Optimal. Leaf size=146 \[ -\frac {(a-2 b) \cot (e+f x)}{a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {4 (a-2 b) b \tan (e+f x)}{3 a^3 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {8 (a-2 b) b \tan (e+f x)}{3 a^4 f \sqrt {a+b \tan ^2(e+f x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3744, 464, 277,
198, 197} \begin {gather*} -\frac {8 b (a-2 b) \tan (e+f x)}{3 a^4 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {4 b (a-2 b) \tan (e+f x)}{3 a^3 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(a-2 b) \cot (e+f x)}{a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 277
Rule 464
Rule 3744
Rubi steps
\begin {align*} \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-2 b) \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {(a-2 b) \cot (e+f x)}{a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(4 (a-2 b) b) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx,x,\tan (e+f x)\right )}{a^2 f}\\ &=-\frac {(a-2 b) \cot (e+f x)}{a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {4 (a-2 b) b \tan (e+f x)}{3 a^3 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {(8 (a-2 b) b) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a^3 f}\\ &=-\frac {(a-2 b) \cot (e+f x)}{a^2 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {\cot ^3(e+f x)}{3 a f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {4 (a-2 b) b \tan (e+f x)}{3 a^3 f \left (a+b \tan ^2(e+f x)\right )^{3/2}}-\frac {8 (a-2 b) b \tan (e+f x)}{3 a^4 f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.78, size = 140, normalized size = 0.96 \begin {gather*} \frac {\sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)} \left (-\cot (e+f x) \left (2 a-8 b+a \csc ^2(e+f x)\right )+\frac {2 b \left (-3 a^2+2 a b+4 b^2+\left (-3 a^2+7 a b-4 b^2\right ) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{(a+b+(a-b) \cos (2 (e+f x)))^2}\right )}{3 \sqrt {2} a^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 245, normalized size = 1.68
method | result | size |
default | \(\frac {\left (2 \left (\cos ^{6}\left (f x +e \right )\right ) a^{3}-18 \left (\cos ^{6}\left (f x +e \right )\right ) a^{2} b +32 \left (\cos ^{6}\left (f x +e \right )\right ) a \,b^{2}-16 \left (\cos ^{6}\left (f x +e \right )\right ) b^{3}-3 \left (\cos ^{4}\left (f x +e \right )\right ) a^{3}+30 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b -72 \left (\cos ^{4}\left (f x +e \right )\right ) a \,b^{2}+48 \left (\cos ^{4}\left (f x +e \right )\right ) b^{3}-12 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2} b +48 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{2}-48 \left (\cos ^{2}\left (f x +e \right )\right ) b^{3}-8 a \,b^{2}+16 b^{3}\right ) \left (\cos ^{5}\left (f x +e \right )\right ) \left (\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}\right )^{\frac {5}{2}}}{3 f \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{4} \sin \left (f x +e \right )^{3} a^{4}}\) | \(245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 209, normalized size = 1.43 \begin {gather*} -\frac {\frac {8 \, b \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{3}} + \frac {4 \, b \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {16 \, b^{2} \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{4}} - \frac {8 \, b^{2} \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {3}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \tan \left (f x + e\right )} - \frac {6 \, b}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} \tan \left (f x + e\right )} + \frac {1}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \tan \left (f x + e\right )^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 95.74, size = 250, normalized size = 1.71 \begin {gather*} -\frac {{\left (2 \, {\left (a^{3} - 9 \, a^{2} b + 16 \, a b^{2} - 8 \, b^{3}\right )} \cos \left (f x + e\right )^{7} - 3 \, {\left (a^{3} - 10 \, a^{2} b + 24 \, a b^{2} - 16 \, b^{3}\right )} \cos \left (f x + e\right )^{5} - 12 \, {\left (a^{2} b - 4 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{3} - 8 \, {\left (a b^{2} - 2 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{6} - 2 \, a^{5} b + a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{6} - a^{4} b^{2} f - {\left (a^{6} - 4 \, a^{5} b + 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{4} - {\left (2 \, a^{5} b - 3 \, a^{4} b^{2}\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\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 {\csc ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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