Optimal. Leaf size=314 \[ \frac{2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}-\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 (7 a B+A b) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)} \]
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Rubi [A] time = 1.3429, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3608, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}-\frac{\sqrt{-b+i a} (-B+i A) \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 (7 a B+A b) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}-\frac{\sqrt{b+i a} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 3608
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2}{7} \int \frac{\frac{1}{2} (-A b-7 a B)+\frac{7}{2} (a A-b B) \tan (c+d x)+3 A b \tan ^2(c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{4 \int \frac{\frac{1}{4} \left (-35 a^2 A-4 A b^2+7 a b B\right )-\frac{35}{4} a (A b+a B) \tan (c+d x)-b (A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{35 a}\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{8 \int \frac{\frac{1}{8} \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right )-\frac{105}{8} a^2 (a A-b B) \tan (c+d x)-\frac{1}{4} b \left (35 a^2 A+4 A b^2-7 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}+\frac{16 \int \frac{\frac{105}{16} a^3 (a A-b B)+\frac{105}{16} a^3 (A b+a B) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{105 a^3}\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}+\frac{1}{2} ((a-i b) (A-i B)) \int \frac{1+i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} ((a+i b) (A+i B)) \int \frac{1-i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}+\frac{((a-i b) (A-i B)) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{((a+i b) (A+i B)) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}+\frac{((a-i b) (A-i B)) \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 )}{d}+\frac{((a+i b) (A+i B)) \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 )}{d}\\ &=-\frac{\sqrt{i a-b} (i A-B) \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\sqrt{i a+b} (i A+B) \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 (A b+7 a B) \sqrt{a+b \tan (c+d x)}}{35 a d \tan ^{\frac{5}{2}}(c+d x)}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{105 a^3 d \sqrt{\tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 3.76991, size = 265, normalized size = 0.84 \[ \frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left (a \left (35 a^2 A-7 a b B+4 A b^2\right ) \tan ^2(c+d x)+\left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \tan ^3(c+d x)-3 a^2 (7 a B+A b) \tan (c+d x)-15 a^3 A\right )}{a^3 \tan ^{\frac{7}{2}}(c+d x)}+105 (-1)^{3/4} \sqrt{-a-i b} (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )-105 \sqrt [4]{-1} \sqrt{a-i b} (B+i A) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.735, size = 2183144, normalized size = 6952.7 \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(-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(-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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