Optimal. Leaf size=232 \[ \frac{\left (3 a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{5/2} d}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{-b+i a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{b+i a}} \]
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Rubi [A] time = 0.989346, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {3566, 3647, 3656, 6725, 63, 217, 206, 912, 93, 205, 208} \[ \frac{\left (3 a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{5/2} d}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{-b+i a}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d \sqrt{b+i a}} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3647
Rule 3656
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 912
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx &=\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\int \frac{\sqrt{\tan (c+d x)} \left (-\frac{3 a}{2}-2 b \tan (c+d x)-\frac{3}{2} a \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 b}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\int \frac{\frac{3 a^2}{4}+\frac{1}{4} \left (3 a^2-8 b^2\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 b^2}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3 a^2}{4}+\frac{1}{4} \left (3 a^2-8 b^2\right ) x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 a^2-8 b^2}{4 \sqrt{x} \sqrt{a+b x}}+\frac{\frac{3 a^2}{4}+\frac{1}{4} \left (-3 a^2+8 b^2\right )}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 b^2 d}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (3 a^2-8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{8 b^2 d}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{\operatorname{Subst}\left (\int \left (\frac{i}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{i}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left (3 a^2-8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{4 b^2 d}\\ &=-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left (3 a^2-8 b^2\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 )}{4 b^2 d}\\ &=\frac{\left (3 a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{5/2} d}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{i \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 )}{d}+\frac{i \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 )}{d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{i a-b} d}+\frac{\left (3 a^2-8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 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 )}{\sqrt{i a+b} d}-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{4 b^2 d}+\frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}\\ \end{align*}
Mathematica [A] time = 6.19809, size = 295, normalized size = 1.27 \[ \frac{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{2 b d}+\frac{-\frac{3 a \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{2 b d}-\frac{-\frac{\sqrt{a} \left (3 a^2-8 b^2\right ) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{-a-i b}}+\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{a-i b}}}{2 b d}}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.36, size = 945945, normalized size = 4077.4 \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{7}{2}}}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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