Optimal. Leaf size=501 \[ \frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {3 b \tan (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a^2 e}+\frac {\tan ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{2 a e}-\frac {\tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e} \]
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Rubi [A] time = 0.82, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3701, 6725, 744, 806, 724, 206, 1036, 1030, 208} \[ -\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tanh ^{-1}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \cot (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {3 b \tan (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{4 a^2 e}+\frac {\tan ^2(d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}{2 a e}-\frac {\tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 724
Rule 744
Rule 806
Rule 1030
Rule 1036
Rule 3701
Rule 6725
Rubi steps
\begin {align*} \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 \left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^3 \sqrt {a+b x+c x^2}}-\frac {1}{x \sqrt {a+b x+c x^2}}+\frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\cot (d+e x)\right )}{e}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}-\frac {\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{e}\\ &=\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3 b}{2}+c x}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 a e}+\frac {\operatorname {Subst}\left (\int \frac {-b+\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\operatorname {Subst}\left (\int \frac {-b+\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}\\ &=-\frac {\tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}-\frac {\left (3 b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\cot (d+e x)\right )}{8 a^2 e}+\frac {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\left (b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-2 b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}\\ &=-\frac {\tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}+\frac {\left (3 b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \cot (d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{4 a^2 e}\\ &=-\frac {\tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \tanh ^{-1}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e}\\ \end {align*}
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Mathematica [C] time = 40.14, size = 134907, normalized size = 269.28 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan \left (e x + d\right )^{3}}{\sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 26.22, size = 1829775, normalized size = 3652.25 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tan}\left (d+e\,x\right )}^3}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{3}{\left (d + e x \right )}}{\sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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