Optimal. Leaf size=78 \[ \frac {(a+b)^2}{4 a^3 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {a+b}{a^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\log \left (b+a \cos ^2(e+f x)\right )}{2 a^3 f} \]
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Rubi [A]
time = 0.08, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4223, 455, 45}
\begin {gather*} \frac {(a+b)^2}{4 a^3 f \left (a \cos ^2(e+f x)+b\right )^2}-\frac {a+b}{a^3 f \left (a \cos ^2(e+f x)+b\right )}-\frac {\log \left (a \cos ^2(e+f x)+b\right )}{2 a^3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 4223
Rubi steps
\begin {align*} \int \frac {\tan ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {x \left (1-x^2\right )^2}{\left (b+a x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x)^2}{(b+a x)^3} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {(a+b)^2}{a^2 (b+a x)^3}-\frac {2 (a+b)}{a^2 (b+a x)^2}+\frac {1}{a^2 (b+a x)}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {(a+b)^2}{4 a^3 f \left (b+a \cos ^2(e+f x)\right )^2}-\frac {a+b}{a^3 f \left (b+a \cos ^2(e+f x)\right )}-\frac {\log \left (b+a \cos ^2(e+f x)\right )}{2 a^3 f}\\ \end {align*}
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Mathematica [A]
time = 2.37, size = 136, normalized size = 1.74 \begin {gather*} -\frac {2 \left (a^2+4 a b+3 b^2\right )+(a+2 b)^2 \log (a+2 b+a \cos (2 (e+f x)))+a^2 \cos ^2(2 (e+f x)) \log (a+2 b+a \cos (2 (e+f x)))+2 a \cos (2 (e+f x)) (2 (a+b)+(a+2 b) \log (a+2 b+a \cos (2 (e+f x))))}{2 a^3 f (a+2 b+a \cos (2 (e+f x)))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 80, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {-2 a -2 b}{2 a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}+\frac {a^{2}+2 a b +b^{2}}{4 a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 a^{3}}}{f}\) | \(80\) |
default | \(\frac {\frac {-2 a -2 b}{2 a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}+\frac {a^{2}+2 a b +b^{2}}{4 a^{3} \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {\ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 a^{3}}}{f}\) | \(80\) |
risch | \(\frac {i x}{a^{3}}+\frac {2 i e}{a^{3} f}-\frac {4 \left (a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+a b \,{\mathrm e}^{6 i \left (f x +e \right )}+a^{2} {\mathrm e}^{4 i \left (f x +e \right )}+4 a b \,{\mathrm e}^{4 i \left (f x +e \right )}+3 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+a^{2} {\mathrm e}^{2 i \left (f x +e \right )}+a b \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{a^{3} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{4 i \left (f x +e \right )}+\frac {2 \left (a +2 b \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{a}+1\right )}{2 a^{3} f}\) | \(196\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 116, normalized size = 1.49 \begin {gather*} \frac {\frac {4 \, {\left (a^{2} + a b\right )} \sin \left (f x + e\right )^{2} - 3 \, a^{2} - 6 \, a b - 3 \, b^{2}}{a^{5} \sin \left (f x + e\right )^{4} + a^{5} + 2 \, a^{4} b + a^{3} b^{2} - 2 \, {\left (a^{5} + a^{4} b\right )} \sin \left (f x + e\right )^{2}} - \frac {2 \, \log \left (a \sin \left (f x + e\right )^{2} - a - b\right )}{a^{3}}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.35, size = 122, normalized size = 1.56 \begin {gather*} -\frac {4 \, {\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{2} - a^{2} + 2 \, a b + 3 \, b^{2} + 2 \, {\left (a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \log \left (a \cos \left (f x + e\right )^{2} + b\right )}{4 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b f \cos \left (f x + e\right )^{2} + a^{3} b^{2} f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 921 vs.
\(2 (66) = 132\).
time = 83.21, size = 921, normalized size = 11.81 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x \tan ^{5}{\left (e \right )}}{\sec ^{6}{\left (e \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\\frac {\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {\tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {\tan ^{2}{\left (e + f x \right )}}{2 f}}{a^{3}} & \text {for}\: b = 0 \\\frac {- \frac {\tan ^{4}{\left (e + f x \right )}}{2 f \sec ^{6}{\left (e + f x \right )}} - \frac {\tan ^{2}{\left (e + f x \right )}}{2 f \sec ^{6}{\left (e + f x \right )}} - \frac {1}{6 f \sec ^{6}{\left (e + f x \right )}}}{b^{3}} & \text {for}\: a = 0 \\\frac {x \tan ^{5}{\left (e \right )}}{\left (a + b \sec ^{2}{\left (e \right )}\right )^{3}} & \text {for}\: f = 0 \\- \frac {2 a^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 a^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {2 a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {a^{2} \tan ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 a^{2} \tan ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {4 a b \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {4 a b \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {4 a b \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 a b \tan ^{2}{\left (e + f x \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 b^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 b^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {2 b^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 531 vs.
\(2 (74) = 148\).
time = 2.05, size = 531, normalized size = 6.81 \begin {gather*} -\frac {\frac {2 \, \log \left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{a^{3}} - \frac {4 \, \log \left ({\left | -\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {3 \, a^{2} + 6 \, a b + 3 \, b^{2} + \frac {20 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {8 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {12 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {50 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {28 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {18 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {20 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {8 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {12 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {6 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {3 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}{{\left (a + b + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {2 \, b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{2} a^{3}}}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.60, size = 166, normalized size = 2.13 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {4\,b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2}{8\,b^2+\frac {8\,b^3}{a}+4\,b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {8\,b^3\,{\mathrm {tan}\left (e+f\,x\right )}^2}{a}}\right )}{a^3\,f}+\frac {\frac {-a^3+3\,a\,b^2+2\,b^3}{4\,a^2\,b^2}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a^2-b^2\right )}{2\,a^2\,b}}{f\,\left (2\,a\,b+a^2+b^2+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (2\,b^2+2\,a\,b\right )+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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