Optimal. Leaf size=231 \[ -\frac {(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac {\sqrt {b} \left (15 a^2-70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{11/2} f}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\left (5 a^2-30 a b+27 b^2\right ) \cot (e+f x)}{5 a^5 f}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3663, 462, 456, 1259, 1261, 205} \[ -\frac {\sqrt {b} \left (15 a^2-70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{11/2} f}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\left (5 a^2-30 a b+27 b^2\right ) \cot (e+f x)}{5 a^5 f}-\frac {(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 456
Rule 462
Rule 1259
Rule 1261
Rule 3663
Rubi steps
\begin {align*} \int \frac {\csc ^6(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^6 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {10 a-9 b+5 a x^2}{x^4 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \operatorname {Subst}\left (\int \frac {4 \left (\frac {9}{a}-\frac {10}{b}\right )+4 \left (\frac {10}{a}-\frac {5}{b}-\frac {9 b}{a^2}\right ) x^2+\frac {3 \left (5 a^2-10 a b+9 b^2\right ) x^4}{a^3}}{x^4 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{20 a f}\\ &=-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-8 a (10 a-9 b) b-8 b \left (5 a^2-20 a b+18 b^2\right ) x^2+\frac {b^2 \left (35 a^2-110 a b+99 b^2\right ) x^4}{a}}{x^4 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{40 a^4 b f}\\ &=-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \left (-\frac {8 (10 a-9 b) b}{x^4}-\frac {8 b \left (5 a^2-30 a b+27 b^2\right )}{a x^2}+\frac {5 b^2 \left (15 a^2-70 a b+63 b^2\right )}{a \left (a+b x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{40 a^4 b f}\\ &=-\frac {\left (5 a^2-30 a b+27 b^2\right ) \cot (e+f x)}{5 a^5 f}-\frac {(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}-\frac {\left (b \left (15 a^2-70 a b+63 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^5 f}\\ &=-\frac {\sqrt {b} \left (15 a^2-70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{11/2} f}-\frac {\left (5 a^2-30 a b+27 b^2\right ) \cot (e+f x)}{5 a^5 f}-\frac {(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac {\cot ^5(e+f x)}{5 a f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (5 a^2-10 a b+9 b^2\right ) \tan (e+f x)}{20 a^4 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b \left (35 a^2-110 a b+99 b^2\right ) \tan (e+f x)}{40 a^5 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.68, size = 346, normalized size = 1.50 \[ \frac {-960 \sqrt {b} \left (15 a^2-70 a b+63 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )-\frac {2 \sqrt {a} \cot (e+f x) \csc ^4(e+f x) \left (-128 a^4 \cos (6 (e+f x))+64 a^4 \cos (8 (e+f x))+1600 a^4+1788 a^3 b \cos (6 (e+f x))-863 a^3 b \cos (8 (e+f x))-165 a^3 b-8800 a^2 b^2 \cos (6 (e+f x))+2479 a^2 b^2 \cos (8 (e+f x))+637 a^2 b^2+4 \left (416 a^4-447 a^3 b-1400 a^2 b^2+13125 a b^3-13230 b^4\right ) \cos (2 (e+f x))-4 \left (32 a^4-257 a^3 b-2821 a^2 b^2+8925 a b^3-6615 b^4\right ) \cos (4 (e+f x))+14700 a b^3 \cos (6 (e+f x))-2625 a b^3 \cos (8 (e+f x))-28875 a b^3-7560 b^4 \cos (6 (e+f x))+945 b^4 \cos (8 (e+f x))+33075 b^4\right )}{((a-b) \cos (2 (e+f x))+a+b)^2}}{7680 a^{11/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.58, size = 1199, normalized size = 5.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 4.13, size = 263, normalized size = 1.14 \[ -\frac {\frac {15 \, {\left (15 \, a^{2} b - 70 \, a b^{2} + 63 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{\sqrt {a b} a^{5}} + \frac {15 \, {\left (7 \, a^{2} b^{2} \tan \left (f x + e\right )^{3} - 22 \, a b^{3} \tan \left (f x + e\right )^{3} + 15 \, b^{4} \tan \left (f x + e\right )^{3} + 9 \, a^{3} b \tan \left (f x + e\right ) - 26 \, a^{2} b^{2} \tan \left (f x + e\right ) + 17 \, a b^{3} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2} a^{5}} + \frac {8 \, {\left (15 \, a^{2} \tan \left (f x + e\right )^{4} - 90 \, a b \tan \left (f x + e\right )^{4} + 90 \, b^{2} \tan \left (f x + e\right )^{4} + 10 \, a^{2} \tan \left (f x + e\right )^{2} - 15 \, a b \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )}}{a^{5} \tan \left (f x + e\right )^{5}}}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.73, size = 380, normalized size = 1.65 \[ -\frac {7 b^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \,a^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {11 b^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{4 f \,a^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \,a^{5} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {9 b \tan \left (f x +e \right )}{8 f \,a^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {13 b^{2} \tan \left (f x +e \right )}{4 f \,a^{3} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {17 b^{3} \tan \left (f x +e \right )}{8 f \,a^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \,a^{3} \sqrt {a b}}+\frac {35 b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{4 f \,a^{4} \sqrt {a b}}-\frac {63 b^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \,a^{5} \sqrt {a b}}-\frac {1}{5 f \,a^{3} \tan \left (f x +e \right )^{5}}+\frac {b}{f \,a^{4} \tan \left (f x +e \right )^{3}}-\frac {2}{3 f \,a^{3} \tan \left (f x +e \right )^{3}}-\frac {1}{f \,a^{3} \tan \left (f x +e \right )}+\frac {6 b}{f \,a^{4} \tan \left (f x +e \right )}-\frac {6 b^{2}}{f \,a^{5} \tan \left (f x +e \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.73, size = 212, normalized size = 0.92 \[ -\frac {\frac {15 \, {\left (15 \, a^{2} b^{2} - 70 \, a b^{3} + 63 \, b^{4}\right )} \tan \left (f x + e\right )^{8} + 25 \, {\left (15 \, a^{3} b - 70 \, a^{2} b^{2} + 63 \, a b^{3}\right )} \tan \left (f x + e\right )^{6} + 8 \, {\left (15 \, a^{4} - 70 \, a^{3} b + 63 \, a^{2} b^{2}\right )} \tan \left (f x + e\right )^{4} + 24 \, a^{4} + 8 \, {\left (10 \, a^{4} - 9 \, a^{3} b\right )} \tan \left (f x + e\right )^{2}}{a^{5} b^{2} \tan \left (f x + e\right )^{9} + 2 \, a^{6} b \tan \left (f x + e\right )^{7} + a^{7} \tan \left (f x + e\right )^{5}} + \frac {15 \, {\left (15 \, a^{2} b - 70 \, a b^{2} + 63 \, b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}}}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.31, size = 199, normalized size = 0.86 \[ -\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (15\,a^2-70\,a\,b+63\,b^2\right )}{15\,a^3}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (10\,a-9\,b\right )}{15\,a^2}+\frac {5\,b\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (15\,a^2-70\,a\,b+63\,b^2\right )}{24\,a^4}+\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^8\,\left (15\,a^2-70\,a\,b+63\,b^2\right )}{8\,a^5}}{f\,\left (a^2\,{\mathrm {tan}\left (e+f\,x\right )}^5+2\,a\,b\,{\mathrm {tan}\left (e+f\,x\right )}^7+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^9\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a}}\right )\,\left (15\,a^2-70\,a\,b+63\,b^2\right )}{8\,a^{11/2}\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________