Optimal. Leaf size=456 \[ \frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}+\frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d} \]
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Rubi [A] time = 0.703612, antiderivative size = 456, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3566, 3647, 3631, 3485, 700, 1129, 634, 618, 206, 628} \[ \frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}+\frac{b \log \left (-\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}-\frac{b \log \left (\sqrt{2} \sqrt{\sqrt{a^2+b^2}+a} \sqrt{a+b \tan (c+d x)}+\sqrt{a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt{2} d \sqrt{\sqrt{a^2+b^2}+a}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a^2+b^2}+a}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} d \sqrt{a-\sqrt{a^2+b^2}}}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 3566
Rule 3647
Rule 3631
Rule 3485
Rule 700
Rule 1129
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \tan ^4(c+d x) \sqrt{a+b \tan (c+d x)} \, dx &=\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{2 \int \tan (c+d x) \sqrt{a+b \tan (c+d x)} \left (-2 a-\frac{7}{2} b \tan (c+d x)-2 a \tan ^2(c+d x)\right ) \, dx}{7 b}\\ &=-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{4 \int \sqrt{a+b \tan (c+d x)} \left (2 a^2+\frac{1}{4} \left (8 a^2-35 b^2\right ) \tan ^2(c+d x)\right ) \, dx}{35 b^2}\\ &=\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\int \sqrt{a+b \tan (c+d x)} \, dx\\ &=\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{a+x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{d}\\ &=\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}\\ &=\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{2 d}+\frac{b \operatorname{Subst}\left (\int \frac{-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 x}{\sqrt{a^2+b^2}-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 x}{\sqrt{a^2+b^2}+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} x+x^2} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}\\ &=\frac{b \log \left (a+\sqrt{a^2+b^2}+b \tan (c+d x)-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \log \left (a+\sqrt{a^2+b^2}+b \tan (c+d x)+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}+\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{2 \left (a-\sqrt{a^2+b^2}\right )-x^2} \, dx,x,-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 \sqrt{a+b \tan (c+d x)}\right )}{d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{2 \left (a-\sqrt{a^2+b^2}\right )-x^2} \, dx,x,\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}}+2 \sqrt{a+b \tan (c+d x)}\right )}{d}\\ &=\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\sqrt{a^2+b^2}}-\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} \sqrt{a-\sqrt{a^2+b^2}} d}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\sqrt{a^2+b^2}}+\sqrt{2} \sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{a^2+b^2}}}\right )}{\sqrt{2} \sqrt{a-\sqrt{a^2+b^2}} d}+\frac{b \log \left (a+\sqrt{a^2+b^2}+b \tan (c+d x)-\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}-\frac{b \log \left (a+\sqrt{a^2+b^2}+b \tan (c+d x)+\sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} \sqrt{a+b \tan (c+d x)}\right )}{2 \sqrt{2} \sqrt{a+\sqrt{a^2+b^2}} d}+\frac{2 \left (8 a^2-35 b^2\right ) (a+b \tan (c+d x))^{3/2}}{105 b^3 d}-\frac{8 a \tan (c+d x) (a+b \tan (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \tan ^2(c+d x) (a+b \tan (c+d x))^{3/2}}{7 b d}\\ \end{align*}
Mathematica [C] time = 19.2857, size = 167, normalized size = 0.37 \[ \frac{\frac{2 \sqrt{a+b \tan (c+d x)} \left (-2 b \left (2 a^2+25 b^2\right ) \tan (c+d x)+8 a^3+3 b^2 \sec ^2(c+d x) (a+5 b \tan (c+d x))-38 a b^2\right )}{b^3}-105 i \sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )+105 i \sqrt{a+i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.129, size = 556, normalized size = 1.2 \begin{align*}{\frac{2}{7\,d{b}^{3}} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}}-{\frac{4\,a}{5\,d{b}^{3}} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}}{3\,d{b}^{3}} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{2}{3\,bd} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4\,bd}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\ln \left ( b\tan \left ( dx+c \right ) +a+\sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}+\sqrt{{a}^{2}+{b}^{2}} \right ) }-{\frac{1}{4\,bd}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\sqrt{{a}^{2}+{b}^{2}}\ln \left ( b\tan \left ( dx+c \right ) +a+\sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}+\sqrt{{a}^{2}+{b}^{2}} \right ) }+{\frac{b}{d}\arctan \left ({ \left ( 2\,\sqrt{a+b\tan \left ( dx+c \right ) }+\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}}-{\frac{a}{4\,bd}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\ln \left ( b\tan \left ( dx+c \right ) +a-\sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}+\sqrt{{a}^{2}+{b}^{2}} \right ) }+{\frac{1}{4\,bd}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\sqrt{{a}^{2}+{b}^{2}}\ln \left ( b\tan \left ( dx+c \right ) +a-\sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}+\sqrt{{a}^{2}+{b}^{2}} \right ) }+{\frac{b}{d}\arctan \left ({ \left ( 2\,\sqrt{a+b\tan \left ( dx+c \right ) }-\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \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 \sqrt{b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.633, size = 3893, normalized size = 8.54 \begin{align*} \text{result too large to display} \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 \sqrt{a + b \tan{\left (c + d x \right )}} \tan ^{4}{\left (c + d x \right )}\, 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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