Optimal. Leaf size=342 \[ \frac{2 a (A b-a B)}{3 b d \left (a^2+b^2\right ) \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{b^2 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{b^{5/2} d} \]
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Rubi [A] time = 2.45785, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {4241, 3605, 3645, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac{2 a (A b-a B)}{3 b d \left (a^2+b^2\right ) \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a B \left (a^2+3 b^2\right )\right )}{b^2 d \left (a^2+b^2\right )^2 \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{(-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac{(B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}}+\frac{2 B \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{b^{5/2} d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3605
Rule 3645
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\tan ^{\frac{5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{\tan (c+d x)} \left (-\frac{3}{2} a (A b-a B)+\frac{3}{2} b (A b-a B) \tan (c+d x)+\frac{3}{2} \left (a^2+b^2\right ) B \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{4} a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )-\frac{3}{4} b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+\frac{3}{4} \left (a^2+b^2\right )^2 B \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3}{4} a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )-\frac{3}{4} b^2 \left (a^2 A-A b^2+2 a b B\right ) x+\frac{3}{4} \left (a^2+b^2\right )^2 B x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 b^2 \left (a^2+b^2\right )^2 d}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{3 \left (a^2+b^2\right )^2 B}{4 \sqrt{x} \sqrt{a+b x}}-\frac{3 \left (b^2 \left (2 a A b-a^2 B+b^2 B\right )+b^2 \left (a^2 A-A b^2+2 a b B\right ) x\right )}{4 \sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{3 b^2 \left (a^2+b^2\right )^2 d}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{b^2 \left (2 a A b-a^2 B+b^2 B\right )+b^2 \left (a^2 A-A b^2+2 a b B\right ) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d}+\frac{\left (B \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{b^2 d}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{-b^2 \left (a^2 A-A b^2+2 a b B\right )+i b^2 \left (2 a A b-a^2 B+b^2 B\right )}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{b^2 \left (a^2 A-A b^2+2 a b B\right )+i b^2 \left (2 a A b-a^2 B+b^2 B\right )}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d}+\frac{\left (2 B \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{b^2 d}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a+i b)^2 (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left ((a-i b)^2 (A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left (2 B \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{b^2 d}\\ &=\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{b^{5/2} d}+\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}-\frac{\left ((a+i b)^2 (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(-a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left ((a-i b)^2 (A+i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}\\ &=-\frac{(i A-B) \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{\sqrt{i a-b} (a+i b)^2 d}+\frac{2 B \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{b^{5/2} d}-\frac{(i A+B) \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d \cot ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}+\frac{2 a \left (2 A b^3-a \left (a^2+3 b^2\right ) B\right )}{b^2 \left (a^2+b^2\right )^2 d \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 32.6799, size = 250233, normalized size = 731.68 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 3.756, size = 76827, normalized size = 224.6 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cot \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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