Optimal. Leaf size=199 \[ \frac{2 a \sqrt{\tan (c+d x)}}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
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Rubi [A] time = 0.676233, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3567, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 a \sqrt{\tan (c+d x)}}{3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3567
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{3}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{a}{2}-\frac{3}{2} b \tan (c+d x)-a \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{3 \left (a^2+b^2\right )}\\ &=\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{4 \int \frac{\frac{3}{4} a \left (a^2-b^2\right )-\frac{3}{2} a^2 b \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a \left (a^2+b^2\right )^2}\\ &=\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{\int \frac{1+i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac{\int \frac{1-i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}\\ &=\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a-i b)^2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-(i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(a-i b)^2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-(-i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(a+i b)^2 d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(i a-b)^{5/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}+\frac{2 a \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{4 \left (a^2-2 b^2\right ) \sqrt{\tan (c+d x)}}{3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.1588, size = 198, normalized size = 0.99 \[ \frac{\frac{2 \sqrt{\tan (c+d x)} \left (2 b \left (a^2-2 b^2\right ) \tan (c+d x)+3 a \left (a^2-b^2\right )\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{3/2}}+\frac{3 (-1)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(-a-i b)^{5/2}}+\frac{3 (-1)^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{(a-i b)^{5/2}}}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.906, size = 1487425, normalized size = 7474.5 \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{\tan \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \tan \left (d x + c\right ) + a\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{\frac{3}{2}}{\left (c + d x \right )}}{\left (a + b \tan{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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