Optimal. Leaf size=307 \[ -\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b c d^2 \left (149+100 m+15 m^2\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.34, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {276, 5921, 12,
534, 1281, 470, 372, 371} \begin {gather*} \frac {c^4 d^2 (f x)^{m+5} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (m+5)}-\frac {2 c^2 d^2 (f x)^{m+3} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (m+3)}+\frac {d^2 (f x)^{m+1} \left (a+b \cosh ^{-1}(c x)\right )}{f (m+1)}-\frac {b c d^2 \left (15 m^2+100 m+149\right ) \sqrt {1-c^2 x^2} (f x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (m^2+13 m+38\right ) \left (1-c^2 x^2\right ) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d^2 \left (1-c^2 x^2\right ) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 276
Rule 371
Rule 372
Rule 470
Rule 534
Rule 1281
Rule 5921
Rubi steps
\begin {align*} \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-(b c) \int \frac {d^2 (f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{f \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{f}\\ &=\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {1}{1+m}-\frac {2 c^2 x^2}{3+m}+\frac {c^4 x^4}{5+m}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{f \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b d^2 \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m} \left (\frac {c^2 (5+m)}{1+m}-\frac {c^4 \left (38+13 m+m^2\right ) x^2}{(3+m) (5+m)}\right )}{\sqrt {-1+c^2 x^2}} \, dx}{c f (5+m) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {-1+c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {\left (b c d^2 \left (149+100 m+15 m^2\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {(f x)^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{f (1+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (38+13 m+m^2\right ) (f x)^{2+m} \left (1-c^2 x^2\right )}{f^2 (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 (f x)^{4+m} \left (1-c^2 x^2\right )}{f^4 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {d^2 (f x)^{1+m} \left (a+b \cosh ^{-1}(c x)\right )}{f (1+m)}-\frac {2 c^2 d^2 (f x)^{3+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^3 (3+m)}+\frac {c^4 d^2 (f x)^{5+m} \left (a+b \cosh ^{-1}(c x)\right )}{f^5 (5+m)}-\frac {b c d^2 \left (149+100 m+15 m^2\right ) (f x)^{2+m} \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{f^2 (1+m) (2+m) (3+m)^2 (5+m)^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.48, size = 315, normalized size = 1.03 \begin {gather*} d^2 x (f x)^m \left (\frac {a}{1+m}-\frac {2 a c^2 x^2}{3+m}+\frac {a c^4 x^4}{5+m}+\frac {b \cosh ^{-1}(c x)}{1+m}-\frac {2 b c^2 x^2 \cosh ^{-1}(c x)}{3+m}+\frac {b c^4 x^4 \cosh ^{-1}(c x)}{5+m}-\frac {b c x \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},1+\frac {m}{2};2+\frac {m}{2};c^2 x^2\right )}{\left (2+3 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b c^3 x^3 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},2+\frac {m}{2};3+\frac {m}{2};c^2 x^2\right )}{\left (12+7 m+m^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 x^5 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},3+\frac {m}{2};4+\frac {m}{2};c^2 x^2\right )}{(5+m) (6+m) \sqrt {-1+c x} \sqrt {1+c x}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )\, dx\]
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]
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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{2} \left (\int a \left (f x\right )^{m}\, dx + \int b \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx + \int \left (- 2 a c^{2} x^{2} \left (f x\right )^{m}\right )\, dx + \int a c^{4} x^{4} \left (f x\right )^{m}\, dx + \int \left (- 2 b c^{2} x^{2} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{4} \left (f x\right )^{m} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \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 \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2\,{\left (f\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________