )-b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) x}{2 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{b \left (a^2+b^2\right ) f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{b \left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right )-b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b \left (a^2+b^2\right ) f}+\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{b f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \left (\frac{b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right )+i b \left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{-b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right )+i b \left (a c^3+3 b c^2 d-3 a c d^2-b d^3\right )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b \left (a^2+b^2\right ) f}+\frac{\left (2 d^3\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 )}{b^2 f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}-\frac{(c-i d)^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 (i a+b) f}+\frac{\left ((i a+b) (c+i d)^3\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 \left (a^2+b^2\right ) f}+\frac{\left (2 d^3\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 )}{b^2 f}\\ &=\frac{2 d^{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 )}{b^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}-\frac{(c-i d)^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 )}{(i a+b) f}+\frac{\left ((i a+b) (c+i d)^3\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 )}{\left (a^2+b^2\right ) f}\\ &=-\frac{i (c-i d)^{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 )}{(a-i b)^{3/2} f}+\frac{i (c+i d)^{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 )}{(a+i b)^{3/2} f}+\frac{2 d^{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 )}{b^{3/2} f}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 6.14188, size = 1198, normalized size = 4.39 \[ \frac{i (c+i d) \left ((c+i d) \left (\frac{2 \sqrt{c+d \tan (e+f x)}}{(-a-i b) \sqrt{a+b \tan (e+f x)}}-\frac{2 (c+i d) \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 )}{(-a-i b) \sqrt{a+i b} \sqrt{-c-i d}}\right )-\frac{2 d \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 (1-\frac{\sqrt{b} \sqrt{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{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}} \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 )}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{a+b \tan (e+f x)} \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 )}\right )}{2 f}-\frac{i (i d-c) \left (\frac{2 d \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 (1-\frac{\sqrt{b} \sqrt{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{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}} \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 )}{b \sqrt{\frac{b}{\frac{b^2 c}{b c-a d}-\frac{a b d}{b c-a d}}} \sqrt{a+b \tan (e+f x)} \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 )}-(i d-c) \left (\frac{2 \sqrt{c+d \tan (e+f x)}}{(a-i b) \sqrt{a+b \tan (e+f x)}}-\frac{2 (i d-c) \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 )}{(a-i b) \sqrt{i b-a} \sqrt{c-i d}}\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 ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( a+b\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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