Optimal. Leaf size=165 \[ -\frac {(a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3751, 485, 597,
12, 385, 209} \begin {gather*} -\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}-\frac {(a-b)^{3/2} \text {ArcTan}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 209
Rule 385
Rule 485
Rule 597
Rule 3751
Rubi steps
\begin {align*} \int \cot ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {\text {Subst}\left (\int \frac {-a (5 a-6 b)-(4 a-5 b) b x^2}{x^4 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {\text {Subst}\left (\int \frac {-a \left (15 a^2-20 a b+3 b^2\right )-2 a (5 a-6 b) b x^2}{x^2 \left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a f}\\ &=-\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}+\frac {\text {Subst}\left (\int -\frac {15 a^2 (a-b)^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{15 a^2 f}\\ &=-\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}-\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}\\ &=-\frac {(a-b)^{3/2} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f}-\frac {\left (15 a^2-20 a b+3 b^2\right ) \cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a f}+\frac {(5 a-6 b) \cot ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 f}-\frac {a \cot ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{5 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 4.67, size = 140, normalized size = 0.85 \begin {gather*} -\frac {\cos (e+f x) \left (b+a \cot ^2(e+f x)\right )^2 \left (a \left (-2 b+3 a \cot ^2(e+f x)\right ) \, _2F_1\left (1,1;-\frac {1}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )+2 (a-b) (a+b+(a-b) \cos (2 (e+f x))) \, _2F_1\left (2,2;\frac {1}{2};\frac {(a-b) \sin ^2(e+f x)}{a}\right )\right ) \sin (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{15 a^3 f} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.43, size = 10026, normalized size = 60.76
method | result | size |
default | \(\text {Expression too large to display}\) | \(10026\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.97, size = 405, normalized size = 2.45 \begin {gather*} \left [-\frac {15 \, {\left (a^{2} - a b\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{3} - a \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b}}{\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{5} + 4 \, {\left ({\left (15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - 6 \, a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{60 \, a f \tan \left (f x + e\right )^{5}}, -\frac {15 \, {\left (a^{2} - a b\right )} \sqrt {a - b} \arctan \left (-\frac {2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {a - b} \tan \left (f x + e\right )}{{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - a}\right ) \tan \left (f x + e\right )^{5} + 2 \, {\left ({\left (15 \, a^{2} - 20 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{4} - {\left (5 \, a^{2} - 6 \, a b\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2}\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{30 \, a f \tan \left (f x + e\right )^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^6\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________