Optimal. Leaf size=395 \[ \frac {b \tanh ^{-1}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left (\frac {b-\left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left (\frac {b-\left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e} \]
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Rubi [A] time = 0.76, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3700, 6725, 730, 724, 206, 987, 1030, 205} \[ -\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left (\frac {b-\left (-\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \tan ^{-1}\left (\frac {b-\left (\sqrt {a^2-2 a c+b^2+c^2}+a-c\right ) \tan (d+e x)}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {b \tanh ^{-1}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 724
Rule 730
Rule 987
Rule 1030
Rule 3700
Rule 6725
Rubi steps
\begin {align*} \int \frac {\cot ^2(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^2 \sqrt {a+b x+c x^2}}+\frac {1}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\operatorname {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 a e}-\frac {\operatorname {Subst}\left (\int \frac {-a+c-\sqrt {a^2+b^2-2 a c+c^2}-b x}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\operatorname {Subst}\left (\int \frac {-a+c+\sqrt {a^2+b^2-2 a c+c^2}-b x}{\left (-1-x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}\\ &=-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e}+\frac {b \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a 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 {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^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 {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\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}} \tan ^{-1}\left (\frac {b-\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^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}} \tan ^{-1}\left (\frac {b-\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {b \tanh ^{-1}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2} e}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a e}\\ \end {align*}
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Mathematica [C] time = 1.57, size = 264, normalized size = 0.67 \[ \frac {\frac {b \tanh ^{-1}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{a^{3/2}}+\frac {i \tanh ^{-1}\left (\frac {2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}-\frac {i \tanh ^{-1}\left (\frac {2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}-\frac {2 \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{2 e} \]
Antiderivative was successfully verified.
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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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 21.38, size = 636822, normalized size = 1612.21 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (e x + d\right )^{2}}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}}\,{d x} \]
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 \[ \int \frac {{\mathrm {cot}\left (d+e\,x\right )}^2}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\left (d + e x \right )}}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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