Optimal. Leaf size=188 \[ \frac{2 a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A+2 a b B-A b^2\right )}{d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}} \]
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Rubi [A] time = 0.400631, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3591, 3529, 3539, 3537, 63, 208} \[ \frac{2 a (A b-a B)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A+2 a b B-A b^2\right )}{d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3591
Rule 3529
Rule 3539
Rule 3537
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan (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 (a+b \tan (c+d x))^{3/2}}+\frac{\int \frac{A b-a B+(a A+b B) \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx}{a^2+b^2}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{\int \frac{2 a A b-a^2 B+b^2 B+\left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{(i A-B) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac{(i A+B) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(i (A+i B)) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i a}{b}-\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{(a+i b)^2 b d}+\frac{(i A+B) \operatorname{Subst}\left (\int \frac{1}{-1-\frac{i a}{b}+\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{(a-i b)^2 b d}\\ &=-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{(a-i b)^{5/2} d}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{5/2} d}+\frac{2 a (A b-a B)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (a^2 A-A b^2+2 a b B\right )}{\left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.62852, size = 325, normalized size = 1.73 \[ \frac{\frac{2 a \left (a^2+b^2\right ) (A b-a B)}{(a+b \tan (c+d x))^{3/2}}+\frac{6 b \left (a^2 A+2 a b B-A b^2\right )}{\sqrt{a+b \tan (c+d x)}}+\frac{3 b \left (a^2 \left (-\left (A \sqrt{-b^2}+b B\right )\right )+2 a b \left (A b-\sqrt{-b^2} B\right )+b^2 \left (A \sqrt{-b^2}+b B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right )}{\sqrt{-b^2} \sqrt{a-\sqrt{-b^2}}}-\frac{3 b \left (a^2 A \sqrt{-b^2}-a^2 b B+2 a A b^2+2 a \sqrt{-b^2} b B+A \left (-b^2\right )^{3/2}+b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right )}{\sqrt{-b^2} \sqrt{a+\sqrt{-b^2}}}}{3 b d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.1, size = 12841, normalized size = 68.3 \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(-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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \tan{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )}{{\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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