Optimal. Leaf size=397 \[ \frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2}+\frac{b^2}{a d \left (a^2+b^2\right ) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{2 a^2+5 b^2}{3 a^2 d \left (a^2+b^2\right ) \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 d \left (a^2+b^2\right ) \sqrt{\tan (c+d x)}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2} \]
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Rubi [A] time = 1.00065, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {3569, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d \left (a^2+b^2\right )^2}+\frac{b^2}{a d \left (a^2+b^2\right ) \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^2}-\frac{2 a^2+5 b^2}{3 a^2 d \left (a^2+b^2\right ) \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 d \left (a^2+b^2\right ) \sqrt{\tan (c+d x)}}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3569
Rule 3649
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^2} \, dx &=\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{\int \frac{\frac{1}{2} \left (2 a^2+5 b^2\right )-a b \tan (c+d x)+\frac{5}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{2 \int \frac{\frac{3}{4} b \left (4 a^2+5 b^2\right )+\frac{3}{2} a^3 \tan (c+d x)+\frac{3}{4} b \left (2 a^2+5 b^2\right ) \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{4 \int \frac{-\frac{3}{8} \left (2 a^4-4 a^2 b^2-5 b^4\right )+\frac{3}{4} a^3 b \tan (c+d x)+\frac{3}{8} b^2 \left (4 a^2+5 b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{3 a^3 \left (a^2+b^2\right )}\\ &=-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{4 \int \frac{-\frac{3}{4} a^3 \left (a^2-b^2\right )+\frac{3}{2} a^4 b \tan (c+d x)}{\sqrt{\tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}+\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \int \frac{1+\tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{2 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}+\frac{8 \operatorname{Subst}\left (\int \frac{-\frac{3}{4} a^3 \left (a^2-b^2\right )+\frac{3}{2} a^4 b x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{3 a^3 \left (a^2+b^2\right )^2 d}+\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{2 a^3 \left (a^2+b^2\right )^2 d}\\ &=-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac{\left (b^4 \left (9 a^2+5 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{a^3 \left (a^2+b^2\right )^2 d}\\ &=\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}\\ &=\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}-\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2-2 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}\\ &=\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^2 d}+\frac{b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} \left (a^2+b^2\right )^2 d}+\frac{\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^2 d}-\frac{2 a^2+5 b^2}{3 a^2 \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x)}+\frac{b \left (4 a^2+5 b^2\right )}{a^3 \left (a^2+b^2\right ) d \sqrt{\tan (c+d x)}}+\frac{b^2}{a \left (a^2+b^2\right ) d \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.16759, size = 230, normalized size = 0.58 \[ \frac{-\frac{2 a^2+5 b^2}{a \tan ^{\frac{3}{2}}(c+d x)}+\frac{3 b \left (4 a^2+5 b^2\right )}{a^2 \sqrt{\tan (c+d x)}}+\frac{3 \left (b^{7/2} \left (9 a^2+5 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )+\sqrt [4]{-1} a^{7/2} (a+i b)^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )+\sqrt [4]{-1} a^{7/2} (a-i b)^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\tan (c+d x)}\right )\right )}{a^{5/2} \left (a^2+b^2\right )}+\frac{3 b^2}{\tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))}}{3 a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 595, normalized size = 1.5 \begin{align*} \text{result 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(-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 [A] time = 1.37818, size = 543, normalized size = 1.37 \begin{align*} \frac{b^{4} \sqrt{\tan \left (d x + c\right )}}{{\left (a^{5} d + a^{3} b^{2} d\right )}{\left (b \tan \left (d x + c\right ) + a\right )}} - \frac{{\left (\sqrt{2} a^{2} - 2 \, \sqrt{2} a b - \sqrt{2} b^{2}\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac{{\left (\sqrt{2} a^{2} - 2 \, \sqrt{2} a b - \sqrt{2} b^{2}\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right )}{2 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} - \frac{{\left (\sqrt{2} a^{2} + 2 \, \sqrt{2} a b - \sqrt{2} b^{2}\right )} \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac{{\left (\sqrt{2} a^{2} + 2 \, \sqrt{2} a b - \sqrt{2} b^{2}\right )} \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{4 \,{\left (a^{4} d + 2 \, a^{2} b^{2} d + b^{4} d\right )}} + \frac{{\left (9 \, a^{2} b^{4} + 5 \, b^{6}\right )} \arctan \left (\frac{b \sqrt{\tan \left (d x + c\right )}}{\sqrt{a b}}\right )}{{\left (a^{7} d + 2 \, a^{5} b^{2} d + a^{3} b^{4} d\right )} \sqrt{a b}} + \frac{2 \,{\left (6 \, b \tan \left (d x + c\right ) - a\right )}}{3 \, a^{3} d \tan \left (d x + c\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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