Optimal. Leaf size=399 \[ \frac {(A+i B) \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a-b)^{5/2} d}+\frac {(A-i B) \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{(i a+b)^{5/2} d}+\frac {2 b \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\right )}{3 a^3 \left (a^2+b^2\right ) d \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^{3/2}}+\frac {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\right )}{3 a^4 \left (a^2+b^2\right )^2 d \sqrt {\cot (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.16, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4326, 3690,
3730, 3697, 3696, 95, 209, 212} \begin {gather*} \frac {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (-3 a^3 B+7 a^2 A b-4 a b^2 B+8 A b^3\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 {2 b \left (-3 a^5 B+8 a^4 A b-17 a^3 b^2 B+30 a^2 A b^3-8 a b^4 B+16 A b^5\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 {(A+i B) \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 {(A-i B) \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}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 95
Rule 209
Rule 212
Rule 3690
Rule 3696
Rule 3697
Rule 3730
Rule 4326
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {5}{2}}(c+d x) (A+B \tan (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 {A+B \tan (c+d x)}{\tan ^{\frac {5}{2}}(c+d x) (a+b \tan (c+d x))^{5/2}} \, dx\\ &=-\frac {2 A \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 {\frac {3}{2} (2 A b-a B)+\frac {3}{2} a A \tan (c+d x)+3 A 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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \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 A-8 A b^2+4 a b B\right )-\frac {3}{4} a^2 B \tan (c+d x)+3 b (2 A b-a B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \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 A-14 a^2 A b^2-16 A b^4+9 a^3 b B+8 a b^3 B\right )+\frac {9}{8} a^3 (A b-a B) \tan (c+d x)+\frac {3}{4} b \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 A-A b^2+2 a b B\right )+\frac {9}{16} a^4 \left (2 a A b-a^2 B+b^2 B\right ) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 (A-i B) \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 (A+i B) \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 {(A+i B) \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 {(A-i B) \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 \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\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 {2 (2 A b-a B) \sqrt {\cot (c+d x)}}{a^2 d (a+b \tan (c+d x))^{3/2}}-\frac {2 A \cot ^{\frac {3}{2}}(c+d x)}{3 a d (a+b \tan (c+d x))^{3/2}}+\frac {2 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\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 = 2.66, size = 385, normalized size = 0.96 \begin {gather*} \frac {\sqrt {\cot (c+d x)} \left (\frac {6 (6 A b-3 a B)}{a (a+b \tan (c+d x))^{3/2}}-\frac {6 A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}+\frac {6 b \left (7 a^2 A b+8 A b^3-3 a^3 B-4 a b^2 B\right ) \tan (c+d x)}{a^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\sqrt {\tan (c+d x)} \left (9 (-1)^{3/4} a^4 \left (\frac {(a+i b)^2 (A-i B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {(a-i b)^2 (A+i B) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right )+\frac {6 b \left (8 a^4 A b+30 a^2 A b^3+16 A b^5-3 a^5 B-17 a^3 b^2 B-8 a b^4 B\right ) \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{a^3 \left (a^2+b^2\right )^2}\right )}{9 a d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 29.98, size = 82067, normalized size = 205.68
method | result | size |
default | \(\text {Expression too large to display}\) | \(82067\) |
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 )}{{\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]
________________________________________________________________________________________