Optimal. Leaf size=338 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.88, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4326, 3650,
3730, 3731, 3697, 3696, 95, 209, 212} \begin {gather*} \frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 d \left (a^2+b^2\right )^2 \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 d \left (a^2+b^2\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \text {ArcTan}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 209
Rule 212
Rule 3650
Rule 3696
Rule 3697
Rule 3730
Rule 3731
Rule 4326
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {5}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx\\ &=-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}-\frac {\left (2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {3 b+\frac {3}{2} a \tan (c+d x)+3 b \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx}{3 a}\\ &=\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {\left (4 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{4} \left (a^2-8 b^2\right )+6 b^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}} \, dx}{3 a^2}\\ &=\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {\left (8 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {3}{8} \left (3 a^4-14 a^2 b^2-16 b^4\right )+\frac {9}{8} a^3 b \tan (c+d x)+\frac {3}{4} b^2 \left (7 a^2+8 b^2\right ) \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx}{9 a^3 \left (a^2+b^2\right )}\\ &=\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {\left (16 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-\frac {9}{16} a^4 \left (a^2-b^2\right )+\frac {9}{8} a^5 b \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{9 a^4 \left (a^2+b^2\right )^2}\\ &=\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left ((a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )^2}-\frac {\left ((a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left ((a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}-\frac {\left ((a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {\left ((a-i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {\left ((a+i b)^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\left (a^2+b^2\right )^2 d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac {2 b^2 \left (7 a^2+8 b^2\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {4 b \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {4 b^2 \left (4 a^4+15 a^2 b^2+8 b^4\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.30, size = 504, normalized size = 1.49 \begin {gather*} \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-\frac {2}{3 a d \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{3/2}}-\frac {2 \left (-\frac {6 b}{a d \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}-\frac {3 \left (\frac {a b \sqrt {\tan (c+d x)}}{3 (i a-b) (a+b \tan (c+d x))^{3/2}}+\frac {16 b^2 \sqrt {\tan (c+d x)}}{3 a (a+b \tan (c+d x))^{3/2}}-\frac {a b \sqrt {\tan (c+d x)}}{3 (i a+b) (a+b \tan (c+d x))^{3/2}}+\frac {32 b^2 \sqrt {\tan (c+d x)}}{3 a^2 \sqrt {a+b \tan (c+d x)}}-\frac {-\frac {3 \sqrt [4]{-1} a^2 \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(-a+i b)^{3/2}}+\frac {(5 a-2 i b) b \sqrt {\tan (c+d x)}}{(a-i b) \sqrt {a+b \tan (c+d x)}}}{3 (i a+b)}+\frac {-\frac {3 \sqrt [4]{-1} a^2 \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^{3/2}}+\frac {(5 a+2 i b) b \sqrt {\tan (c+d x)}}{(a+i b) \sqrt {a+b \tan (c+d x)}}}{3 (i a-b)}\right )}{2 a d}\right )}{3 a}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 37.55, size = 41735, normalized size = 123.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(41735\) |
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 [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]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________