Optimal. Leaf size=106 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3751, 457, 87,
162, 65, 214} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}-\frac {b}{a f (a-b) \sqrt {a+b \tan ^2(e+f x)}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{f (a-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 87
Rule 162
Rule 214
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \frac {\cot (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (1+x) (a+b x)^{3/2}} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {a-b-b x}{x (1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 a (a-b) f}\\ &=-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 a f}-\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x}} \, dx,x,\tan ^2(e+f x)\right )}{2 (a-b) f}\\ &=-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{a b f}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^2(e+f x)}\right )}{(a-b) b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} f}-\frac {b}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.09, size = 91, normalized size = 0.86 \begin {gather*} \frac {-a \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan ^2(e+f x)}{a-b}\right )+(a-b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};1+\frac {b \tan ^2(e+f x)}{a}\right )}{a (a-b) f \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32887\) vs.
\(2(92)=184\).
time = 0.97, size = 32888, normalized size = 310.26
method | result | size |
default | \(\text {Expression too large to display}\) | \(32888\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs.
\(2 (95) = 190\).
time = 3.46, size = 952, normalized size = 8.98 \begin {gather*} \left [-\frac {{\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) + 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) + {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {a} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (f x + e\right )^{2}}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {2 \, {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) - {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {a - b} \log \left (\frac {b \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} + 2 \, a - b}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{2 \, {\left ({\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f\right )}}, \frac {{\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a}}{a}\right ) + {\left (a^{2} b \tan \left (f x + e\right )^{2} + a^{3}\right )} \sqrt {-a + b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{a - b}\right ) - {\left (a^{2} b - a b^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} f}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\cot {\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.64, size = 1922, normalized size = 18.13 \begin {gather*} \frac {b}{f\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (a\,b-a^2\right )}-\frac {\mathrm {atanh}\left (\frac {2\,a^2\,b^8\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {12\,a^3\,b^7\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}+\frac {30\,a^4\,b^6\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {38\,a^5\,b^5\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}+\frac {24\,a^6\,b^4\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}-\frac {6\,a^7\,b^3\,f^2\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}}{\sqrt {a^3}\,\left (-6\,a^6\,b^3\,f^2+24\,a^5\,b^4\,f^2-38\,a^4\,b^5\,f^2+30\,a^3\,b^6\,f^2-12\,a^2\,b^7\,f^2+2\,a\,b^8\,f^2\right )}\right )}{f\,\sqrt {a^3}}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7\,f^4-2\,a^4\,b^8\,f^4-28\,a^6\,b^6\,f^4+32\,a^7\,b^5\,f^4-18\,a^8\,b^4\,f^4+4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3}+\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8\,f^4-12\,a^5\,b^7\,f^4+28\,a^6\,b^6\,f^4-32\,a^7\,b^5\,f^4+18\,a^8\,b^4\,f^4-4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3}}{2\,a^3\,b^6\,f^2-6\,a^4\,b^5\,f^2+6\,a^5\,b^4\,f^2-2\,a^6\,b^3\,f^2-\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (12\,a^5\,b^7\,f^4-2\,a^4\,b^8\,f^4-28\,a^6\,b^6\,f^4+32\,a^7\,b^5\,f^4-18\,a^8\,b^4\,f^4+4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}}{f\,{\left (a-b\right )}^3}+\frac {\left (\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\left (-4\,a^8\,b^2\,f^3+16\,a^7\,b^3\,f^3-26\,a^6\,b^4\,f^3+22\,a^5\,b^5\,f^3-10\,a^4\,b^6\,f^3+2\,a^3\,b^7\,f^3\right )}{2}+\frac {\sqrt {{\left (a-b\right )}^3}\,\left (2\,a^4\,b^8\,f^4-12\,a^5\,b^7\,f^4+28\,a^6\,b^6\,f^4-32\,a^7\,b^5\,f^4+18\,a^8\,b^4\,f^4-4\,a^9\,b^3\,f^4+\frac {\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}\,\sqrt {{\left (a-b\right )}^3}\,\left (16\,a^{11}\,b^2\,f^5-88\,a^{10}\,b^3\,f^5+200\,a^9\,b^4\,f^5-240\,a^8\,b^5\,f^5+160\,a^7\,b^6\,f^5-56\,a^6\,b^7\,f^5+8\,a^5\,b^8\,f^5\right )}{4\,f\,{\left (a-b\right )}^3}\right )}{2\,f\,{\left (a-b\right )}^3}\right )\,\sqrt {{\left (a-b\right )}^3}}{f\,{\left (a-b\right )}^3}}\right )\,\sqrt {{\left (a-b\right )}^3}\,1{}\mathrm {i}}{f\,{\left (a-b\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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